mathematical analysis : functions, limits, series, continued fractions

MATHEMATICAL ANALYSIS Functions, Limits, Series,

Continued Fractions

E D I T E D BY

L. A. LYUSTERNIK AND A. R. YANPOL'SKÏÏ

TRANSLATED BY

D . E . BROWN

TRANSLATION EDITED BY

E. SPENCE

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Copyright © 1965 PEROAMON PRESS LTD.

First Edition 1965

Library of Congress Catalog Card Number 63-19330

This is an edited translation of the original Russian volume entitled MameMamunecKau αηαΜΐ3—Φγηκμιΐίΐ, npedeAU,

pndbi, qeriHue dpoou (Matematichekii analiz—Funksiyi, predeli, ryady, tsepnye droby),

published in 1961 by Fizmatgiz, Moscow

FOREWORD

THE PRESENT book, together with its companion volume devoted to the differential and integral calculus, contains the fundamental part of the material dealt with in the larger courses of mathematical analysis. Included in this volume are general problems of the theory of continuous functions of one and several variables (with the geometrical basis of this theory), the theory of limiting values for sequences of numbers and vectors, and also the theory of numerical series and series of functions and other analogous infinite processes, in particular, infinite continued fractions.

Chapter I, "The arithmetical linear continuum and functions defined there" (authors: L. A. Lyusternik and Ye. K. Isakova), is devoted to real numbers, the arithmetical linear continuum, limiting values, and to functions of one variable. The material of this chapter is more or less that which is usually called the introduction to mathematical analysis.

Chapter II, "«-dimensional spaces and functions defined there" (L. A. Lyusternik), effects the transition from functions of one variable to functions of « variables, which, geometrically, corresponds to the transition from the arithmetical linear continuum to «-dimensional space En, the fundamental theory of which is given. §1 is devoted to the fundamentals of «-dimensional geometry and, in particular, of the theory of orthogonal systems of vectors in En9 which serves as a simpler model for the theory (Chapter IV) or orthogonal systems of functions. §2 is devoted to limiting values in En9 to continuous functions of « variables and their systems (transformations in E^). In this chapter also §3 deals with a subject which plays an important part in pure and applied mathematics, the theory of «-dimensional convex bodies.

Chapter III, "Series" (authors G. S. Salekhov and V. L. Danilov), consists of the theory of series and practical methods of summation.

The theory of numerical series is dealt with in §1 including ques-

xiii

XIV FOREWORD

tions relating to infinite products, double series and the summation of convergent series. Side by side with the classical material the reader will find new results about the general tests for the convergence of series and estimations of the remainder.

The more important classes of series of functions are considered in §2: power, trigonometrical, and also asymptotic power series, and their convergence. At this point some methods for the general summation of divergent series are added. In §3 are to be found various devices useful in calculations connected with the theory of series.

Chapter IV "Orthogonal series and orthogonal systems" (authors A. N. Ivanova and L. A. Lyusternik), contains the general problems of the reduction of functions to orthogonal (and also biorthogonal) series. Here, also, general orthogonal systems of polynomials and the classical systems of Legendre, Chebyshev, Hermite Polynomials, and others, are considered.

Chapter V "Continued fractions" (author A. N. Khovanskii), deals with that branch of analysis which occupied the attention of the greatest mathematicians of the eighteenth and nineteenth centuries, but which was afterwards unjustly forgotten. Continued fractions did not find a place in the contemporary larger courses of analysis; on the other hand comparatively recently, some elements of the theory of continued fractions were studied even in middle school. In the past few years the interest in continued fractions has revived in connection with their application in computation and other topics in applied mathematics.

Chapter VI, "Some special constants and functions" (authors L. A. Lyusternik, L. Ya. Tslaf and A. R. Yanpol'skii), has more of the nature of a manual (in the narrow sense of the word). The material here concerns various constants, the most important systems of numbers, including Bernoulli and Euler numbers, some discontinuous functions, and the simpler special functions (elliptic integrals, integral functions, the gamma and beta functions, some Bessel functions, etc.). These functions, together with orthogonal polynomials, after the elementary ones, are the most widely used in applications of mathematics. We would like to mention that these special functions will be dealt with more fully and in the complex domain in one of the following issues.

CHAPTER I

THE ARITHMETICAL LINEAR CONTINUUM AND FUNCTIONS

DEFINED THERE

§ 1. Real numbers and their representation

1· Real numbers

All real numbers can be split into two classes: rational and irrational. All integers and fractions (positive, negative and zero) are rational numbers, while the remaining numbers are irrational.

The set of all rational numbers is everywhere dense, i.e. between any two distinct rational numbers a and b (a<6) there is at least one further rational number c (a<c<b), i.e. in fact, an infinity of rational numbers.

Examples of irrational numbers are:>/2= 1-41421356..., π = = 3-14159..., e = 2-7182818... — the base of natural logarithms, andso on. Irrational numbers consist of algebraic and transcendental numbers. Algebraic irrational numbers are defined as all non-integral real roots of the algebraic equation

xn + axxn-1 + . . . + an_xx + an = 0,

where ax ( /= 1, 2,...,n) are integers; for example, the roots x =l/To, x =V8* of the equations ΛΓ'-ΙΟ = 0, * 5 - 8 = 0, the roots of the equation x5 — 3x* — 2x* -f- x2 -f1 = 0, and so on. The remaining irrational numbers are described as transcendental; examples of these are π, e, é1, 2^2, log n (where n is any integer not equal to 10Ä) and so on.

1

2 MATHEMATICAL ANALYSIS

2. The numerical straight line

We choose on a straight line El9 an origin of measurement — the origin of coordinates 0, a scale the unit of length, and a direction — the orientation. We associate with every real number x a point A(x) on the Une El9 having the coordinate (abscissa), x, conversely, with every point A(x) on El9 we associate a real number x — its abscissa. Ex is called a numerical straight line or a one-dimensional coordinate space (see Chapter II for w-dimensional coordinate spaces i s j .

The number — a for a < 0,

for a ^ 0 is called the absolute or numerical value of the number Û.

The following relationships hold: |β+6 |^ |β | + | Η | e - ô | * l l * H ô | | , |β .* ΙΗ*|. |Η

a I | α | *"!"" W '

The number [0 — i ] is called the distance between the points a and 6 on the straight line E1 (see Chapter II, § 1, sec. 1).

3. p-adic systems

Every real number is expressible as a decimal fraction, i.e. has a definite expansion in the decimal system of numeration. The decimal system is a particular case of a positional system to the base p , where any positive integer p > 1 can be taken as the base. The numbers 0, 1, 2, . . . , />— 1 are called the digits of this system, while pk

(fc = 0, ± 1, ± 2 , . . . ) are units of the &th order in the system. Every positive integer N is uniquely expressible in the form

n Ν=αί>ρΡ + α1ρι + ... + αη!Ρ = Σ^ρ\ (1.1)

where ^ are digits. Equation (1.1) is written as N = αηαη^αη^2... a±a0. (1.10

THE ARITHMETICAL LINEAR CONTINUUM 3

Similarly, any positive real number 5, rational or irrational, is expressible as a fraction to the base p:

S= Σ ahp\ (1.2) ft«-oo

which is written as 5 = anan^1... axaQ, a^a^2a_z (1.20

If 5 is an irrational number, it is uniquely expressible by an infinite non-periodic fraction to the base p of the form (1.2) (or (1.2') ). ·

If 5 is rational, it is expressible as an infinite periodic fraction to the base p, e.g. the number 5 ·= 1/6 is written in the decimal system as

5 = 0-1666... = 0-1(6).

In the binary system, 5 = 1/6 is expressed by the infinite fraction

s = o-ooioioi... = o-o (oi) = 1 + - L + . . . .

Rational numbers to the base/? are numbers expressible as fractions with denominator pk (k = 1, 2, —2· . . . ); each such number has two forms in the system to the base p: one with 0 repeated, the other with p — 1 repeated.

For example, the number 5 = — is written in the binary system as

5 = 0-1000 . . . = 0-1 (0), 5 = 0-0111 . . . = 0-0(1);

and in the decimal system as 5 = 0-5000... = 0-5(0), 5 = 0-4999... = 0-4(9).

Having selected one of these forms for rational numbers to the base p, say the first — with 0 repeated, we obtain a unique form for rational numbers to the base p as infinite periodic fractions to the base p, and at the same time a unique form for every real number.

The elements of various /abased systems were to be found in antiquity in different nations, and traces have been preserved into modern times in certain languages, e.g. of p = 12 (dozens and grosses), p = 20 (traces of this system have

been preserved in French), p = 40 ("forty times forty" refers to number of churches in Moscow (Trans.)), etc. The system to the base 60 was well developed; it originated in ancient Babylon (traces are retained in measurement of angles and time). The 60-base system must have competed with the decimal in the Middle Ages in the Near East and Central Asia. The decimal system originated in India, was further developed in Central Asia, and passed from there into Europe.

At the present time the binary system is widely used in computers (together with the related systems having powers of two as base: p = 2k, k > 1 an integer). A system of numeration to the base three is used in the Moscow State University "Setun" computer. A set of numbers different from 0, 1, 2, . . . , p -1 is occasionally used for the p digits of the system to the base p. For instance, a convenient choice of digits in the system to the base 3 is - 1 , 0, 1. The digits -1 and 1 can be used in the binary system.

Non-homogeneous positional systems are more general; here, the ratios of units of different orders are different numbers. Such systems were used (before the introduction of the metric system) for representing "denominate" numbers, i.e. for representing magnitudes such as length, weight, etc. For example, the following system of units was used for measuring weight in pre-revolutionary Russia: 1 pud (« 16 kg) — 40 funtov, 1 funt («400 g ) « 32 lota, etc.

4. Sets of real numbers

We shall discuss various sets of real numbers — for example, the set of natural numbers: 1, 2, 3, 4, . . . , n, . . . , the set of all proper fractions, the set of all rational numbers, the set of all real numbers between 0 and 1, etc.

The numbers are called the elements of the set in question.

One can consider sets of elements of any kind, and not merely sets of real numbers. For instance, the set of points of a plane, the set of trees in a district, etc. The elements of these sets are respectively points of a plane, trees, etc.

Sets are denoted in this book by capital letters: M, N, A, B, X, Y, etc., or by the symbol {xn}, where xn are the elements of the set (countable sets). The set of numbers satisfying the inequalities a< x< ò (a, b are numbers) is called an interval and is written as (a, b). The sets of numbers satisfying the inequalities x<a,x>b, are called

infinite intervals and are written as (— «>, a) and (è, + <») respectively. The set of numbers* satisfying the inequalities a ^ x ^b is called a segment (or c/aserf interval), and is written as [a, b ].

The sets of points x satisfying the inequalities a ^ x < è, a < x ^ 6,

are called semi-intervals and are written respectively as [a, è), (a, b ]. The infinite semi-intervals ( — ©o, a] and [6, + <») are similarly

defined. The interval (x — ε, x+ ε), (ε > 0) is called an ^-neighbourhood of

the point x. If an element x belongs {does not belong) to the set X, this is written

symbolically as χζ Χ(χζ X or x $ X). If all the elements of a set X are simultaneously elements of a set

7, X is said to be a MOsei of the set 7, and we write symbolically: X cY. Otherwise, X is not a subset of 7, and we write this symbolically as X c 7(or ^ φ 7). For example,

y 6(0, 1), a €[«,*), β?(β, 6), *€[* ,*) ;

(0, 1) e [0, 1), [1, 2] e (0, 1), (a, b) e [a, 0).

The set M of all the elements that belong both to a set A and a set 5 is called the intersection or product of the sets /I and 5, and is written symbolically as

M=A(]B ( M = ΑχΒ = Α.Β = ΑΒ). For example,

(0, l] = T - y , l l n ( 0 , 2), 6 = (a, 6]n[é, c)

and so on. The set M consisting of all the elements that belong either to a set

A or to a set B is called the union or sum of sets 4 and B9 and is denoted symbolically by

M = A{jB (M= A + B). For example,

(0, 2 ) u H j , + - ) = ( 0 , +oo), ( -3 , 7]U(5, 8] = ( -3 , 8].


The set M consisting of the elements of a set B that do not belong to a set A is called the complement of the set A with respect to the set B or the difference between the sets A and B, and is written symbolically as

M = B\A (M = B-A). For example,

(7, 8] = (5, 8 ] \ ( - 3 , 7], (0, 2)\ |~0, ±\ = Γ-ί, 2^ etc.

The notation B\A is employed in more general cases.

5. Bounded sets, upper and lower bounds

A set X of real numbers is said to be bounded from above (below) if there exists a number M (m), not less (not greater) than all the numbers J C Ç I The number M (m) is called an upper (lower) bound of the set X.

A set A" is said to be bounded if it is bounded from above and below. For example, the set ( — » , 0) is bounded from above, the set (0, + oo) bounded from below, while (0, 1) is a bounded set.

The least (greatest) of all the upper (lower) bounds of a set X is called the strict upper (lower) bound M* (m*) of the set X and is written symbolically as

M* = sup* (m* = inf x).

The numbers M* and m* possess the following properties: (1) The inequalities hold for all x € X:

M* ^ x, m* -^ x. (2) Whatever the number ε > 0, a number x0 £ X can be found

for which, respectively,

x0 ^ M* — ε, ;c0 ^ m* + e. For example,

sup x = 0, inf x = 0. «€(-«>, 0) *€(0,*)

(3) If the set X = {*} is bounded from above (below), it has a strict upper (lower) bound.

The concepts introduced below (§2, sec. 2) of strict upper and lower bounds of a function are particular cases of the strict upper and lower bounds of a set.

6. The theory of irrational numbers

Precise irrational number theories, due to Dedekind, Cantor and Weierstrass, made their appearance in the second half of the nineteenth century, in connection with the critical consideration of the fundamental concepts of analysis.

Dedekind's theory. The set of all rational numbers with all their properties is assumed

given. The set of all rational numbers is divided into two classes A and A'. This division is called a section in the rational number domain if the following conditions are satisfied:

(a) every rational number falls into one and only one of the sets A and A'.

(b) every number a of the set A is less than every number a' of the set A'.

The set A' is called the upper class, and the set A lower; 2L section is denoted by A\A'.

Sections can be of three types: (1) either there is no greatest number in the lower class A, while

there is a least number r in the upper class A'; (2) or there is a greatest number r in the lower class A, while

there is no least in the upper class A'; (3) or there is no greatest in the lower class, and no least in the

upper class. We say in the first two- cases that the section is performed by the

rational number r (which is the boundary between the classes A and A'), or that the section defines the rational number r. In the third case the section A\A' defines no rational number; we say that a section of type (3) defines some irrational number oc.

For example, if all the numbers a ^ 0 are referred to class A, as also the numbers a > 0 for which a2 < 2, while all the remainder are in class A\ the section A\A' defines the irrational number yJ2.

s MATHEMATICAL ANALYSIS

AU real numbers can be ordered in the following way: two irrational numbers a and ß, defined by the sections A\A' and B\B' respectively, are reckoned equal if the sections A \ A' and B \ B' are identical, and conversely, if the sections A\A' and B \ B' coincide, the respective irrational numbers are said to be equal. We say that the number a >/?if the class A wholly contains the class B9 which does not coincide with A, and a >- r, where r is any rational number of the class A. Hence only one of the following relationships is possible for any two real numbers a and ß : a = /8, a -< ß> a>/?.

If sections are performed as described above in the domain of real numbers, it turns out that there always exists for any such section A | A' a real number accomplishing the section. This is the essence of Dedekind's basic theorem. This property of the set of all real numbers is described by saying that it is complete or continuous.

We can introduce for real numbers the concepts of the arithmetical operations and laws (addition, multiplication, division by a non-zero number, etc.). For instance, the sum of two real numbers a and ß is taken to be the real number y = a -{- ß which satisfies the relationship a+b < y ·< a'+b\ where a, a\ b and b' are all possible rational numbers satisfying the inequalities: a < a < a\ b << ß << b'.

All the other arithmetical operations can be similarly introduced, while retaining the fundamental properties.

Cantor's theory. We take all possible fundamental sequences (see § 3, sec. 2) of rational numbers. A sequence of rational numbers, convergent to a rational limit, is fundamental. At the same time, there exist fundamental sequences of rational numbers which do not have rational limits, as for instance the sequence of decimal approximations {1; 1*4; 1*41;... } of the square root of two.

Two infinite sequences {xn} and {yn} are said to be equivalent or confinal if \xn—yn\ tends to zero as n -* ».This means that two equivalent fundamental sequences {xn} and {j>n} can only have the same rational limit x as n-+ » , All equivalent fundamental sequences of rational numbers are referred to one class — the equivalence class, and the set of all fundamental sequences of rational numbers is divided into equivalence classes.

There are two possibilities: either there exists a rational number r, the common limit as n -+ «> of all sequences {jcn} of the same


equivalence class X, or there is no such number among all the rational numbers.

We say in the first case that the equivalence class X defines the rational number r; in the second case, we say that the equivalence class X defines an irrational number x (which is also regarded as the limit of the sequences of the class l a s n -*~ oo). Every equivalence class defines a real (rational or irrational) number.

Arithmetical operations may also be introduced for all real numbers. For example, the sum x+y.of two real numbers x and y is taken to be the number which is defined by the class X+ 7, where X is the equivalence class defining the number x, and Y the equivalence class defining the number y, the sum X+ Y being understood as the class to which sequences of the form {xn+yn} belong, where {xn} is any sequence of X, and {yn} any sequence of Y.

All the other arithmetical operations on real numbers may be similarly introduced.

A real number x is said to be positive (x >0), if the corresponding equivalence class X contains a fundamental sequence of positive rational numbers not convergent to zero.

The inequality a > ß} where a and ß are two real numbers, means that α -β > 0.

The concepts of fundamental sequence and equivalence class can also be defined on the set ΕΎ of all real numbers. It turns out that all fundamental sequences are convergent on Ex (see § 3, sec. 2), so that any equivalence class on Ex defines a real number — the common limit of sequences convergent to it. No new numbers can be obtained with such a completion of the set Ex\ in this sense, the set Ex is complete.

Thus the real number set Ex is obtained as a result of completion of the set of rational numbers by the limits of all possible fundamental sequences, of rational numbers. This idea of completion has acquired great value in functional analysis.

As regards other irrational number theories, we may mention, in addition to Weierstrass's, A.N. Kolmogorov's argument (see ref. 6, p. 269) and the axiomatic construction of the real numbers (see ref. 4, p. 157, and ref. 15, p. 180).

§ 2. Functions. Sequences

1. Functions of one variable

If we are given a set of real numbers X = {*}, and each number x £ X is associated with a corresponding number y, where 7 = {y} is the set of all such y, the function y = /(JC) is said to be defined on the set X. The set Zis called the domain of definition of the function f(x)9 and every number x Ç X the argument. The set 7 is called the range of the function/(x). If the argument x is given, the value y = /(*) of the function is given. For instance, for the function y = xz the sets X and Y coincide with the real axis Ελ; for the function y = tan x the set Y is the whole of the real axis El9 while X = Ex — M, where

Af is the set of all numbers of the form—π+rar(« = 0, ±1 , ±2 , . . . ) ;

for the function >>=x\ the sets 7 and JJf are the sets of natural numbers; for the function y = E(x) (the integral part of JC), X is the real line El9 and Y is the set of natural numbers. For the function

sign* = f. — 1 for x < 0,

0 for x = 0, 1 for x > 0

the set Z is the real line, while the set Y consists of three numbers: - 1 , 0, 1.

Any finite set of numbers {ÛJ (/ = 1, 2, . . . , « ) can be regarded as a function, given on the finite set of natural numbers X < {1, 2, . . . ,«} and associating with each of these numbers i the value of the function /(/) = at (/ = 1, 2, . . . , ri).

The concept of a function has been subjected to wide generalization. X can be a set of arbitrary elements. A numerical function is said to be given on this set if a number/(x) is associated with every element x of the set.

We shall consider in Chapter II functions defined on a set of points (or vectors) of n-dimensional space. The area bounded by a polygon, or its perimeter, can be regarded as functions defined on a set of plane polygons; physical magnitudes, such as the mass of

a body, its charge, etc. are defined on the set of corresponding physical bodies, etc.

The elements of a set X, on which a function is defined, are occasionally called points.

2. Upper and lower bounds of a function

An upper {lower) bound of a function f{x), defined on a set X, is a number M{m) such that f{x) ^ M{f{x) ^ m) for all x £ X. If this number exists, the function f{x) is said to be bounded from above {below) on X. A function, bounded from above and below on X, is said to be bounded on X.

The least (greatest) of all the upper (lower) bounds of a function f{x) is called the strict upper {lower) bound M*, {m*) of the function and is written as

Λ/* = supf{x) (m* = inf fix)). x€X \ x£X )

If there exists an element xQ (χχ) of X, for which

f{x0) = sup f{x) (ftpcj = inf f{x))9 χζΧ I x<=X I

then sup f{x) = f{x0) ( inf fix) = f{xj\ χ€Χ ΚχξΧ 1

is called the absolute maximum {minimum) of the function f{x)is written as

f{x0) = sup f{x) = max f{x) (f{x^) = inf f{x) = min fix)). χζΧ xÇX V x€X x£X 1

In this case, we say that/(x) attains its absolute maximum {minimum) at the point x0 {χλ)

Given the finite set al9 a2,..., an, we write max {al9 a2, . . . , an} /min {al9 a2, . . ., an}\

for the maximum {minimum) of the numbers al9 a29 ..., an. For example,

inf — = 0, min sin x = — 1, max sin x = 1, x€(0, oo) X x€Ei xeEi

max{4, 3, 7, 11, 8} = 11, min {4, 3, 2, 10, 17}= 2.

The following inequalities hold:

(1) sup/(*)+ supf{x) ^ sup (M+Mx))9 (1.3) x£X XÇ.X χζΧ

inf /(*)+ inf f(x) ^ inf (f(x)+f(x)). (1.4) χζΧ XÇ.X xÇX

If/(*), f(x) and Λ(χ)+/(*) attain a maximum (minimum) on X, then we have

max f(x) + max f(x) ^ max ( f(x) +f(x)), (1.3a) Ä€^: xtx xtx min /(*) + min f(x) ^ min (/(*) +/i(*)). (1.4a)

χζΧ χξΧ χξΧ

The sign of equality holds in (1.3a) (or in (1.4a)) when/(x) and /i(*) attain a maximum (minimum) at the same point.

For example,

max sin x+ max cos x = 2 > max (sin x + cos x) = 2 . x€JEi x€.Ei Ä € £ I

(2) IfYczX (the set 7 is part of the set X), we have

sup f{x) s sup /(x), inf /(y) s: inf /(*). yer *€* ycy x^x

For example,

max sin x = 2— < max sin x = 1,

max (3, 11, 15, 8) = 15 > max (3, 11, 8) = 11.

The following notation is often used,

(/(*) = *), (/(*)<*), (/(*)£ *), (f(x)>a\ (f(x)^a)

and similar ones, to denote the sets of points x for which the respective inequalities are satisfied (they are called Lebesgue sets).

For instance, (x2< 2) is the interval ( —V 2, >/2); (sin * = 1) is

the set of numbers \(4n+ 1)-π[, where « is any integer. | ( 4 /2+1) ίπ | ,


3. Even and odd functions

Let us consider functions defined on the real axis or on the segments [ — a, a] and intervals ( — a, a) (in general on a set M9 symmetric with respect to the origin).

A function f{x) is said to be even if/( — x) = f(x) for any x of its domain of definition, and odd if/( — x) = —fix). All even powers x2n are even functions, and all odd x2n+1 are odd functions. Other examples of even functions are cos x, \x\, and of odd: sin x, tan x, etc.

A sum of even functions is even, and of odd, odd. The product of even functions is even; the product of an even number of odd functions is even, and of an odd number, odd. For example, sin x tan x is an even function, x sin x tan x is odd. The product (and quotient) of an odd and even function is odd. For example, \x\ sin x is odd.

A constant is an even function. Any function of an even function is an even function, e.g. e'x',

sin (cos x) are even functions. An even function of an odd function is an even function, e.g. cos (sin x). An odd function of an odd function, is an odd function, e.g. tan (sin x).

Any function f(x) is expressible as the sum of an even function /λ(χ) and an odd function /2(x):

/ ϊ*)=/ ι (χ)+£(*) , where

Λ(*) = ^ [/(*)+/(-*)] +C,

Λ(*) = ^ [ / ( * ) - / ( - * ) ] - £

(C is a constant).

4. Inverse functions

Given two sets X and Y, each element x £ X being associated with some element y = A(x) 6 Y, a mapping (or correspondence) A of the set X into the set Y is said to be given.

If y = A(x\ y ζ Y is called the //wage of the element x Ç.X, while # is the pre-image of >\

If a unique element x g X corresponds as pre-image to each element y £ Y, the mapping (correspondence) A is called a one-to-one mapping of X into 7.

EXAMPLE 1. Let all the houses of a street be numbered by the integers from 1 to 80. We have a one-to-one correspondence between the set of houses and the set of the first 80 natural numbers.

EXAMPLE 2. Let E' be the set of all non-zero real numbers; one of the two signs + or — is associated with each number x Ç JE". We have a mapping of E' into the set of signs consisting of two elements; the pre-images of the sign 4- (—) are all positive (negative) numbers.

Let the sets X and Y be sets of the numerical axis Ej ; the mapping A = fix) of the set X into Y is some function y = f(x), defined on the set X, with the domain of values Y.

If the function y = f(x) is a one-to-one mapping of the set X into the set Y9 we say that there exists for the function f(x) the inverse function x = <p(y\ which maps the set Y into the set X. The set Y is the domain of definition of the function x = <p(y), while the set X is its range.

EXAMPLE 3. y = sin x9 x = arc sin y, X = π, —-π

EXAMPLE 4. y = tan x, x = arc tan j , X = π, — π

EXAMPLE 5. y = e^ (k ^ 0), * = (l/fc)log y, X = £i-

5. Periodic functions

A function f(x) is called periodic if a number co>0 exists such that, for any x,

f(x+a>)=f(x). (1.5) The number co is called a period of the function /(*). If G^ and

co2 are periods of/(x), ω1-\-ω2 is also a period of/(x). The least of all such positive numbers co is called the least period

(or simply the period) of the function/(x).

■

THEOREM I. If a continuous (see § 3, sec. 11) function f(x) is periodic and differs from a constant, then it has a least period coQ > 0, and all its remaining periods are multiples of ω0.

For instance, when f(x) = sin x we have ω0 = 2π, when f(x) = = | sin x\ the period ω0 = π, and when/(x) = E(x) (see § 2, sec 1) the period ω0 = 1.

6. Functional equations

A functional equation is an equation connecting different values of functions; we say that a function, for which such an equation holds, is a solution of the functional equation, or, that the functional equation is satisfied by it.

EXAMPLE 6. Solutions of the functional equation

Λ*+Λ>Λ*)+Λ>0 (1.6) include the linear functions

f(x) = kx (k is a constant).

It can be shown that they are the only continuous functions satisfying this functional equation.

EXAMPLE 7. Solutions of the functional equation

Âx)-/(y)]=Ax+y) 0.7) include the exponential functions a* (a 0 ) , these being the only continuous functions that satisfy this equation.

Notice that periodic functions satisfy the functional equation

/(*+ω) =/(*). We can also discuss systems of functional equations. For instance,

the pair of functions

f(x) = sin xy ψ(γ) = cos y (and the pair of functions f{x) = 0, φ(χ) = 0) is the unique continuous solution of the system of functional equations

Ax+y)^Axyv(y)+m^x\ <p(x+y) = <p(x)<p(y)-f(x)f(y),


if we required in addition that the functions be positive in the interval

0, — π j and that the conditions / ( — n 1 = 1, ψ l —π ) = 0 be

satisfied.

7. Numerical sequences

A function, defined on the set of positive integers, is called an infinite numerical sequence if for each positive integer n(n = 1 , 2, 3 , . . . ) there is a corresponding number xn — the n-th term of the sequence {xn} = {*i> *2> · · · 9 xn> · · · } · The number n in the expression xn

is called the index (or subscript). We sometimes consider sequences {xn} where, in addition to the natural values, n can be zero or have any integral value.

We shall say that, when m>n, the term xm follows the term xn

(xnprecedes x„)9 independently of the actual magnitude of the numbers xn and xm. A sequence is regarded as given if a relation is known for forming the terms of the sequence. It is often possible to find an expression (formula) for the general term xn of a sequence.

EXAMPLE 8. Arithmetical progression

*n = β + (Λ—l)fl.

EXAMPLE 9. Geometrical progression

xn = aq*-\

EXAMPLE 10. Decimal approximations to the numberrc=3-14159... :

xx = 3; x2 = 3·1; x2 = 3-14; JC4 = 3-141; . . . .

EXAMPLE 11. Let n^axa^ . . . ak be a given number in the decimal system, where al9 a29..., ak are digits; then the numbers xn = = 0 · axa29... ak and yn = 0 · a^^ . . . ax form numerical sequences when n = 1, 2, For example, when

n = 15 we have x15 = 0-15; y15 =0-51.

EXAMPLE 12. We decompose the positive integer n into prime factors

n = / f i ./ft . . . /f* ./f*î1+1 . . . P**"*1


and define a sequence {xn} as follows:

with n = 1, 2, 3, For example, Λ = 18 = 21.32, x18 = 3.2"1 = 3/2. It is possible to discuss sequences of vectors, functions, etc., as well

as numerical sequences.

8· Upper and lower bounds of a sequence

A sequence X = {*n} is said to be bounded from above by a number M if xn ^ M for all indices n. In accordance with the definition given in § 1, sec. 5, the number M is called an upper bound of the sequence {*n}. The sequence X = {*π} is said to be bounded from below by the number /w, if xn ^ m for all n. The number m is called a lower bound of the sequence {xn}. The sequence is said to be bounded if it is bounded from above and below.

The least M*(greatest m*) of all the upper bounds M (lower bounds m) is called the strict upper {lower) bound of the sequence {xn} and is denoted, as before, by

sup xn = M* and inf xn = m*. n n

The propositions of § 2, sec. 2, hold for the strict upper and lower bounds of a sequence {xn}.

9. Maximum term of a sequence

Given a sequence {wt} (k « 0, 1 ,2, . . . ) , we sometimes want to find max uk -k

the maximum term of the sequence {uk}, provided such a term exists; let us call it μ: : ' μ =» maxHjfc.

ft The subscript of the maximum term of {w*} is called the central subscript and

is denoted by v. If the terms of {uk} include several that are equal to μ, the greatest of the subscripts of these terms is taken as v.

ifuH * un(x) are functions of x, x € X c El9 then v » v(*), μ » μ(χ) are also functions of x.

EXAMPLE 13. Given the sequence {uk(x) » xk/k\} (k » 0,1,2, . . . ) , the number v(x) = E(x) = n, while the maximum term μ(χ) = χηΙηΙ for non-integral x; if * is


an integer, x « m, then v(x) « m, while μ(χ) - um_x « wm. For example, with x = 6 we have v(6) « 6, μ(Θ - uh - 65/5! « 66/6!

EXAMPLE 14. Given the sequence {«*(*) = χ*/(2£+1)!} (A: « 0 1, 2, ...)» the index of the maximum term is defined by the inequality

2Λ(2Λ + 1) ^ x ^ ( 2 Λ + 2 ) ( 2 Λ + 3 ) ,

and, with fixed x > 0, the maximum term wn corresponds to the value n = E(x).

10. Monotonie sequences

A sequence {xn} = {^, x2,..., xn} is said to be monotonically increasing (decreasing), if

and to be non-decreasing(non-increasing), if *! ^ *2 ^ JC3 ^ . . . ^ x n ^ . . .

v^l £= ^2 — · · · ~ xn — · · · ·

Sequences of the form (1.8) and (1.9) are described as monotonie, sequences of the form (1.8) being monotonie in the strict sense (or strictly monotonie). For example, the sequences {jcn}, where

*n = 2 r-, χη = 5 - ζ ^ , xn = 2 + 5 ( Λ - 1 ) 2 n/8+i .

are monotonically increasing in the strict sense; the sequences {xn}, where

__1^ 1

are monotonically decreasing in the strict sense; the sequence {xn}, where

_ l _ ! - 1 - 1

is non-increasing; the sequence {xn}, where Xi = 0, x2 = 1, 3 = 2, #4 = 2, x5 = 3, xe = 4, . . .

is non-decreasing. All the above-mentioned sequences are monotonie.

(1.8)

(1.9)


11. Double sequences

A double sequence {anm} is a function of a pair of integral subscripts n and m: to each pair of integers n and m there corresponds a number anm — the term of the sequence {anm}.

EXAMPLE 15. {anm = 1/(Λ2+™2)} (ny m = 1, 2, . . . ) . If the numbers n and m are represented in the decimal system by

n = tfbtfb-i . . . ufi, 1

« Λ Λ (L10)

(it can be assumed that n and m are represented by the same number of digits; this can always be arranged by putting a suitable number of zeros in front), we can form a double sequence {iVnm}, where

N s Nnm = tfftMft-A-i . . . aLbv (1.11)

EXAMPLE 16. If n = 103, m = 27 = 027, then/\f10327 = 100 237. Conversely, we can associate with every integer N, represented

by the right-hand side of (1.11) ( if the number of digits (2k— 1) in N is odd we put the digit a2Ä= 0 in front), a corresponding pair of integers n and m in accordance with (1.10).

We thus obtain a one-to-one correspondence between the set of pairs of integers and the set of all non-negative integers.

EXAMPLE 17. Given the double sequence {anm} (n, m = 0, 1, 2,...), it can be related thus to an ordinary sequence {bN} (N = 0, 1, 2 , . . . ) , when N = Nnm, we have bN = anm.

The double sequence {anm} is said to be arranged as the ordinary sequence {bN}.

See Chapter III for the summation of double sequences (double series).

§ 3. Passage to the limit

1. The limit point of a set

We call xQ a limit point of a set X if there is at least one more point of X (different from x0) in any ε-neighbourhood of the point x0 (e is any positive number). Alternatively, x0 is a limit point of a set

X if an infinity of points of X is contained in any e-neighbourhood of *0. A set may have no (or even infinitely many) limit points, which may or may not belong to the set. For instance, the set X = = {l//i}, where n ^ 1, has one limit point 0, which does not belong to the set; every point of any interval (a, b) (as also the points a and b) is a limit point of the interval.

THEOREM 2 (Bolzano-Weiestrass). Every bounded infinite set X ofΕλ has at least one limit point.

A set M is said to be closed if it contains all its limit points.

For example, Λ/ = < 0, —-, — , . . . , — , . . . > is a closed set; [ 2 3 n J any segment [a, b] is a closed set. A set M in Ex is described as open or as a domain if any point

appears, in M along with its e-neighbourhood. For example, any interval (α, β) is an open set; any system of

intervals is an open set. The real axis is an open and a closed set; on the other hand, any

semi-interval (α, β], [a, β) is neither an open nor a closed set. The complement of an open (closed) set M with respect to Σ^ is a

closed (open) set. For instance, the complement of the open set (— «>, 0) U (1,

+ oo) is the closed set [0, 1].

2. The limit point and limit of a sequence

The number x is called a limit point of the sequence X = {xn} if any e-neighbourhood of x contains at least one term xm of X different from x (i.e. an infinity of terms of the sequence).

A bounded infinite sequence X = {*n} has at least one limit point. The constant number x0 is called a limit of the sequence {xn} if,

given any number e > 0, however small, there is a subscript ^(depending only on the choice of e) such that, for all n > JV:

We say in this case that {xn} has a finite limit x0 and write lim xn = x0,

or that {xn} is convergent to x09 and write xn-»x0. The sequence {xn} is said to be convergent in this case. This means geometrically that,

given any ε > 0, from some subscript Ne onwards, all the points xn lie in the neighbourhood (x0—ε, χ0+ε). In particular, xn is described as an infinitesimal if lim xn = 0. A sequence that has no limit is

71—»-00

said to be divergent. A sequence {xn}, having the property that, given any ε < 0, there

exists a subscript N such that | xm — xn | < ε for all m > JV, n > JV, is described as fundamental.

A general criterion for the convergence of a sequence {xn} is: THE BOLZANO-CAUCHY CRITERION. A necessary and sufficient

condition for the sequence {xn} to be convergent is that it be fundamental.

We say that the limit of the sequence {x j is equal to infinity, lim nn = oo,

n—>· oo

if, no matter how large the number K > 0, there exists N such that, for all n>-N,

\*n\>K. If the numbers xn(n >iV) are positive (negative), we write

lim xn = + oo / lim xn = — «Λ . 71-*· oo \ n — > o e /

Some limits:

1. Ummk + a^'1+ • • •+ f l n = £ ί 6 ? ί 0 ) .

71-»· o© 2

3. lim ! ^ = 1 for α > 1. 71—*o©

4. lim V«"= 1·

5. lim J L - = e [cf. (6.36)].

, r Λ e 6. lim π-~7(Η+1)(Λ + 2 ) . . . 2 * 4

7. l im l + >/2+VS"+ ^ + ^ = 1.

Um


8. For p > 1

lim J r

9. For a > 0.

PK-f2y+-]K· 1β-ι_2α-ι + 3 , - ι_4α-ι + . . . + (-l)»-i*"-i Λ

lim - - - = 0.

10. If we write

An = ^-p-j and Ο Λ - 1 = v^o^i · · · *n-i »

we have, with ÖÄ = C*:

lim yTn = 2, lim V ^ = %/*"·

11. If the above notation is retained for An and Gn, we have with ak = a+kd(k = 0, 1, 2, . . . ) , e > 0 , i > 0:

hm -~ = —.

3· Fundamental theorems concerning limits

1°. A sequence can have only one limit. 2°. If a sequence has a finite (infinite) limit, it is bounded (non-

bounded). 3°. If a sequence {xn} has a unique limit point x0, the sequence is

convergent andx0 is its limit. Conversely, if a sequence ( x j is convergent to x0, x0 is its unique limit point.

4°. If x is a limit point of a sequence {an}, there exists a subsequence {anJ, convergent to x. Conversely, if y is the limit of some subsequence {anj}9 y is the limit point of the sequence {an}.

5°. Similarly, if x is a limit point of a set M of the numerical axis Ex, M contains a sequence {xn} of numbers different from x that converges to x.

6°. On the assumption that lim xn and lim yn exist, we have:

THE ARITHMETICAL LINEAR CONTINUUM

lim (xn ± yn) = lim xn ± lim y,

lim (χη.^η) = lim xn. lim ^ η—► οο

lim xn

lim -2- = —^ , where lim yn yt 0; n-»ooyn km 7n n-*~ n

ifXn^y» then lim χΛ ^ lim >>n.

4. Some propositions on limits

Γ . Tjf lim αη = α a/zrf α// iAe ακ > 0 (ζ = 1 ,2 , . . . ) , ίΑβ/ι

lim ν α ι α 2 . . . ση = α, lim — - = η - > «> η—>» οο η

2°. 7f 4 ö the greatest of the numbers

al9 a2, . . . , αη {αχ ^ 0) and

Pi > 0 (i = 1, 2, 3, . . . , «), we have

lim 1y/p1a?+iw?+ . . . + ; n a « = Λ,

3°. //*, /or a sequence {an},

and

lim Xfa^ = a

lim ^ an

both exist, then 5 = a.

23


4°. / /

lim —■ ί-2 .— = 0 , pi > 0, n->oo/?o+/?l+ · · · +Pn

and, as n -+ » , sn has a limit equal to s, we have

j i m SoPn + SiPn-.1 + S2Pn-2+ · * - + *n/>0 _ n-^oc Λ + Α + Α + · · · + A i

5. Upper and lower limits of a sequence

The upper (lower) limit of a sequence {xn} is the strict upper (lower) bound of the set numbers which are limits of the sequence, and is written as

lim xn ( lim x :n / l im xn\.

For example,

lim | " ( _ l ) n + l l = i,

ta_[(-V4].-.. while at the same time,

».Γί...[(-1)η4]=4· jï..!*-1^]»-1· n - 1 , 2,

Every bounded sequence has an upper and a lower limit. If a sequence is convergent, its limit coincides with its upper and

lower limits; if the upper and lower limits are the same, the sequence is convergent to their common value.

Given any sequence that has an upper and a lower limit, we can readily form monotonie sequences convergent respectively to the upper and lower limits: a monotonically non-increasing {x*^ to the lower, and a monotonically non-decreasing {**} to the upper:

Λη

X*n

lim xn

If {xn} is a monotonically non-decreasing (non-increasing) sequence and a = lim xn, then x*n = xn, x*n = a (x* = a, x*n = *„).

6. Uniformly distributed sequences

Let the sequence {xn} lie on the segment [a, b], Let Nn{cc,ß) denote the number of the points xk(k = 1, 2 , . . . , n)

which lie in the interval (a, /3) c [a, 6]. If the limit

exists, then no matter what the interval (a, ß) c [a, b], the sequence is said to be uniformly distributed on the segment [a, b],

EXAMPLE 18. The sequence {yn} of EXAMPLE 11 (see § 2, sec. 7) is uniformly distributed on [0, 1].

EXAMPLE 19. The sequence {*n}, where

xn = αησ-\αησ\ a > 0, 0 < a < 1 (n = 1, 2, . . . ) ,

([x] is the integral part of x) is uniformly distributed on [0, 1]. EXAMPLE 20. The sequence {*n}, where

xn = α(ίηή)σ-[α(\ηη)σ], a > 0, a > 1 (« = 1, 2, . . . ) ,

is uniformly distributed on the segment [0, 1]. Uniformly distributed sequences have applications in numerical

integration. Obviously, all interior points of the segment [a, b] are limit points for {*η}, and moreover,

a = lim xn, b = lim xn.

Given any function f(x) continuous on [α, 6], and a sequence {xnY

SUp {Xn, Xn+i, Xn+2> · · ·/>

Ulf {Xn, Xn+i, *n+2> · · ·}>

lim **, lim xn = lim x*n. n-)-oo n ^ o n-*oo


uniformly distributed on [a, b\ the following relationship holds: 1 n 1 Cb

Conversely, if this equation is satisfied for all functions continuous on [a, b], then the sequence {xn} is uniformly distributed.

7. Recurrent sequences

A sequence {xn} is said to be given by a recurrence formula if the first few terms are given and a formula is known, with the aid of which xn is expressible in terms of the preceding terms:

xn = f(xn-l* *n-2> · · · > *n-p)> /? ^ 1 (rt = 1, 2 , . . . ) .

For instance, χλ = 1, x2 = 2, *n = x n e l = xn_2 (n = 3, 4, . . . ) . The sequence itself is sometimes described as recurrent. The simplest example of a recurrent sequence is an iterative sequ

ence {xn}: Xn = / ( *n - l ) ·

Iterative, and in general recurrent, sequences are of great value in approximation methods — for example, in the method of successive approximations and Newton's method.

(a) The method of successive approximations for solving the equation x = /(*), where/(JC) is a continuous function, leads to an iterative sequence {xk}:

**+i = / {**} (* = 0, 1, 2, . . . ) , where some arbitrary number is taken as x0. Here, if the sequence {xk} is convergent to x, χ proves to be a solution of the equation:

(b) Newton's method (or method of tangents) for finding the roots of the equation f(x) = 0, where f(x) is a differentiate function, also leads to an iterative sequence {xk} :

v _/X*fe)*fe--/(*fe) ffr - o 1 2 ì xk+1 — /.,/ N v* — v l, z, . . . ; ,

where some number is taken as x0; here, if {xk} is convergent, then lim xk = 3c is the required root, i.e. /(3c) = 0.

fe-*oo

(See above, for a sufficient condition for the convergence of the sequence {xn} indicated in method (a).).

8. The symbols 0(0^ and 0(a^)

A variable that takes a certain sequence {an} of values is called a variant. For instance, the variable term of any progression is a variant.

If ccn and ßn are given variants, while their ratio <χη/βη tends to zero as n —00/ lim ocn/ßn = 0\, we say thatan (β^ is an infinitesimal

\η-*Όο )

(infinitely large quantity) with respect to βη (α^ and we write symbolically:

*π = θ(βη).

For example, i/n2 = o(l/n), n = o(n2). If αη, βη are infinitesimals and an = oGSJ, we say that an (/?J

is an infinitesimal of higher (lower) order with respect to βη (αη), or αη (/?„) decreases faster (more slowly) than ßn (α„). If an, /?n are infinitely large quantities, and <xn = ο(β^), we say that an (ß j increases more slowly (faster) than /?n (<%„).

If I ocn I ^ C I /3n I, C> 0 (C is a constant), we say that βη has a rate of decrease not faster than ocn or that an has a rate of increase not faster than βη, and we write symbolically:

*n = 0(βη). In particular, if lim <xjßn = C ^ 0 , then an = O (ßj or

For example, n = 0 ( V « 2 + l ) . The equation

an = 0(1) implies that the sequence {an} is bounded, i.e. that |an | C for all n.

THEOREM 3. Given an arbitrary sequence {Xn} = {xJJJ of the following sequences:

X1 = {χΐ9 χ2, χ^, . . . , x m , . . . } ,

A2 = \-^i> «^2» * 3» · · · > *^m> · · ·/>

y / Y 7 1 Y 7 1 Y 7 1 Y 71 \ Λ η — \ Λ ι > ·*2 > ^ 3 ' * * · » ^τη» · · ·/>

there exists a numerical sequence X = {xk} = {xX9 x2,..·, Χ& · · ·}> increasing faster (decreasing faster) than any of the sequences {xJJJ.

For example, if Xn = {nm} = {n\ n2, w3 , . . . , wm, . . .} , the sequence X = {m\} = {1 !, 2!, 3!, . . . , ml,...} increases at a faster rate than any sequence {Xn}: lim /2n/w! = 0 for any n.

9. Limit of a function

Let the function f(x) be defined on some set X. We say that the number A is the limit of/(x) as x -* JC0:

Λ = lim /(*),

or that/(;c) tends to A as x -+ x09 if, given any sequence {xn} € A convergent to x0 ( lim xn = x0), the sequence {fix,)} is convergent

to A. Or, in other words, the number A is the limit of the function fix) a s ^ - x0:

A = lim f(x) ifix) -+■ A as x -* x0)>

if, given any positive number ε, there exists ô > 0 such that |/(x)—A \ < <ε for all x £ A' such that 0 < | x—x01 < ό. For example,

sin* , ,. tan x , ,. 1— cos* 1 lim = 1, lim = 1, lim X ϊ->.(1 Λ ^

lim xü In x = 0 for any a > 0. Ä - * 0

10· Right and left continuity of a function

We shall consider functions y =/(*) , defined on sets X of J^; X will usually be a set such as

[a, b], (a, 0), [*, b\ (a, fl, ( - - , + co), ( - co, 0], [0, + - ) , ( -co, 0), (0, +oo),

etc. Let/(x) be defined on an interval (x0, a). The number Λ=/(χ0+0)

is called the limit of the function fix) from the right at the point x0=x0

if, given any sequence {xn} of (x0, a), convergent to χ0>/(*π) *s c o n" vergent to A. We can similarly define at the point x = x0 the limit from the left f(x0 — 0) = B of f(x), defined in an interval (b, x0). For example, we have for the function y = E(x) (see § 2, sec. 1) and for the integral argumentx = n: E(n — 0) = H— 1, is(7z-f-0) = n. When the argument x0 = 0, the limits from the left and right off{x) are written respectively as/( — 0) and/(-f 0). For example, if/(x) = = sign x (see § 2, sec. 1), then/( -0) = - l , / ( + 0 ) = + 1 .

A function f(x), defined at a point x = xQ, is continuous there from the right (left) if/(xo + 0) [ffro — 0)1 exists, equal to f(x0).

11. Continuous and discontinuous functions

If /(*o-0) =/(*o + 0) = /(x0), (1.12)

the function f(x) is said to be continuous at the point x = xQ. lff(x) is not defined in the interval (6, JC0) or (x0, a), the left- (or right-) hand term in (1.12) is ignored. (See also p. 59). lff(x), defined on the set X, is continuous at every point x ζ X, it is said to be continuous on the set X.

For instance, f(x) = V(x) is continuous for all x ^ 0. Otherwise, f(x) is described as discontinuous. We say that (f(x) has

a discontinuity of the first kind or jump at the point x = ;c0 if/(.x0 — 0) and /(JCO + 0) exist, but (1.12) is not satisfied. In all other cases of a discontinuous function, the point x = x0 is called a point of discontinuity of the second kind. For instance, the function

»-{;; - 1 for | x | < 1, for \x ^ 1

has a discontinuity of the first kind at the points χλ = — 1 and x2 = +1.

Certain commonly encountered functions y = /(x) are equal, at a point of discontinuity of the first kind x = x0, to the arithmetic mean value

/(XQ) = / (*o-0H/(*o + 0)

For example, in the case off(x) = sign x (see § 2, sec. 1): / ( - 0 ) + / ( + 0 )

/(0) = 2

We say that/(;c) is uniformly continuous on the set X if, given any ε > 0, there exists a i = δ(ε) > 0 such that, for any pair of points x\ x" Ç X for which/(*) has a meaning, | ΛΓ' — JC" | < δ implies |/(x0 — —/(*") I < β· A function f{x\ defined and continuous on a bounded segment [a, b], is uniformly continuous on this segment (Cantor's theorem).

12· Functional sequences

An important role is played in analysis by functional sequences {fn(x)} (n = 1, 2, . . . ) , defined on some set X of the numerical axis Ev Various definitions can be given of passage to the limit for such sequences. It is natural to start from passage to the limit at each point, when {fn(x)} becomes a numerical sequence for any fixed χζΧ. If the sequence {fn(x)} is convergent as n — «> for any x £ X, its limit depends on the point* £ X, i.e. is a function/(;c) (a limit function), and this is written as

fn(x) «^=-*Λ*> 0Γ l i m /»(*) s Λ*> * € * EXAMPLE 21. f £

"2 hm arc tan nx = — sign £

n-> +o© 2 o

for

for

for

J C < 0 ,

x = 0,

J C > 0 . T This example shows that the limit of continuous functions may be

a discontinuous function. As a result of a double passage to the limit, functions can be

obtained in the limit that have even more complicated discontinuities, for instance,

lim lim (cos2jtm\x)n = χ(χ)9

where χ(χ) is Dirichlef s function: rational irrational.

f 1 if x is ratic X [0 if x is irrat


REMARK. Together with functional sequences, we often encounter in analysis sequences of numbers, dependent on functions (functionals) For instance, the mean values of functions are defined by the limits of such sequences.

If the function f(x) is integrable on [a, b] a n d / ^ = Λα+νδ^ b-a

6n = , then 1 n 1 Cb

n-*«, n kmtl o a j a

is the arithmetic mean of f(x) on [a, b];

lim \lfmf2n · · -fnn = exP

is the geometric mean of f(x) on [a, b], and

is the harmonic mean of f(x) on [a9 b]. The following also holds: THEOREM 4. If functions fix) and g(x) are continuous and positive

on [a, b]9 then

-Si-VJT (1) lim / g(x)[f(x)]ndx = max M n - ^ e o \ J a x€[o,6]

\bg(x)[f(x)]n+1dx (2) lim ^fb = max f(x).

n " ~ g(x)[f(x)]ndx *€[a'03

13. Uniform convergence of functions

The concept of a uniformly convergent sequence of functions {/n(*)} plays an exceptionally important role in analysis.

DEFINITION. A sequence of functions {fn(x)}> defined on a set X c Εχ, is said to converge uniformly as n — o© to the function f(x)9 also defined on the set X, if given any positive number ε, an integer

N = N(e) can bejound, independent o/x E X, such that,for all n > N:

Ifn(x)-j(x) 1-< e. (1.13)

If {fn(x)} is convergent as n -. 00 at every point x E X, but doesnot satisfy the uniformity condition (1.13), we say that {fn(x)}converges non-uniformly to f(x) on the set X. With non-uniformconvergence, the number N depends not only on the choice of e,but also on the number x E X:

N = N(e, x).

THEOREM 5. If all the functions fn(x) of a sequence {fn(x)} , uniformly convergent to .f(x) on X as n -+- 00 are continuous, the limitfunction f(x) is also continuous on X.

It follows from this that, if the limit function is discontinuous,the convergence as n -. 00 of the sequence {fn(x)}. is non-uniform.

EXAMPLE 22. The sequence {fn(x)} = {xn} is uniformly convergentas n -+- 00 to f(x) = 0 on any segment [0, q], 0 <: q <:: 1; on thesegment [0, 1] this sequence converges non-uniformly to the function

{0 for 0 ~ x <:: 1,

f(x) = 1 for x = 1.

y

O-----a-------~b ......)(-FIG.!.

CAUCHY'S TEST. A necessary and sufficient conditionfor a sequenceoffunctions fn(x) , defined on a set X eEl' to be uniformly convergentas n -+- 00 to a function f(x) is that, givell any e >- 0, there exists an N,depending only on €, such that Ifn(x) - f m(x) I <: e for a/I x E X,provided only that n >- Nand m :> N.

Geometrical interpretation of uniform convergence. Let fn(x) (n = = 1,2,...) be continuous in [a9 b], and let the sequence {fn(x)} be uniformly convergent as n -*· <» to the function f(x\ continuous in [a, b]. Now, all the curves y = fn(x) fall within an ε-neighbourhood of the curve y = f(x) when n > N (see Fig. 1), i.e. they are contained in the strip between the curves y = f(x) — ε and y = f(x) + ε.

14. Convergence in the mean

A functional sequence {fn(x)} is convergent in the mean on the segment [a, b] c Ελ as n — <» to the function /(x) if, given any ε > 0, there is a number N such that, for all n > JV, we have

P [AM-Λ*)ΡΛ^« (it is assumed that this integral exists).

Use is often made of convergence in the mean in various branches of analysis (for instance, in the approximation methods of analysis),

Convergence, defined by the norm of a space (see Chapter II, §1, sec. 2), is a generalization of convergence in the mean.

15. The symbols o(x) and 0(x)

Given two functions x(t) and y(t), defined on a set X, if their ratio x(t)/y(t) tends to zero as t -* a ζ X ( lim x(t)/y(t) = 0), we say that

x(t) [KO] >s a n infinitesimal (infinitely large quantity) with respect to y(t) [x(i)] and we write symbolically:

*(0 = o(y(t)). For example, t2 = o(sin t) as ί — 0, ίΛ = ö(e*) as t -+ <» for

any Λ > 0. If x(0 and KO are infinitesimals as ί -* a, and x(0 = o(K0)>

we say that x(0 (KO) *s a n infinitesimal of higher (lower) order with respect to y(t) (x(t)\ or: x(t) CKO) decreases faster (more slowly) than y(t) (x(t)).

If x (0 ,K0 a r e infinitely large quantities as / — a, andx(0= o(K0)> we say that x(0 (KO) increases more slowly (faster) than j> (/) (KO).

If IX(t) I~ C Iyet) I (c. is a positive constant), we say that x(t) doesnot decrease at a faster r~te than y (t) or that x(t) does not increaseat a faster rate than yet), and we write symbolically:

xCt) = O(y(t».

For instance, t = OCt sin(l/t» and t = O(tan 2t) as t ..... 0,

et = o(ei t) and et = O(eYt2+1) as t _ 00.

In particular, if lim x(t)/y(t) = C ;:c 0 (C is a constant), thent-+oa

xCt) = O(y(t) and yet) = O(x(t).

THEOREM 6. Whatever the sequence of functions fn(x), defined in·the neighbourhood of a point x = a, there always exists a functionq;>(x) EM, decreasing (increasing) faster then any oJthefn(x) as x ..... a.

16. Monotonic functions

A function f(x) is described as monotonically non-increasing (nondecreasing) on a set X (e.g. on [a, b]), if, given any points Xl' X2 E Xsuch that Xl <: X2' we have f(xJ ~ f(X2) ([(Xl) ~ f(x2). A functionf(x) is said to be monotonically increasing (decreasing) in the strictsense (or to be strictly monotonic) if I(xl ) <: f(X2) (f(xl ) :> !(X2»for any Xl' X2 E X for which Xl <: X2.

Functions monotonically non-increasing, monotonically nondecreasing, and strictly monotonic are all described as monotonic.

When f(x) is monotonic, it always has limits from the left andright at a point of discontinuity x = Xo; if f(x) is non-increasing, wehave

if f(x) is non-decreasing,

Lety = f(x} be a monotonically increasing (decreasing) continuousfunction, defined in a segment interval or semi-interval X E.Et, andmapping X into a segment, interval or semi-interval Y c .Et ;an inverse X = q;>(y) of f(x} now exists, defined on Y. The function

<p(y) is continuous on Y and is monotonically increasing or decreasing along mthf(x). For example, the function y = x2 in the semi-interval (0, + oo) has an inverse x = yfy in (0, + <»).

17· Convex functions

A function f(x), defined on a set X (a finite or infinite interval, semi-interval, segment) is described as convex if, given any numbers *i» *2 € X, we have

f(x) is described as concave if the reverse inequality is satisfied:

V j Ç i ^ N 1.[Λχύ_Λχ2)1

This means geometrically that, if f(x) is convex, no arc of the curve y = f(x) lies above the chord subtending it, whereas iff(x) is concave, no arc of the curve y = f{x) lies below the chord subtending it.

For example, "the functions | x |, x2, e*, x+ \ x | are convex, while the functions— |*|, \]x, — e~*are concave; the function y = xa(x > 0) is convex a ^ 1 and a < 0 and concave for 0 < a < 1. A necessary and sufficient condition for a function f(x) to be convex on [a, b] is that, given any numbers xlf x2, x$ satisfying a ^ x± < x2 ·< xz — > the following inequality holds:

*i / (* i ) 1 *2 /(*2) 1 *3 Λ*ζ) 1

^ο.

If/(*) is continuous (even only from the right) and/(;c) = —· [/(*-

— A) +y(x+A)], it is convex.

If/(x) is monotonie in [a, b], /(£) αξ is a convex function.

It must be borne in mind that the graph of a convex function faces downwards, i.e. it is a concave curve in the accepted geometrical terminology.

Convex functions have the following properties: 1°. A convex (concave) function/(*) is continuous at every point

of its domain of definition. 2°. If/(x) is convex, —fix) is concave on the set X. 3°. A convex (concave) function f(x), not equal to a constant on

the segment [a, b], cannot attain a maximum (minium) inside [a, b]. 4°. If/(*) is convex (concave) on [a, b], and l(x) is a linear function,

where 1(a) = f(a) and 1(b) = f(b\ then either/(*) < l(x) (Ax) < /(*)) at every point of (a, b), orf(x) = l(x).

5°. A linear combination of convex functions with positive coefficients is a convex function; in particular, the sum of a finite number of convex functions is a convex function.

6°. If /(«) is a non-decreasing convex function, while u = φ(χ is a convex function, f[<p(x)] is a convex function of x.

7°. The inverse of a decreasing (increasing) convex function is a convex (concave) function.

8°. If f(x) is convex on [a, b], given any positive numbers pi >- 0 (i = 1, 2 , . . . , n) such that /?ι+/>2+· · ·+Αι =: 1 anc* anY points xl9 *2> · · ·> *n o n tke segment [a, b], we have

Λ Λ * 1 + Α * 2 + · · · +PnXn)^Plf(Xl)+Pzf(X2)+ . . . +/>n/(*n).

9°. Any convex function /(*), satisfying the condition /(*0) = 0, is expressible in the form

/(*)= f* P(t)dt9 Jxo

where p(t) is a non-decreasing function, continuous from the right. A function φ(χ)9 defined on a set X<zEl9 is described as logarithm

ically convex if log <p(x) is a convex function, i.e. if

for any χνχ2 6 X- A logarithmically convex function is a convex function. For instance, the function φ(χ) = x log x is logarithmically convex for x>0; φ(χ) = e*2 is logarithmically convex for all x£


ç ( -oo , f oo); φ(χ) = xk (Jç is an integer) is logarithmically convex for x ç (— oo, + oo) and even k & 0, for x £ (0, + oo) and any k ^ 0;

Γ(χ) = tx~'1e~t dt is logarithmically convex for x> 0 (concerning Jo

the function «TOc), see Chapter VI, § 4, sec. 5).

CHAPTER II

«-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE

Introduction

We considered in Chapter I functions of a single variable as functions of a point of the numerical axis Ev Similarly, functions/(*, y) of two variables can be regarded as functions of a point of the plane E2 with coordinates x, y, and functions/(x, y, z) of three variables as functions of a point of space £3 with coordinates x9 y9 z.

About the middle of the last century, coordinate space of n dimensions was introduced into mathematics, and functions of n variables came to be regarded as functions of a point of τζ-dimensional space. At the same time, various concepts of ordinary two- and three-dimensional geometry were generalized to «-dimensional spaces.

This generalization proved to be not merely formal. Our geometrical intuition can be carried over to multi-dimensional entities, and the treatment in geometrical terms of the problems of analysis and algebra of n dimensions led to greater clarity. Geometrical intuition sometimes enables us to discover the facts in w-dimensional geometry, which can be interpreted as corresponding facts of analysis and algebra.

In the present chapter § 1 deals with the theory of w-dimensional spaces, and in particular with the theory of orthogonal systems of vectors, which is an elementary analogue of the more compUcated theory of systems of orthogonal functions, discussed in Chapter IV.

Section 2 is devoted to passage to the limit, continuous functions and continuous (in the generalized sense) operators on them in n~ dimensional space. It is directly related to Chapter I, in which we discussed the same topics for the particular case n = 1.

Section 3 gives a treatment of one of the branches of w-dimensional

38

«-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE 39

geometry — the theory of convex «-dimensional bodies, which, apart from its geometrical interest, has acquired importance in a number of applied mathematical problems. It should be remarked that no great simplification is obtained by confining the discussion of convex solids to three-dimensional rather than «-dimensional space; and in fact, it is precisely the theory of convex bodies in space of a large number of dimensions that has acquired practical applications.

§ 1. «-dimensional spaces

1. «-dimensional coordinate space

An element X of an n-dimensional space En is defined as a set of « numbers xl9 x29 . . . , xn. It is written as X{x^ x2, . . . , *n) or X = = (*i> *2> · · ·> Χτυ· The space En is the set of such elements.

Elements of the space En are treated in two ways: on the one hand, they are regarded as points with coordinates xl9 x2, . . . , xn, on the other, as vectors with coordinates xl9 x29 . . . , xn {n-dimensional vector space). We shall start by using the first treatment.

There are various ways of introducing distance or a metric into the space En (see, for example below, § 3, sec. 3). The most common is the following (Euclidean) metric: the distance ρ(Χ9 Υ) between the points Χ(χχ, x2, . . . , *n) an^ Y CVi> Λ · · ·» Λι) °f^n IS defined as the number

Q(X, Y) = Jtbi-yp- i2·1) y t « i

The space En with a distance introduced in this way is called n-dimensional Euclidean space.

Formula (2.1) reduces to the formula for the distance between two points in analytical geometry when « = 1, 2, 3.

We write Θ for the point with zero coordinates: Θ = (0,0, . . . , 0) (the origin of coordinates). We have

ρ(Χ,θ)= /£*?. (2.2) V i=»i

The distance thus introduced has the following properties: (1) ρ(Χ9 Y) ^ 0, where ρ{Χ9 Y) = 0 only when X = Y; (2) ρ(Χ9 Y) = ρ(Χ9 Y); (3) ρ(Χ9 Υ)^ρ(Χ, Υ)+ρ(Υ, Ζ)

(the triangle inequality). An attempt is generally made to preserve these properties in

generalized concepts of distance.

2. -dimensional vector space

The «-dimensional space £n can also be regarded as a vector space. The operations of addition and multiplication by a scalar (number) can be performed on vectors in the plane £2

an<* *η three-dimensional space £8. Every element X (xl9 x29..., x j of £n will be regarded as a vector with coordinates xl9 x29 . . . , xn.

The sum of two vectors X (xl9 x29 ..., x j and Y (yl9 y29 . . . , y^ is the vector X+ Y of £n with coordinates x-L+y^ *2+^2> · · ·> *π+>π· The difference X— Y is similarly defined.

If λ is a number (scalar), and X(xl9 x2, . . . , x j is a vector of £n, XX denotes the vector of £n with coordinates λχΐ9 λχ29 ..., λχη. Vectors X and λΧ are described as collinear. Linear operations on vectors reduce to linear operations on their coordinates.

The zero point 0 corresponds to the zero vector Θ with components equal to zero. Obviously, X+ Θ = X9 λθ = 0.

The vectors ^(1, 0, 0, . . . , 0), e2(09 1, 0, . . . , 0), . . . , en(09 0, . . . ; 0, 1)

or alternatively, ei(&11> i2> · · · > δφ . . . , Ó in),

where ôtJ is the Kronecker delta: ( 0 for Ï V j ,

for 1 = j , are called 101ft vectors. The following equation holds for any vector X = (χ^ x29 . . . , x^ :

n

z = Σ *^· t - 1

The norm \\X\\ of the vector X is the number n

11*11= /£* î . (2.3)

equal to the distance ρ(Χ, 0) from the point X to Θ. The norm of a vector satisfies the conditions:

(1) \\χ\\ ^ 0, where | |Z | | = 0 only when X= Θ; ] (2) \\λΧ\\ = \λ\\\Χ\\; (2.4) 0) i|jr+riiss||jrii + |iri|. J

The distance ρ(Χ, Y) between elements X and 7, introduced in accordance with (2.1), is the same as the norm of the difference between the corresponding vectors

Q(X9 7 ) = ' | | * - 7 | | . (2.5)

The sphere S(X0, r) in En of radius r with centre at X0 is the set of points (vectors) X, for which

ρ(Χ, X0) = \\X-X0\\^r. If we introduce for sets in En the notation introduced in Chapter I, § 2, sec. 2, we have

S{X» r) = (ρ(Χ, Χ0) < r).|

3. Scalar product

In analogy with the two- and three-dimensional case, the scalar or inner product of two vectors X(xl9 x2,..., x^) and Y(yl9 y2,..., ^n) of En is defined by the equation

XY=(XY) = fW. (2.6)

The elementary properties of scalar products are:

(1) XY=YX; (2) XX = \\X\\2^0. CAUCHY'S INEQUALITY. Given any two vectors X(xl9 x2, . . . , xn) and

\XY\^\\X\\\\Y\\ (2.7)

0Γ

Σ χΰ>χ i - 1

Σ A Σ >i · (2.7') i - 1 i - l

Equality holds in formulae (2.7) arcd (2.7') */ and only if the vectors X and Y are collinear (i.e. if X = 0 or if Y = λΧ, i.e. if all the *{ = 0 or yi = Axf (/ = 1,2, . . . , «)).

JAe a^/e between two vectors. Let -ifo, x2> · · ·> *n) an<* CVi> J>2> · · ·» yn) be two vectors of En different from 0; the angle ψ between them is given by

C O S Ψ = H u^\ ii · ( 2 · 8 ) 11*11 IIJMI

In view of (2.7), | cos φ | ^ 1. The projection of the vector X on to the vector Y is given in magnitude

by xy 11*11 = || x| | cos 7. (2.9)

When || Γ|| = 1, the projection of vector X on to 7is equal in magnitude to their scalar (inner) product:

XY= »I'll cosy. (2.10)

EXAMPLE 1. The magnitude of the projection of the vector X(xl9 *2> · · ·> *n) o n to the unit vector e{(z = 1, 2, . . . , 72) is equal to the coordinate x, of the vector X.

4. A linear system and its basis

An w-dimensional vector space is a particular case of an «-dimensional linear system.

A linear system L is a set of elements to which we can apply the operation of addition, i.e. the operation of finding a new element c£L from two elements a, b ÇL, c = a+b9 and the operation of multiplication by a real number λ, i.e. the operation of finding from an element a € L and a number λ an element d = λα Ç L. These operations have the following properties:


(1) (a + 6) + c = a + (i + c); (2) a+b = b + a; (3) λ(λλα) = (λλ^; (4) Ma + b) = Αα+Aè; (5) (Α + λ^α = λα + λ^; (6) 1. α = α.]

The system L possesses a zero element 0, for which α+θ = a, 0.# = 0 for a € L.

Linearly independent elements. Elements xl9 x2, · · ·» *Λ °f a linear system are said to be linearly dependent if there exists a set of numbers cl9 c2, ..., ck, not all zero, such that

C1X1 + C2PC2+ . . . H-CfcXft = Θ; (2.11)

if there is no such set of numbers, i.e. if (2.11) is satisfied only with

cx = c2 = . . . = ck = 0,

the elements xl9 x2, ..., xk are said to be linearly independent. EXAMPLE 2. On a plane, the vectors Xx (1, 1) and X2 (2, 3) are line

arly independent, while the vectors Α\(1,1) and Xz (3, 3) are linearly dependent (since 3X1—XZ = Θ). In three-dimensional space, the vectors Υλ (1,0,0), Y2 0> !> °)and Yz 0> 1> 1) a r e linearly independent; the vectors Zt (1,0,0), Z2 (2,1,1) and Z3 (3,2,2) are linearly dependent since

Z1-2Z2^ZZ = 0 A linear system is described as n-dimensional if it contains n line

arly independent vectors, and any « + 1 elements of it are linearly dependent. A linear system is said to be infinite-dimensional if it contains any number of linearly independent elements. An «-dimensional vector space is an «-dimensional linear system.

The set of solutions of a linear homogeneous differential equation forms a linear system. In the case of an ordinary «th-order equation this system is «-dimensional; in the case of a partial differential equation the system is infinite-dimensional.

The basis of an «-dimensional linear system Ln is any set (el9 e29..., e„) of « linearly independent elements.


Any element I of the system Ln is linearly (and uniquely) expressible in terms of the elements of the basis:

I = x1e1 + x2e2 + . . . + xnen; (2.12)

the elements xiei (i = 1, 2, . . . , n) are called the components with respect to the basis (el9 e2,..., e^ while the numbers xn are the coordinates with respect to this basis.

EXAMPLE 3. The unit vectors ei form a basis in τι-dimensional vector space:

e1 = (1, 0, 0, . . . , 0), e2 = (0, 1, 0, . . . , 0), . . . . . . , en = (0, 0, . . . , 0, 1).

An w-dimensional linear system can be regarded as an w-dimensional space, in which the elements of the basis play the part of unit vectors.

Examples of «-dimensional linear systems. Linear systems of functions, for which addition and multiplication by a number are understood in the ordinary sense, are aften encountered in analysis. We shall quote some examples.

(1) The (/i + l>dimensional system of polynomials

P00 « £ cS te-0

of degree not higher than n. The system of powers

1» X> X » · · · » X^t

can be taken as the basis, as also can any system of polynomials

P0 - 1, Px{x) - X+Û 1 0 ,

Ä - 1 P2(x) - x*+a2lx+at0, . . . , Pk(x) « x*+ £ akix\

t-o n- i

. . . , P . ( x ) - * " + £ aHtx{. i-o

(2) The system of homogeneous polynomials Pk (x„ *2, . . . , xn) of degree k in n variables xlt x2i..., xni i.e. sums of terms of the form

where

k,+k2 + . . . +fcn ~ k.

7Z-DIMENSI0NAL SPACES AND FUNCTIONS DEFINED THERE 45

The basis of this system consists of all single terms of the form χ\ιχ**... xk» of degree k « kx + kt+ . . . ±kn. The number of such single terms (i.e. the number of dimensions of the system) is

(zi+fc-1)! Λ ! ( £ - 1 ) 1 '

For instance, with n = 3, k = 3, the number of them is equal to 5!/3!2! - 10 (the single terms are x\, x\xZi x\xZi χγχ\, ΧχΧ^χ^ xxx\, xj, x\xZi x^c\y χ%).

If a system Ln has one basis (el9 e2, . . . , e^), it has an infinite set of bases. Let (e'v e'v...9 e^) be another basis; then the elements ei of the old basis are expressible in terms of the elements e\ of the new in accordance with

n *i = Σ αΰ*ί (/ = 1, 2, . . . , TI),

where the determinant | «gli;* ^ 0. An element/is expressible in terms of the elements of the new basis in accordance with

/ = Σ Ae\. i - 1

The connection between the coordinates xi and x\ of an element / with respect to bases (ej and (e^ is given by

*i= i*ax) 0 '= U 2 , . . . , » ) . (2.13) j - i

It follows from what has been said that the elements of any /i-dimen-sional linear system Ln with basis (ex, e2, . . . , e^) can be regarded as vectors of an Λ-dimensional space in which the elements eK of the basis play the role of unit vectors. We shall confine our attention to such spaces in the present chapter.

5. Linear functions

A linear function or linear form in En is a function/pf) = f(xl9 x2,..., x j , satisfying the conditions:

(1) f(X+Y)=f(X)-f(n (2.14) (2) f(XX) = Xf(X\ where λ is any number. (2.15)


If y*( i) = y% for the unit vectors e{ (i = 1, 2, . . . , n), then

/(Z) = f(xl9 xl9 . . . , *n) = / f £ Xiet) = £ Λχ4. (2.16) i « l

The numbers yi are the coefficients of the linear form fi These coefficients can be regarded as the coordinates of the vector

Y = (yl9 y2, . . . , yj of En; we now have from (2.16):

f(X) = YX. (2.17)

Γ/ie /wear function f{X) is equal to the scalar product of the fixed vextor Y with the variable vector X. We usually say that the function f(X) is generated by the vector Y.

The scalar product XY is a bilinear function of X and Y: it is a linear function of X if Y is fixed, and a linear function of Y if X is fixed.

We have the equation

1711 = maxTZ. (2.18)

i.e. the norm of the vector Y generating the linear form YX is equal to its maximum value on the unit sphere \\X\\ ^ 1. Criteria for linear independence. The Gram determinant for vectors. Let

*i(*ii» *i2> · · > *in) (i = 1, 2, . . . , ri)

be vectors of the space En. From them we form the following determinant:

Δ{Χΐ9 X29 . . . , Xn) —

xn x12

*21 *22

xln x2n

xnl xn2

(2.19)

THEOREM 1. A necessary and sufficient condition for vectors Xl9

X2, . . . , Xn ofEn to be linearly independent is that

A(Xl9 X2, . . . , Xn) * 0. Gram's determinant Γχ x^ t Xm of the vectors Xl9 X2 . . . , Xm

of En is given by

tf-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE 47

X\Xz . . . Xm

(χ,χθ (χχχύ . . . (XJJ {χ^) (χ2χ2) . . . (XM

\XmXl) (XynXz) . . . {ΧγηΧγγ^

(2.20)

Gram's determinant has the properties: ( 1 ) ^ Λ . . . χ . δ 0 ; (2) A necessary and sufficient condition for linear independence

of the vectors Xl9 X2, . . . , Xm is that Γ χ Λ . . . χ . = > 0 ; (2.21)

(3) When m = n, 1 X\Xt ... Xn = [A(Xl9 X29 . . . , JQ]2.

See Chapter IV, § 2, sec. 3 for the Gram determinant for functions. Manifolds. Let Xl9 X2> ..., Xk be linearly independent elements

o f £ n ( l ^ k < Λ). A set of elements Z of the form

X = Σ c f * (2.22)

with arbitrary real ci is called a k-dimensional linear manifold. A one-dimensional linear manifold, i.e. a set of elements of the

form X = Axt (jq 5* 0) is a straight line passing through Θ and xv

Part of this straight line — the set of elements of the form X = λχΐ9

λ > 0, — is called a ray. A displaced k-dimensional manifold or k-dimensional plane is a

set of elements X of the form

Z = X0+ Σ *Λ (2·23)

with fixed and linearly independent Xl9 Xl9 . . . , Xk and arbitrary values of the numbers c1? c2, . . . , cfe. It is obtained by displacement of the manifold {2.22) along the vector XQ. A one-dimensional displaced manifold is a straight line:

X= X0+tX1 ( - o o < r < + co ) . (2.24)

We have for a straight line through the points Xx and X2:

X= (1 - 0*i + *** ( - - < t < + oo), (2.25)

The segment of the straight line joining points X1 and X2 is the set of elements of the form

X = (1 - 0ΑΊ+ tX2 (0 ^ t ^ 1). (2.26)

An (w—l)-dimensional displaced manifold is called a hyperplane. Any hyperplane is given by an equation (i.e. is a set of points Χ(Χχ>

x29 , Xj)9 satisfying the equation)

W = I W = 4 (2.27) j - i

where YXis & linear form generated by the non-zero vector Y(yl9y2f

. . . , y^. Conversely, every such equation defines a hyperplane. In general, any k-dimensional linearly displaced manifold (1 ^ k < ri)

is given by a system of equations

ΆΧ = f yirt = d{ (/ = 1, 2, . . . , n-k\ X2.28)

where the vectors Yi(yivyi2,.. .,^in) (/ = 1, 2, . . .,n-k\ generating corresponding linear forms, are linearly independent; in other words, every such manifold is given by n—k linear equations. Conversely, Ti—k equations (2.28) (with linear independence of the vectors Y{) define a displaced k-dimensional manifold. If all the right-hand sides di = 0, we get a fc-dimensional (non-displaced) manifold. In particular, the straight line (2.24) is given by 72—1 equations (2.28).

6. Linear envelope

The linear envelope of a set M of En is the least linear manifold containing M : in other words, the linear envelope of the set M is the set of all linear combinations of any finite number of elements of My i.e. of elements of the form

£ tiX» Xi € M.

In particular, if Xl9 X2, ..., Xk are linearly independent elements of £„, their linear envelope is the /^-dimensional linear manifold con-

W-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE 49

sisting of all elements of the form

Σ hXi-

Notice that, if the vectors Xl9 X29 . . . , Xk form a basis of the A>dimensional linear manifold Ek of En9 we can supplement them by vectors Zl9 Z2, . . . , Zn_k in such a way that the vectorsXl9X2 .. .9 Xk, Zl9 Z2, . . . , Zn_kform a basis in En. Let En_k denote the linear envelope of vectors Lx. Every element Y οΐΕη can be expressed (uniquely) as

7 = X + Z , (2.29) where

Here, the space £n is the direct sum of the manifolds Eh and £n_Ä, this being written symbolically as

En = Ek+En_k. (2.30)

X and Z in (2.29) are called the components of the vector Y in Ek and £n_fe.

7. Orthogonal systems of vectors

Two (non-zero) vectors X and Y are said to be orthogonal if XY = 0 (i.e. if cos 9? = 0). The vectors Xl9 Xl9 ..., Xm form an orthogonal system if they are orthogonal in pairs, i.e.

XiXj^O for i*j. (2.31)

The vectors of an orthogonal system are linearly independent. If, in addition, ||Jf4|| = 1, the vectors are said to form an orthonormal system. In this case,

0 for i 7* /, - . (2.32) for ι = j .

f ° XiXj = ôij = < j

The unit vectors ex, e2> · · · > £n f °r m a n

orthonormal system. In /?-dimensional space En there exists an infinite set of orthogonal,

and in particular orthonormal, systems of n elements, but there are

no orthogonal systems of (n+l) elements. A system of n orthogonal vectors in En forms a basis in En, called an orthogonal basis.

THEOREM 2. If vectors ev ev . . . , enform an orthonormal system in En, any vector X of En is expressible {uniquely) in the form

X = f xi<> x'i = Xe\> (2·33) t » l

l l*ll2 = E^ i 2 · (2.34)

The passage from unit vectors elf e2, . . . , en in En to another orthonormal basis ev e'2, . . . , e'A in En is called an orthogonal transformation. An inner product of vectors is unchanged in an orthogonal transformation, i.e. if

x = Σ*A = Σ*ί«ί. ^ Σ Μ - Σ y ' i < -i « l t « l i - 1 i - 1

then

XY = f *Λ = Σ Φί·

In particular, the norms || X\\ = %JXX (see (2.34)) of all elements of £n and the distance between any pair of such elements: ρ(Χ, Υ) = = || X— Y11, are unchanged by an orthogonal transformation.

In general, if vectors ev eT ..., e'n form an orthogonal system, we have

n -y / X = 2 xieU xi = H v i|2 ' (2.33 )

t « i l ib i l i

l l * l l 2 = îx?\ie'i\\2= liXe'if. (2.34')

In analogy with the theory of orthogonal series (see Chapter IV), the coefficients x[ in equations (2.33) and (2.33') can be termed Fourier coefficients.

8. Biorthogonal systems of vectors

Two vector systems Xl9 X2, . . . , Xn and Yl9 Y2, . . . , Yn of En form a biorthogonal system if

XiYj = 0 for i T£ J\ XiYj ja 0 for i = j . (2.35)

Multiplying the vectors Xt or Yi by constants, we can arrange that

X& = 1 (/ = 1, 2, . . . , /z) where the old notation is retained for the new vectors. In this case, every vector Z of En can be written as

n Z = Σ *tfi, *i = ZXi( (2.36)

or as

Z = f ViXu Λ = ZF,. (2.36')

G/ve/i a system ofn independent vectors Xl9 X2, . . . , Χή in En, we can form a system of vectors Yl9 Y2, . . . , Yn, such that the two systems {Xj} and {FJ are biorthogonal If

* i = (*il> xi2> · · · > *in) a n d 7 i = CVil* 7i2» · · · » Jin),

formation of the system {7J from the system {XJ amounts to formation of the inverse \\y{j \\}^ of the matrix libili;]1. (See Chapter IV regarding orthogonal and biorthogonal systems of functions).

i EXAMPLE 4. Let Xi = £ e fi = 1, 2, . . . , τζ), where e^ are unit

j = i vectors in En; Y{ = ei—eiJf.1 (i = 1, 2, . . . , w— 1), 7n = en. In this case (ZJ and {FJ are biorthogonal systems in En.

9. The projection of a vector on to a manifold

Let Ek be a ^-dimensional manifold in «-dimensional space En (1 ^ /c < «). The vector X ofEn is said to be orthogonal to Ek (written as X ± Ε£) if Xis orthogonal to any vector Y of Ek, i.e.

XY = 0 if . Y J_ Ee.

A vector X0 of Eh is called the projection of the vector X on to Ek if X—X0 is orthogonal to Ek9

Xo= VT.EkX (2.37) We have:

X=X0 + (X-X0\ (X-X0)±Ek> X*ZEh. when X € Ek, to pr.£t X = X.

lfX£Ek,pv.EkX=X. Any vector X of En has a unique projection on to Eh. If Z0 is the projection of the vector X on to Ek, we have

(1) i m i 2 = \\X0\\*+\\X-X0\\2, (2.38) (2) \\Χ-Υ\\*ίζ\\Χ-Χ0\\\ (2.39)

where Y is any vector of £ft, and equality holds only when Y = X0. In other words, the minimum of || J5f— I II2, where Y is any vector of Eh9 is attained when

Y=X0 = pr.EkX

Let U0 7* Θ be a vector of iTfc. We shall write JE for the set of vectors of the form tU0 ( - oo c * < + oo), i.e. the straight line Ex = *£/<;. 7%e projection of the vector X on to the vector U0 is defined as the projection of X on to the straight line El9 i.e. we have:

and in particular, if || J70|| = 1, then

' pr.^o X = (XU0)U0 = (||JT|| cos cp)U0,

where φ is the angle between the vectors X and U0. Expansion (2.33) of a vector X into a system of orthogonal vectors

e^{j = 1, 2, . . . , //) implies the representation of X as the sum of its projections on to the e^

Let Xl9 X2, . . . , Xk be orthonormal vectors in En (k < ri); they form an orthogonal basis in Eh — their linear envelope. We can supplement them by the vectors Xk+j (7= 1, 2, . . . , w—fc) such that the 72 vectors A (i = 1, 2, . . . , ri) form an orthogonal basis inEn.

* - Σ « Α 2

attains

Let X be any vector in En; then k

pr X= Σ CtXi, c^XX,.

The sum of the first k terms in expansion (2.33) or (2.33') is the projection of the vector X on to the linear envelope of the first k vectors * , ( / = 1,2, . . . , * ) .

Let the vectors {X,} (i = 1, 2, . . . , &< «) be orthogonal. Let n

us form all possible sums £ c^ ·. The expression

a minimum when q = x·, where q is the "Fourier coefficient"; c{ = ATf for||JTf || = 1, while in the general case cx = XAy||XJ|,

THEOREM 3. Let {L^ (i = 1, 2, . . . , k) form a basis {generally speaking, not orthogonal in E^, and X be any vector ofEn; then

h

Pr.~ X = Σ dfo (2.41)

where d^ are given by the system of linear equations h

Σ (IHLM = XL, · (/ = 1, 2, . . . , k)9 (2.42)

the determinant of which is the same as the Gram determinant Γ. When k = n, the system (2.42) becomes a system for determining the components of the vector X with respect to the basis {LJ ( /= 1, 2 , . . . , n).

§ 2. Passage to the limit, continuous functions and operators

1. Passage to the limit in /z-dimensional space

Let Ln be a linear system with a chosen basis. Every element X ζ Ln is defined by its coordinates xl9 x2, . . . , xn, X = (xl9 x2, . . . , *n)· Let {Xn} = {Ζΐ5 X2, . . .} be a sequence of elements of Ln, where xm = (*mi> xm2» · · ·> O ' T h e element X = (*lf *2, . . . ,*„) of Ln is said to be ί/ze //mfr o/ i/ie sequence {Xm} if

xf = lim xmi (2.43)

(i.e. if all the coordinates of X are the limits of the corresponding coordinates of the terms of the sequence {Xm}). We write conventionally in this case

Z = lim Xm9 (2.44)

and also speak of the sequence {Xm} converging to X. The coordinates of the elements {Xm} and X depend on the choice

of basis in Ln, but equation (2.43), i.e. (2.44) also, does not depend on the choice of basis in Ln: if the coordinates of Xm tend to the corresponding coordinates of Xin one basis asm -► <χ>, this property will hold for any choice of basis.

An «-dimensional linear system, in which passage to the limit is defined in the sense indicated, is called an n-dimensional linear space.

In Euclidean, and more generally, in normed (see § 3, sec. 3) spaces, where the distance between two points, or the norm of a vector, is defined, passage to the limit (2.44) can be defined as follows: X= Urn Xm if

lim o(Xm, X) = lim \\Xm-X\\ = 0. (2.45)

Condition (2.45) is equivalent to condition (2.43). Thus, if we define the distance between two points or the norm of a vector (§ 1, sec. 1 or 7) in a linear space, and make use of the definition of limit in accordance with (2.45), no changes whatever are introduced into the meaning of passage to the limit in Ln. |

There will be no loss of generality if we consider in this chapter passage to the limit in Euclidean spaces En.

As in the one-dimensional case (see Chapter I, § 3, sec. 2), we shall call a sequence {Xm} of En fundamental if, given any ε > 0, here exists N such that

| | X m , - X m | | < £ , for all m! > N, m > N.

A necessary and sufficient condition for a sequence {Xm} to have a limit is that it be fundamental.

The concept of limit point {point of condensation) for a set M c En is an immediate generalization of this concept for Ex : X is called a

t This introduction of a distance is called metrization of Ln.

limit point of the set M c ΕηΊϊ it is the limit of some sequence {Xm} of points of M different from X.

An equivalent definition is : X is called a limit point of the set M if any sphere S(X, e) with centre at X contains a point of M different from X.

An analogue of the Bolzano-Weierstrass Theorem holds (see Chapter I, § 3, sec. 1).

Every bounded infinite set in En has at least one limit point. An open set in En is a set U of points of En such that, if a point

X belongs to U, there exists a sphere with centre at X contained in U. An open set is called a domain if any two points of it can be joined by a step-line, contained wholly in this set.

Examples of a domain are as follows : on a straight line, a finite or infinite interval; on a plane, the interior of a circle, triangle, strip between two parallel straight lines, etc. ; in three-dimensional space, the interior of a sphere, parallelepiped, cone, etc.

An example of a domain in En is a parallelepiped (a parallelepiped in E1 is an interval), i.e. a set of points Χ(χχ, x2, · · . , * n ) , the coordinates of which satisfy tfie inequalities ax -^ xi -< bi (i = 1, 2, . . . , n), where at and bx are given numbers.

The intersection of any finite number of domains is a domain. A boundary point of a domain Q in En is a limit point of Q that

does not belong t o g . The set of boundary points of Q is called the boundary of Q. A domain Q together with its boundary forms a closed domain (or body) in En.

For example, the boundary of an interval (a, b) consists of the points a and 6, while the segment [a, b] is a closed domain; the boundary of a sphere S(X0, r) is the surface of the sphere, i.e. the set of points X for which

A closed set in En is a set in En containing all its limit points; for instance, closed domains are closed sets.

THEOREM 4. The complement in En of an open set (domain) is a closed set, and of a closed set an open set,

For example, the complement pf the sphere (\\X\\ ■< r) is the closed set (|| Z11 ^ r), the complement of the closed sphere (\\X\\ ^ r) is the domain (|| X\\ >» r).

2. Series of vectors

The laws for passage to the limits of vectors (points) of H-dimensional space En are essentially the same as the laws of passage to the limit in Ελ\ the generalization to the /i-dimensional case of the theory of conditionally convergent series (i.e. series which are themselves convergent, while the series formed from the absolute values of their terms are divergent) is of some importance.

A vectot series in En: oo

Y*k= *i + *i + *i + . . . (2.46)

is defined like a numerical series. If Xk = (xkv xk2, . . . , xkn), series (2.46) corresponds to the k

numerical series oo

Σ Xhi = *it + *2i + *3i+ · . · (Ϊ = U 2, . . . , n), (2.47)

the terms of which are the coordinates of vectors Xk> i.e. of the terms of series (2.46).

The partial sum Sm of series (2.46) is the sum of its first m terms:

m

k - 1 Series (2.46) is said to be convergent if its partial sums Sm are con

vergent as m — oo to some vector 5, the sum of (2.46): oo

S = Σ Xk = lim Sm. fe=i m-»*oo

The convergence of series (2.46) is equivalent to the convergence of all the numerical series (2.47). If S = (sX9 s2, . . . , s„)9 then

oo

Si = Σ *fei 0" = 1, 2, 3, . . . , w). k - i


Series (2.46) is described as absolutely convergent if the series of the norms of its terms is convergent:

The absolute convergence of series (2.46) is equivalent to the absolute convergence of the numerical series (2.47).

Series (2.46) is said to be conditionally convergent if it is convergent non-absolutely. We have Steinitz's Theorem, which is a generalization of Riemann's Theorem (see Chapter III, § 1, sec. 2).

(1) If series (2.46) is absolutely convergent, its sum remains unaltered whenever the order of the terms is changed.

(2) If series (2.46) is conditionally convergent, its sum can be altered or it can be made divergent by changing the order of the terms.

(3) The sums of the series obtained by interchanging the terms of a convergent series (2.46) entirely fill some k-dimensional plane, 0 ^ ^k^ n (see § 1, sec. 5).

For an absolutely convergent series, k = 0; for a conditionally convergent series, k > 0.

EXAMPLE 5. Let xk = (—l)Ä(l/fc), and let s be an integer, 1 s ^ n. Every positive integer m can be written in the form m = rs+k9 where r is an integer, Ì ^. k ^ s. Let Xm = Xrs+k = xrek9 where ek (k = 1, 2, . . . , s) are unit vectors.

By changing the order in the series £ Xm9 we can now obtain as m

the sum, any vector of an ^-dimensional manifold Es — the linear envelope of the unit vectors el9 e29 .. .·, es (see § 1, sec. 6).

3. Continuous functions of n variables

Functions of a point (vector) of «-dimensional space. Functions of n variables may naturally be treated as functions of a point (or vector) of «-dimensional space. Let each point X (xl9 x2, . . . , x„) of a set M in «-dimensional space En be associated with a number f(X) =f(xl9X2> ·. ·, *n); we now say that a function f(X) =/(*i, *2>· χ^ of a point {vector) X ox a function of n variables — the coordinates of this point (vector) — is defined on the set M in En.

The concept of passage to the limit is introduced for functions in En as for functions of one variable.

Let/(Z) be a function defined on a set M czEn and let A be a limit point of M. We say that f(X) tends to a number a when X £ M tends to A, if

lim f(Xn) = a. (2.48) 71—>■ <»

for any sequence {Zn} of M, convergent to A. This is written as

a= lim f(X). (2.49)

(If Af is a domain and 4 an interior point, it has to be mentioned that Xn tends to A while remaining in M.)

There is another definition of such a limit, equivalent to the above: (2.49) means that, given any ε > 0, there is a corresponding η > 0 such that, when

0 < \\Χ-Α\\*ζη and ΧζΜ we have

| / ( Ζ ) - 0 | < ε . (2.50)

Continuous functions. We take a function/(A") =/(*!, x2» · · ·> *n) defined on a set Af c En, and a point X Çcl9 3c2, . . . , 3cJ of this set. We say that/(Z) is continuous at the point X ζ Μ if

lim f(Xm)=f(X). (2.51)

This definition is equivalent to the following: a function f(X) is continuous at a point X of a set M if, given any ε > 0, there exists 77 = η(έ) such that, for all points X £ M lying in the sphere

ρ(Ζ, J Ô = | | X - X | | < r ç , (2.52) we have

| / ( * ) - / ( Ζ ) | < ε . (2.52')

A function that is continuous at every point of a set M is said to be continuous in M.

EXAMPLE 6. The function

1 1

is defined and continuous at every point of the sphere | |Z | | 2 < 1. A linear function f(X) (see § 1; sec. 5) can be defined as a function

continuous in En and satisfying the additive condition:

/(Z1 + Z2)=/(Z1)+/(X2).

Uniform continuity. A function f{X) = f(xl9 x2, . . . , x j , defined on a set M c En9 is uniformly continuous there if, given any ε > 0, there exists η > 0 such that, for any pair of points Xl9 Χ2ζ Μ for which

ρ(*ί, X2) = NXk-Xill^t j , (2.53) we have

1 / ( ^ - / ( ^ ) 1 < ε . (2.530 The number η in (2.52) depends on e and on the choice of the point X, whereas in (2.53), defining the uniform continuity of a function, it depends only on ε and not on the choice of the points X± and X2.

Every function, uniformly continuous on M, is continuous on M. The converse is not generally true (for example, the function 1/Vl — 11 112* continuous inside the sphere \\X\\ «< 1, is uniformly continuous there).

As in the one-dimensional case (see Chapter I, § 3, sec. 11), we have

THEOREM 5. A function f(X) = f(xl9 x2, . . . , xj, continuous on a bounded closed set U, is uniformly continuous there. Maximum and minimum. A function f(X), defined on a set M, is said to be bounded from above (below) on M if there exists a constant C such that, for any point X Ç M,

KX)^C (f(X)^C). (2.54)

A function is bounded in M if it is bounded from above and below onM.

ι - Σ 4 i = l


lff(x) is bounded from above (below) on Af, a number C(c) exists which is the strict upper (lower) bound of the numbers f(X)9 i.e. of the values of the function f(X) at the points M :

C = sup f(X) (c = inf f(X)\. (2.55)

If there exists in M a point X0 at which

f(X0) = C = sup f(X), X€M

the upper bound C is called the maximum off(X) on M :

C=/(X0) = max/(jr). (2.56)

Z0 is called the maximum point for /(X) on M. Similarly, if there exists in M a point ΑΊ at which

ΛΧύ = c, twhere

c = inf /(Z),

the lower bound c is called the minimum of /(A") on M9 while A\ is the minimum point forf(X) on M:

c = / ( X 1 ) = min/(Z). (2.57)

We say in this case that f(X) attains its maximum (minimum) on M at the point X0 (Χλ).

THEOREM 6. A function continuous on a closed bounded set attains its maximum and minimum in the set.

Jump functions. We often encounter in problems of mathematical physics the following generalization of the concept of jump to a function of n variables. Let ß i be a domain in En. We say that it is "inside" its boundary Γ, while the domain Qe = En — (Qi+r) is "outside" the boundary Γ. We shall write A{ and Ae for the points of Qi and Q€ respectively.! Let a function / (which may or may not

t / and e are the first letters of "interior" and "exterior".

be defined on Γ) be given in ß i+ß e . Suppose that the limit off(Ae) [[(AJi exists, when Ae (AJ tends to the point A of the boundary Γ; we shall write here:

fe(A) = lim/04e). If f{(A) y£ fe(A)9 we say that the function has a jump

MA)-MA) on passing through the boundary Γ at the point A from Q{ to Qe.

EXAMPLE 7. Let us take a bounded closed convex domain Q{ in E2 with a boundary Γ (a smooth curve), and the domain Qe = E2— —Qi—Γ. Given an arbitrary point ΑζΕ29 we define a number <x{A) > 0, measuring the angle under which the curve Γ is seen from A. (It is formed by the rays joining A to all the points of Γ).

For points Ai 6 Qi9 we have α(^) = 2π; for points Ae ζ ôe, we have 0 < a ^ J ^: π, where <z(Ae) ■— π if Ae tends to a point A ζ -Γ. We have α( 4) = π for points A of the boundary JT.

For the function/(/i) = α(Λ) at the point A £ Γ9 we have/e(^4) = =/(Λ) = π <=/f(^[) = 2π; the j u m p / ^ ) - / ^ ) = - π .

Functions depending on a parameter. Every function φ(χΧ9 x29 ...9xm\ ίχ, ί2> · ■· · > 'η) can be regarded as a family

\fhh . · · **(*l> *2J · · · > xm) = φ(*ΐ> *2> · · · > xm> *l> h> · · · > 'π)}

of functions of /w variables xl5 x2, · · · > *m> e a c^ °f which is defined by a set of« values of another group of variables tl9t29..., tn — called "parameters". We shall confine ourselves to the case of a system {ft(x) = <p(x91)} of functions of one variable x9 depending on one parameter t; t takes any value from a set T of the numerical axis El9 while every function ft(x) is defined for x of the set X 6 Ex (the general case of n parameters and m variables is investigated similarly). Let t0 be a limit-point of T. If, given any t and x ζ X9 the function ft(x) tends tof(x) as t -» tQ (t ζ Τ)9 we say that the functions ft(x) are convergent on the set X to the function f(x). Given any x € Z and any number 77 > 0, we can find a number e9 depending on η and x9 ε = ε(η, x), such that

Ι/ι(*)-/(*)Ι<>7 (2.58) for0< 1 r— r01 <*(* € Γ).

For example, let T be the interval (0, oo), and X = [0, 1]; the family of functions {ft(x)} is convergent tof(x) on [0, 1] as t -* 0 if

[ (a) ft(x) = -^~-, /(*) = x;

I (b) Λ&Ο-Ο + ί*)1'1, /(*) = **; ί θ for 0 < χ < 1 ,

(c) /,(*) = **", /(*) = \ 1 f v [ 1 for x = 1.

The convergence of {ft{x)} to f(x) on X is said to be uniform as * -+ t0 (t ζ T) if, given any η > 0, there exists ε = ε(??), depending on η, such that (2.58) is satisfied for all x 6 X when | /—/01 < ε (ε no longer depends on the choice of x). In examples (a), (b), we have the case of uniform convergence, and of non-uniform in example (c).

The following are properties of the uniform convergence of a family of functions {ft(x)}: 1°. If the family {fix)} is uniformly convergent on X to fix) as

t -* h if € T)9 and {tn} is any sequence of T, convergent to t0, the sequence of functions {f^ix)} is uniformly convergent tof(x).

2°. If all the functions ft(x) are continuous on X, their uniform limit is a function f(x) continuous on X. (See examples (a), (b). In example (c), the limit function is discontinuous, which points to the non-uniform convergence of {ftix)} to f(x).)

3°. Let φ(χ, t) be a function of two variables continuous in the rectangle a ^ x ^ al9 b ^ t ^ bv On writing ft(x) = φ(χ, t) for t £ [b, bj), f(x) = φ(χ9 bx), we find that the functionsfix) are uniformly convergent tof(x) as t -* bx.

4. Periodic functions of n variables. Manifolds of constancy

Let Ek (1 ^ k ^ n) be a linear manifold in £n, and/CJf) a function defined in En. We say that Ek is a manifold of constancy for the function f(X) if, given any X £ En and any Y of Ek, we have

/ (X+7) = / (* ) . (2.59)

For example, in the case of the function fix+y) of two variables x and y, the straight line x = — y is a manifold of constancy. For,


if we add to the vector (x9 y) any vector (x0, — xQ) of this straight line, the function is unchanged.

Let us bring in the basis in En of elements Yl9 Yl9 ..., Yh ; Zl9 Z2, . . . . . . , Zn_Ä, where Yl9 Y2, . . . , Yk form a basis in the manifold of constancy Ek. Every element X ζ En can be written in the form (see (2.29))

X=Y+Z9 \ n-fe I

Y£Ek9 Z= Σ ζ Α , i - l )

where / ( Z ) = / ( 7 + Z ) = / ( Z ) .

The function/OQ — f{Z) reduces to a function of n—k variables — the coordinates Zl9 Z2, . . . , Zn_Ä of the vector Z.

A period of a function of n variables. A period of a function f(X), defined on En9 is vector ω (different from the zero 0) such that, for any X9

• f(X+co)=f(X). (2.60)

It follows from (2.60) that all the elements Y ^ Θ of a manifold of constancy of a function f are periods off A function f(X), having a period that does not belong to the manifold of constancy, is described as periodic.

We shall confine ourselves to periodic functions that have no manifolds on constancy. Notice that every periodic function has an infinite set of periods (in addition to ω, every vector of the form kco9 where k is any integer, is a period).

THEOREM 1. If ω and ωλ are periods of a function f(X), their sum co+col9 is also a period.

If ωΐ9 ω2, . . . , com are periods of f(X), any vector co of the form m

ω = Σ niœi> i « l

where ni are arbitrary integers, is also a period. The periods col9 ω2, . . . , com form a system of fundamental periods

off(X) if all the periods off(X) are expressible as linear combinations with integral coefficients of these periods and are not expressible as such linear combinations of a smaller number of periods.


EXAMPLE 8. Let us take f{X) = f(xl9 x2) = sin χλ cos x2 on the plane E2. It has a system of two fundamental periods, namely, the vectors ωλ (2π, 0) and ω2 (0, 2π). We could also take as fundamental periods ωχ and ω3 = (2π, 2π) (in which case ω2 = ω3—ωχ).

THEOREM 8. A continuous periodic function of n variables, having no manifold of constancy, has a fundamental system of periods consisting of not more than n periods.

EXAMPLE 9. Let col9 ω 2 , . . . , ωη be a system of linearly independent vectors in En. Let us form the set A of all vectors (points) of the form

n Σ ηίωί>

where the n{ are integers. (Such a set is called a net of integers in En.) We shall write q{X9 A) for the distance from the point X ζ Εη to the nearest point of A ; ρ(Χ9 A) is a function of Z. It has a system of fundamental periods consisting of the vectors col9 ω 2 , . . . , ωη.

EXAMPLE 10. A continuous function of a complex variable f(x+iy) = (*> y)+tQ(x> y) (not equal to a constant) cannot have more than two independent periods (Jacobi's theorem).

5. Passage to the limit for linear envelopes

Let al9 a2, . . . , ap be linearly independent elements in E. We shall write L(al9 a29..., ap) for their linear envelope. If bl9 b2, . . . , bp is another basis in the /^-dimensional manifold L(al9..., ap)9 then Lp{bl9 b2,...9 bp) = Lp (al9 a29 . . . , ap).

Let us consider a family of vectors /a, continuously dependent on a parameter a, such that, when o -* a, ax -^(a^aci) , the expression Ca2 ~~ ^oti/fe""^) t ends to a vector which we shall denote by dljdoc. We shall assume that / , and /ao are linearly independent when(% 5*0 . Having chosen another basis in Ζ^(/αι, /α2):

; *α2 "~ 'ai α2 —αχ

we have ί,(/β1, /α2) = L2 ^/β„ ^ _ ^ J ·

H-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE 65

Assuming that the vectors la and dljdoc are linearly independent, we find that, as oc± — a, a2 -*■ a, then the plane L2(lai, /tt2 ) = =Ι2( / α ι , ( / e t - / e i ) / («2- a a ) ) tends to the plane L^la9 dljdoc).

Given similar conditions (assuming that the systems of vectors K > 1^9 - · · >h * w h e r e n o two °f the numbers αχ, a 2 , . . . , ap are equal, and of the vectors /a, dljdx,..., dvla/docp, are linearly independent), we find that, as ai — a ( i = 1, 2, . . . , /?):

^ρίίχρ /<x3, . . . , /ap) — ^p ( L· fa?*' " ' > ^ a p- l '<* ) ·

Given a family of functions fa(x)9 dependent on a parameter a, with the same conditions and with a2· — οφ' = 1 ,2 , . . . , p), we find that

V/atW' /«,W. « « - /.,(*» t e n d S t 0 W«(*>» 3 /aW/^ · · ·. ^ P _ 1 Λ( χ ) /^ α Ρ _ 1 ) · For example, as af — a (i = 1 , 2 , . . .), we have

L.

Lp(e°i* ee**, . . . , e°r*) - Lp(eax,xeaX, . . . , x^e**), 1 1 1

p ' .x—a, ' x—cc»

1 1 1 ' ρ ^ χ - α ' ( x - a ) 2 * ' " , (jc-a)Py

Ip(x«i, xa2, . . . , xcp) - Ιρ(λα, * a In *, . . . , xa In71"1 *), I 2 ( l * - a i l » I* -«» I) - L4(\x-a\, sign |ΛΓ—a | ).

Let Aa be an //-dimensional, matrix, Xta (/ - 1, 2, . . . , /z) be its eigenvalues (simple fora * 0),x{a the corresponding eigenvectors: A a*<a β Λ,αχία, II*,. || = 1. Suppose that £ of the numbers Xia (i = 1, 2, . . . . £ s /i) tend as a - 0 to the eigenvalue λ0 of the matrix Axt, while the eigenvectors x, (/ = 1, 2, . . . , A:) tend to the eigenvector x, of the matrix Au corresponding to the eigenvalue λ0. The linear envelopes Lk (χχ,, j * 2 o t . . . · , xk ) now tend to the linear envelope Lk (xx, x2,... ,*i) where x{ with / ^ 1 are augmented eigenvectors of the matrix Αυ : Α^ — λ^ -- Xi-χ (/ - 2, 3, . . . , A).

6. Operators from £ n into Em

Let Ô be a set in En (which can coincide with En itself); we associate with every element X(xl9 x2, ..., xn) ofQ an element Y(yl9 y2,..., ^m) = = /(Ar) of Em. We say in this case that an operator f is given from En

into £m, Ö being called the domain of definition of the operator. We

can also say that we have a mapping f of the set Q of En into Em or that the operator /performs a mapping of the set Q from the space En into the space Em. If

Y = to» y* · · · > Jm), ^ = (*1> *2> · · · > *n)>

r=/(n (2.61) then

tt = fi(X) = /i(*i> *2> · · · > *n)> (2.62) where / is a function of n variables defined in g, formula (2.62) being the coordinate form of writing the operator/, and (2.61) the vector form. The functions/. (/= 1,2,..., m) are called the components of the operator/, and we write

/ == \fl> ./2> · · · 5 fm)·

Every set of m functions /(χχ, x2, . . . ,*„) (/ = 1, 2, . . . , m) of « variables defines an operator / from En into Em, where/ =

An ordinary function /(χΐ5 ΛΤ2, . . . , *„) is an operator from En into £Ί (into the numerical axis).

An operator f(X) is said to be continuous in Q if

Xn -——► X

Zn and A' belong to Q) implies that

The necessary and sufficient condition for the operator f(X) = (/Ί(Α'), f2(X),.. , ,/n(I)) to be continuous is that all the functions f^X) = / ( ^ , x2, . . . , Χγ) be continuous in g.

For example, the functions

zx = Ji-x*-xl-xl = y/i-uxw*, I Z2 = Xi~l· χ2-\~Χζ J

yield an operator from Ez(xl9 x2, xz) into the plane E2(Zl9 Z2), defined in the three-dimensional sphere | |Z| | 2 < 1 and continuous there. Functions of a complex variable provide us with examples of operators from E2 into E2 (mapping of a plane into a plane).


The term vector function is occasionally used instead of operator from En into Em. An ordinary function of n variables (n >- 1, m = 1) is called a scalar function of a vector variable; an operator from E1

into En where n > 1, is called a vector function of a scalar variable or simply a vector function, and an operator from En into 2im, where « > 1, m > 1, is called a vector function of a vector variable.

Let X = Z(0 be a given vector function, where / runs over the numerical axis (or a segment of it), and let X be a point of Em. The set of points of Em of the form X(t) is called a cwrve or line in i?m.

Linear operators. A //«ear operator {mapping) from -E^fo, χ2>···> *π) into Em(yl9 y2, . . . , y™) is an operator 7(Z)satisfying the conditions:

( 1 ) 7 ^ + ^ ) = 7 ( ^ + 7 ^ ) ; (2) the operator 7(X) is continuous. It follows from conditions (1) and (2) that a linear operator is

homogeneous: Υ{λΧ) = λΥ(Χ).

THEOREM 9. 4 //«ear operator Y(X), where 7 = (yl9 y29 . . . , y„)9

X == (χχ, ΛΓ2, . . . , JC^, Aay the form:

y% = öii^i+flri2x2+ . . . +ainxn (/ = 1, . . . , «), (2.63)

i.e. every coordinate y^X) of the operator Y{X) is a linear function ofX= (xl9 x29 . . . , xj.

Every linear operator Y{X) homEn into Em is defined by a rectangular matrix

^ = II an || (/ = 1, 2, . . . , m\ 7 = 1 , 2 , . . . , «), 7 = iljr.

Conversely, every such matrix defines a linear operator from En into Em (in accordance with (2.63)). When m = n, a linear operator from En into En is defined by a square matrix A = lki;||J; j1

Formula (2.63) is the coordinate form of a linear operator (transformation) from En into Em.

7. Iterative sequences

Let

\Xn} = {^OJ Xl> ^ 2 J · · ·}> ^ n == foil» Λ'η2> · · · » xnk) (2.65)

(2.64)


be a sequence of vectors in Ek> the corresponding k numerical sequences being

{xni} = {xoi, xli9 x2b . . . } (/ = 1, 2, . . . , k). (2.66) The vector sequence (2.65) (or system (2.66) of numerical sequences) is described as iterative if

* n = / Ä - i ) (n= 1,2, 3, ...),] [(2.67) where/is an operator from Ek into Ek. If/ = (/i ,/2, . . .,/&)> equation (2.67) may be written in the coordinate form as

xni = fi\xn-lt 1 xn-l9 2» · · · > * n - l , k) (/ = 1, 2, . . . , k). (2.68)

On putting X0 = (x01, x02, . . . , xoft), we can find successively from (2.67) or (2.68):

Xi = (Χιχ, ^i2> · · · » xlh)> -*2 = (*21> *22> · · · > *2fc)j · · ·

Relationship (2.68) is sometimes written as xni == /iV^nl» *n2 · · · » ·*η, i—1> ·*-η—l, i> ·*π—1, i+l> · · · · ·*π—l. ft)·

(2.69) (When finding the zth component of the nth vector Xn, use is made of the components xnv xnv . . . , xUt imml already found.)

EXAMPLE 11. K.F. Gauss considered a sequence of pairs of posi:ive numbers {αη, βη} (n = 0, 1, 2, . . . ) or of plane vectors, where a0, ß0 are given (α0 ^ β0), and

an-l + Pn-l p __ j~T an = 2 ' "n Va«/V

As 72 — oo, the sequences {an} and {/?n} tend to a common limit a = lim αη = lim ßn — the arithmetico-geometric mean of the www-

ier^a0,/50:

Û = ß(a0, ß0) = — · , G = — 2G a0

EXAMPLE 12. Let us take the sequence of partial sums^n = 1 + -f (*/l !) + . . . + (xn/n !) of the power series for ex. On writing ocn = n,


ßn=(xn/nl) we have: (α0, β0, s0) = (0, 1, 1). We get an iterative sequence (αΛ, βη, s^) of the form (2.69):

The formula provides a unified scheme for obtaining successively (αη, βη, sn)(Le. sn also) for n= 1,2, 3, The unified passage from one term of an iterative sequence to the next term is very convenient for computation on programme-controlled machines.

Iterative processes. We can generalize the concepts of § 2, sec. 6, to operators from En into En.

Let/be a continuous operator from En into En:

Y = f(X), Y = ΟΊ, y29 . . . , yn\ X = (xl9 x2, . . . , xn\

or, in the coordinate form:

y% = Mxu *2> · · · > *n) (i = 1, 2, — , TI).

Let us investigate the equation Χ=ΛΧ) (2.70)

or, in the coordinate form, the system of equations

Xi = /fai, x*> . . .» *n)· (2.71Ì

We form the iterative sequence of elements

^ Ü J ^ l > ^ 2 ' . . · 5 ^ m > · · «j ^ m = (·*τηΐ> ·*τπ2> · · · > ·*7ηπ)>

where ^0C*oi> *02 · · ·» *on) *s a n arbitrary element of En9

* m + i = / W (m = 0, 1, 2, . . .), (2.72)

and in the coordinate form

xm+l> i = Ji\xml9 xm2> · · · > - mri) 0 = U 2, . . . , Π). (2 .73)

If the sequence Xm is convergent to X*, X* is a solution of equation (2.70).

The solution X* is called a. fixed point of the transformation Y=f(X).

8. The principle of contraction mappings

We introduce a norm into the space En (see § 1, sec. 2). An operator(transformation) ffromEn into En is said to be a contraction mappingif there is a constant q (0 <: q <: 1) such that, for any X, Xl EEn'

f1f(XJ-f(X) II ~ qIlXI-XII. (2.74)

The following theorem provides a condition for the existence of afixed point of the transformation Y = f(X), Le. of a solution of equation (2.70), and at the same time, a condition for convergence of theiterative process (2.72).

THEOREM 10. (the principle of contraction mappings). If Y = f(X)is a contraction mapping (i.e. condition (2.74) is satisfied with q <: 1),then:

(1) there exists a solution X = x* of equation (2.70), i.e. a fixedpoint x* of the transformation Y = f(X);

(2) this solution is unique;(3) whatever the initial approximation X o, the iterative process (2.72)

is convergent to the solution X*,

X* = lim X m •m~oo

(2.75)

(2.76)

(4) the sequence X m is convergent to X* asfast as the convergence ofageometric progression with ratio q, in fact,

qnIIX*-Xnll :§ -1-IIXI-Xoll.-q

We shall mention some sufficient conditions for an operator to bea contraction mapping under different norms. It is assumed that thefunctions h(X) =h(x1, x2, ••• , xJ - the coordinates of the. operator(X) - have continuous partial derivatives with respect to all the arguments.

For condition (2.74) to be satisfied, (also the consequences of thetheorem), it is sufficient that:

(a) in the Euclidean metric,

(2.77)

(b) in the metric m(n) (see p. 97),

$/iC*l> x2> · · · > xn) Σ dXj ft < 1 (/ = 1, 2, . . . , ri).

(2.77') Application to the solution of systems of linear algebraic equations. Let us take the system of linear algebraic equations

y% = Σ aHxi + bi

or, in the vector form, 7 = ^LT+tf,

Λ = || ay ||, B = (blf b29 . . . , 6n).

(2.78)

(2.79)

(2.80)

Given any initial vector X0(x0i> *o2> · · ·» *on)> w e f ° r m a system of vectors X0, Xl9 X2, . . . , A"m, . . .

where <&m — V*ml> xm2> · · · > xmnJ9 \2..oi)

X m + 1 = AXm + B (m = 0, 1 , 2 , . . . ) (2.82)

or, in the coordinate form,

n *m + l,i - Σ aiJX7nj + bi (i = 1, 2, . . ., Aï). (2.83)

j = l

It follows from (2.77) and (2.77') that a sufficient condition for convergence of the vector sequence Xm as m — «> to the vector solution Z*(x*, **, . . . , Λ:* ) of system (2.79) or system (2.78) is that

or n

(b) £ | f t j | = ft < 1 / = 1, 2, . . . , Λ) J - l

(fl = max ^ ; i = 1, 2, . . . , n\ j

the accuracy || Jf*—Jfn|| of the nth approximation Xn being defined by (2.76); in case (a) the norm is Euclidean, in case (b) it is m(n).


§3. Convex bodies in τζ-dimensional space

The theory of convex bodies in «-dimensional spaces was developed by G. Minkovskii ; it has found applications in numerous problems of analysis, geometry and number theory, and more recently in new branches of applied mathematics: in the theory of games and in linear programming. This theory provided the basis for developing the theory of «-dimensional normed spaces, the infinite-dimensional generalization of which plays an important role in analysis.

1. Fundamental definitions

A convex set in space En is one which contains, in addition to two points A and B of the set, every segment joining these points.

Particular cases: a convex domain Q is a domain which is a convex set, a convex body Q is the convex domain Q together with its boundary Γ. Points of the domain Q are called interior, and points of Γ boundary points of Q.

On a straight line, an interval is a convex domain, and a segment a convex body; on a plane, the interior of a circle or triangle is a convex domain and the circle and triangle (including their boundaries) are convex bodies; in three-dimensional space, the interior of a cylinder, sphere, cube are convex domains, and the cylinder, sphere and cube (including their boundaries) are convex bodies, the sphere and cube being bounded convex solids, and the cylinder an infinite convex solid.

A closed convex set, lying in a Ä>dimensional plane and not lying in any (k—l)-dimensional plane, is called a k-dimensional convex body (0 ^ k ^ w). A point will be described as a zero-dimensional convex body. (The empty set is also regarded as convex.)

A convex body (set) may be bounded or unbounded (cf. the above examples). We can quote as an example of an unbounded convex body a half-space in En, i.e. the set of points X satisfying the inequality (fX ^ C), where / is a linear functional in En, and C a constant; the whole of the space En and any hyperplane in it are examples of unbounded convex sets.

The intersection of convex sets in En is a convex set, the intersection of convex bodies in En is a convex fc-dimensional body (k ^ w).


A convex polyhedron is a closed convex body—the intersection of a finite number of half-spaces.

The convex envelope of a set M in En is the set Q of points ΧζΕη expressible in the form

X = Σ Mi, (2.84) i = l

where / is any integer, / ^ n,X{ are arbitrary points of M, Ai are arbitrary non-negative numbers satisfying the condition

Σ h = ΐ· 1 = 1

EXAMPLE 13. The convex envelope of a pair of points A and B is the segment joining them; the convex envelope of three points not lying on a straight line is a triangle; the convex envelope of four points, not on the same plane, is a tetrahedron with vertices at these points.

The convex envelope of a set M is the intersection of all convex sets containing M, or the least convex set that contains M. Every bounded convex polyhedron is the convex envelope of a finite number of points — the vertices of the polyhedron.

Limit points. A point A is a limit point of a convex body Q if A is not an interior point of any segment belonging to Q.

Every limit point of Q is a boundary point of Q ; but not every boundary point is a limit point. For example, only the vertices of a polyhedron (polygon) are its limit points. In the case of a circle and sphere, all the boundary points are limit points. In future, a convex body will be taken to mean a bounded convex body.

A convex body Q is the convex envelope of its limit points.

2. Convex functions

A function/(Z) = f(xl9 x2, . . . , x^)9 defined in En (or on a convex set Q ofE^), is described as convex if, given any X and Y οίΕη (of 0 , we have

/ (^H) - T[/w +/(7)1 ( 2 · 8 5 )

(and concave, if the reverse inequality is satisfied).

A convex function also satisfies the more general inequality: /[(l - t)X+ tY] ^ (1 - t)f(X) + tf(Y) (2.85'

for any t € [0, 1]. (See Chapter I, § 3, sec. 17, for convex functions of one variable.) If f(X) is a convex function, c0 is a lower bound in En (in Q), the

set of points ofEn (ofQ), satisfying the inequality AX) si c9

with any c ^ c0, is a convex body (set).

EXAMPLE 14. On the plane Et (xu x2), the functions

//x„*1>-<|*1|» + |*, | ' ) , /* with/? £ 1 are convex functions. Asp- ~,f9(xi9 x») tends to the function foo(xu x2) « «= max(| xx |, | x21) (see Fig. 2). The sketch illustrates the curves /ρ(χί9 x») « 1 for different p. They all pass through the points Ax (1,0), A2 (0, 1), A3 (—1, 0), A< (0, - 1 ) .

FIG. 2.

The curve /"«»(xi» *2> « 1 is the boundary of the square with vertices Al9 Az,

The curves^, « 1, where 7 < /> < «», lie between these two curves. The curve/« (JCX, *2) = 1 is the circle of radius 1.

The figures fp «s 1, bounded by these curves, are convex. When p-<l, such figures are no longer convex. For instance, when p = 2/3, we get a figure bounded by an astroid.

3. Convex bodies and the norm of a vector

Let Q be a convex body in En, having 0 as an interior point. Every point X of En can be written (uniquely, when X ^ 0) as X = AX0, where A > 0, Z0 is a point of the boundary of 0. If XUQS outside Q, then λ > 1 ; if Z lies inside β, λ <: 1 ; if Z lies on the boundary of Q, λ = 1. The point Θ can be written as Ο.Ζ0 (i.e. A = 0 for the point 0).

We now define a function <pQ(X) in En as follows: at the point X = AXQ,

?Q(X) = A;

(^(AO is greater than, equal to, or less than 1, depending on whether X is outside, inside, or on the boundary of Q respectively,

<PQ(0) = 0.

<PQ(X) is a convex function. The body Q is the set of points of En for which <PQ(X) ^ 1, while the boundary of Q is defined by the equation P Q ( * ) = 1 .

9>Q(X) has the following properties: (1) (pQ(X) ^ 0, where cpQ = 0 only when X = 0; (2) When λ ^ 0, (pqihX) = A^Q (Z) (implying a positive homogeneous

function);

(3) 9Q(X+Y)k<pQ{X)+<pQ{Y).

If Q is a central symmetric convex body with centre at 0 (i.e. XÇ.Q implies — Xζ 0 , we have the additional property:

cpcßX) = \X\cpQ{X) for A < 0 . Property (2) becomes the stronger property:

(2') <PQ&X) = \X\cpQ{X) for any real λ. If φ is a given function satisfying conditions (l)-(3), the set Q defined by <pQ(Z) ^ 1 is a convex body, while it is a centrally symmetric convex body with centre at 0 when condition (2') is satisfied.

The Euclidean norm \\X\\ of the vector X € En satisfies conditions (1), (2'), (3). We can generalize the concept of norm of a vector in /i-dimensional space: the norm can be taken as any function <p(X) that satisfies conditions (1), (2), (3). We shall write £n> ψ for such a space.

If || X11 = φ(Χ), conditions (1) - (3) can be rewritten as: (1)||1Ί| ^Ο, and IIXH = 0 only when X = 0; (2) When λ > 0, we have ||λΑΊ| = λΙΙΑΠΙ; (3) \\X+Y\\ ^ ||Jr|| + || Γ|| (the triangle inequality). When φ is even, condition (2) is replaced by the stronger one: (20 IIAATH = |Λ| ΙΙΛΓΙΙ (for any Λ). Spaces En9 in which a norm is introduced, satisfying the above

conditions, are said to be normed. For instance, Euclidean spaces are normed (see § 1, sec. 7).

4. Support hyperplanes n

Let us take the linear form /Χ=Σ hxi *n ^η* ^ β hyperplane (fX = C) and the half-space (fX ^ C). We call (fX = C) a hyperplane of support for the convex body Q if Q lies wholly in the half-space (fX^ C) and the hyperplane has points in common with the boundary ofß.

EXAMPLE 15. In the two-dimensional case, the hyperplanes of support reduce to lines of support. In the case of a circle, the lines of support are the same as the tangents. In the case of a triangle, the lines of support are the three lines along which its sides lie, together with all the straight lines passing through its vertices and lying in its exterior angles. In three-dimensional space, hyperplanes of support become planes of support; for a sphere, the planes of support are its tangent planes, and for a cube, its boundary planes and the other planes that pass through the intersection of two sides, or merely through a vertex and do not cut the interior of the cube.

Let the equation of the hyperplane of support be

fX= C. (2.86)

W-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE 77

THEOREM 11. The number C on the right-hand side Ö/(2.86) is defined by the equation

C = max fX. (2.87) XiQ

The equation of the hyperplane of support therefore has the form

fX = max/X. (2.88) XÇ.Q

THEOREM 12. The linear function fX attains its maximum C on Q at a limit point of Q, i.e. the intersection of (fX = C) and Q contains a limit point ofQ.

COROLLARY. Every hyperplane of support of a body Q passes through one of its limit points (in particular, each hyperplane of support of a polygon passes through one of its vertices).

5. Support functions and conjugate spaces

Let the norm \\X\\ = φη{Χ) be introduced into space En and let Q be the closed unit sphere: Q= ([\X\\ ^ 1); Q is a convex body in En.

If

ix = £ kXi

is a linear form in En = Εηφ, the corresponding hyperplane of support of Q is given by the equation

IX = C(= max IX). (2.89)

Every linear form IX is a vector / with components ll9 /2,..., ln. The set of such forms makes up an Λ-dimensional linear system Ln. A norm can be introduced into Ln, i.e. we can put

||/||* =:φ(/)= max IX. (2.90) ΙΙ*ΙΙΞΐ

This norm satisfies, along with the norm || X\\, conditions (1), (2), (3) (see sec. 3) or (1), (20, (3).

We have : | / X | ^ i | / | | * i m i . (2.91)


This inequality is a generalization of inequality (2.7) for the Euclidean metric. Equation (2.89) can be written as

IX = 11/II*. (2.92) Conjugate Spaces. The space Ln = Env thus introduced, of linear

forms in Εηφ, is described as conjugate to the initial space Εηφ. The following notation is used:

ΕηΨ = Kr (2.93) The conjugate to a Euclidean space is also Euclidean. THEOREM 13 (Minkovskii). If the spaceEny) is regarded as the initial

space, its conjugate becomes Εηφ

ΚΨ = Εηφ (2.94) (the conjugate to the conjugate space is the initial space).

The reciprocal property whereby spaces Εηφ and Env are conjugates of each other is known as reflexivity. Functions y>(X) and ψ(1) are said to be reciprocal; they are connected by the relationships

max IX = ψ(1), max IX = φ(Χ). (2.95) φ(Χ)^ι ?(ϊ)=§ι

Similarly, the convex bodies Q and Q*, defined by the inequalities φΧ ^ 1, ψΐ 1, are also said to be reciprocal. (Minkovskii's theorem is no longer in general true for infinite spaces.)

EXAMPLE 16. Let lntP(p ^ 1) be an «-dimensional space with the norm

. - Vl/P / n \ l

When p > 1, we have /*>p = lnq, where (l/p)+(l/q) = 1. When p = 2, (the case of a Euclidean norm) we have q = 2 (the

«-dimensional Euclidean space / n 2 is self-conjugate: /n,2 = ^12)· EXAMPLE 17. Let /n>1 be an «-dimensional space with the norm

11*11= Zl*il.

and mn an «-dimensional space with the norm

| |Γ | | = π ι κ ( | Λ | , | Λ | , . . . , \yn\)\

n-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE 79

we have:

Every linear function YX in Enrp can be regarded as the scalarproduct of a vector X of Enrp and Y of En".

6. Fundamental theorems on support hyperplanes

THEOREM 14. Let Q be a convex body in n-dimensional space En.A hyperplane of support of Q can be drawn through every point ofthe boundary o.f Q.

The following is a generalization of this theorem:THEOREM 15. Let L k (k <:: n) be a k-dimensional plane drawn through

an interior point A of the n-dimensional convex body ofEn. It cuts fromQ a k-dimensional convex body Qk. Let L k- 1 be the (k-I)-dimensionalplane in Lh, which is a support plane for Qk: An (n-I)-dimensionalhyperplane Ln- 1 can be drawn in En which is a support hyperplane forQ and contains L k - 1•

Let IkX be a linear form in the k-dimensional manifold Ek C En"(1 ~ k <: n). The linear form InX in the whole space En is called anextension of the form Ik if, for X EEk ,

InX = IkX.

The form Ik in L k has the norm

IIlk lILA: = max IkX."XII~l, ~YE EA:

The form In in En has the norm

((In II = II In IlL" = max InX.II.6'Y"~l

It follows from this that II In II ~ II Ik II Lk , i.e. when a linear form isextended, its norm can only increase.

THEOREM 16. Any linear form IkX, defined in a k-dimensional manifoldL k ofEn' 1 ~ k <: n, can be extended to the whole ofspace En withoutchanging the norm.

This theorem can be extended to infinite spaces (the Hahn-BanachTheorem).


7. The connection between reciprocal convex bodies

Let Q and Q* be unit spheres in En and 2Γ* . A duality (not one-to-one) can be established between the points of their boundaries.

Equation (2.89) of a plane of support of Q in En can be written as

In this case (see (2.92)), || Y01|* = 1. Formula (2.92) becomes

Y0X= l|Foll*= 1- (2.96) We associate with every point A^ of £n, for which \\X0\\ = 1 (i.e.

points of the boundary of Q), all the planes of support of Q passing through it, these planes having equations of the form (2.96). We have :

Y0X0=h limoli = || 701|* = 1 . (2.97) Formula (2.97) yields an expression for the Y0 Ç *n, which correspond to a given X0 çEn; it has symmetry with respect to X0 and 70, which points to the reciprocity of this mapping.

Theorem 14 shows that every XQ of the boundary of Q (||Af0||= 1) has at least one associated Y0 of the boundary of Q* Q\Y\\* = 1).

EXAMPLE 18. On the plane m2, the sphere (\\X\\ = m*x(\x1\, | * a l ) s l )

is the square Q = B1B2BZBA; on the plane /12, the sphere (||F|| = = bil + ΙΛΙ = 1) *s * e square Q* = A1A2AZA4. Corresponding to the point X0 (1, 1) = B1 of the boundary of the square Q, we have the Yo(yi, yù f°r which

i = 7o*o = Λ*Ι+Λ*2 = Λ + Λ ; limoli = ΙΛΙ + ΙΛΙ = U i . e . j ^ l ^ l ^ O , ^2=|>;2 |^0.

These points thus fill the side ΑτΑ2 of the square Q*. The side BXB2 of the square Q lies on the support line Χτ = 1, i.e.

lJIfi+OJfi = 1. The corresponding point is 70(1, 0) — the vertex A2 of the square β*. The remaining sides of β* correspond to the remaining vertices of Q, the vertices of Q* to the sides of Q and vice versa.

In three-dimensional space the convex bodies Q and Q* can only be convex polyhedra simultaneously, the vertices of Q being associated

with the faces of Q*9 the ribs of Q with the ribs of Ö*, the faces of Q with the vertices of Q* and vice versa. Such polyhedra are said to be reciprocal. For instance, if Q is a cube, g* is an octahedron (and vice versa); if Q is a dodecahedron, ß* is an icosahedron (and vice versa).

In the «-dimensional case Q and Q* can only be polyhedra simultaneously, the fc-dimensional faces of Q (O^k ^ n-1) being associated with the (n-k- l>dimensional faces of Ö*.

8. The cone. The tangent cone

A cone K in En with vertex at X0 € Enis a set of points of En, different from the whole of En, and such that, if X belongs to K, the entire ray (tX), 0 ^ t -< <~, belongs to ΛΓ. We shall assume without any proviso that the cone K is a convex body. A cone, together with two

FIG. 3. Fia. 4.

of its rays forming an acute angle, contains the entire angle. Cones on a plane are angles not exceeding π. In three-dimensional space examples of cones are provided by dihedral angles not exceeding π, ordinary circular cones or regular pyramids, continued indefinitely, etc.

Let XQ be a point of the boundary of a convex body Q in E; let us write (ΑΌ) for the least cone with vertex at XQ that contains Q; this cone consists of all the rays joining X0 with points of Q and all the limiting rays. We shall call the boundary of this cone the tangent


cone hypersurface to Q at the point X0; two cases are possible: (1) The cone K(X0) coincides with the entire half-space, its

boundary is the unique hyperplane of support at X0 to Q, which "touches" Q at the point X0.

(2) The cone K(X0) is a regular part of the half-space; an infinite set of hyperplanes of support of Q passes through the point X0. Such a point will be called a point of sharpening.

EXAMPLE 19. On a plane, the'cone K(XQ) becomes an angle bounding two tangents to Q at the point X0 = 0 (the rays OA> OB in Fig. 3); if this angle is equal to π, both rays form a unique support line (the line ÄB' in Fig. 4), tangential to the boundary of Q at the point O. If the angle is less than π, the point O is a point of sharpening, and an infinite set of support lines pass through O.

EXAMPLE 20. Let Q be a convex polyhedron in Ez. If the point X0 lies inside a face, K(X0) is the half-space bounded by the plane of the face, which is the unique support plane of Q at X0. If X0 lies inside a rib AB, K(X0) is a dihedral angle less than π, formed by the planes of the faces that intersect in AB; there is an infinite set of support planes to Q at X0, all of which pass through the rib AB. ΙΐΧ0 is a vertex of Ô, K(X0) is a polyhedral angle with vertex at X0, bounded by the planes of the faces that meet in X0 ; points of the ribs, and all the more the vertices of Q, are points of sharpening.

9. Helly's theorem

An interesting theorem on the intersection of convex bodies must be mentioned.

THEOREM 17. Let {Q} be an arbitrary set of convex bodies given in En, at least one of which is bounded. If any n+1 of them have a common point, there is a point common to all the bodies of{Q}.

For instance, if an arbitrary set of segments, such that any pah-has a common point, is given on a straight line, there is a point common to all the segments.

fl-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE 83

10. Linear operations on sets

DEFINITION. Let A and B be any sets of En. The vector sum A+B of these sets is defined as the set (X) of points ofEn expressible in the form X = Χλ+Χ2, where Xx Ç A, X2 6 B.

EXAMPLE 21. If A is an arbitrary set, a a point (vector), then A + a is the set obtained by parallel displacement of the set A along the vector a.

EXAMPLE 22. If A is the x-axis, B the >>-axis, A + B is the entire plane. EXAMPLE 23. If A in En is an «-dimensional closed sphere of radius

ρ with centre at 0, A -f- B is a "layer" of thickness ρ round B, i.e. the set of points of En at a distance less than or equal to ρ from B.

THEOREM 18. The vector sum of convex sets (bodies) is a convex set (body).

DEFINITION. If λ is a number, Aaset in En, XA is the set of all points of the form XX, where X ζ A.

A similitude transformation of the set A with transformation coefficient X > 0 is a transformation of the set A into the set λΑ. A symmetric mapping of A with respect to the centre # is a transformation of A into — A.

When X = 0 the set XA consists of the single point 0. If A is a convex set (body), XA is a convex set (body).

n If Ai (i = 1 , 2 , . . . , /z) are convex sets (bodies), ]T X{AX is a convex

set (body) for any system of non-negative numbers Xl9 X2, . . . , Xn. Let Ti (/ = Ci) and Γ2 (/ = C2) be two parallel hyperplanes of support of the convex bodies Qx and Q2, while (/x ^ Ci), (/χ ^ C2) are the corresponding half-spaces containing Qx and Q2 respectively; tl9t2 are arbitrary positive numbers. Now, t1T1+t2T2 is a hyperplane of support parallel to them (/ = t1C1+t2C2) of the body Q = f ißi+/2ß2 , tying in the half-space (/ ^ / i Q + ^ Q ) .

THEOREM 19 (Brunno-Mmkovskii). Let A0 and Αλ be convex bodies in En, and At = tA0 + (l — t)Alf 0 ^ t ^l, a ''linear system" of convex bodies; the n-dimensional volumes Jn of the bodies AQ, Ax and all the At are connected by the Brunno-Minkovskii inequality

VÜ40 ^ (1 - 0 VÜÄÖ) +1VUAJ. (2.98)


The case of equality occurs if and only ifA0 and ALare homothetic convex bodies, i.e. one is obtained from the other by a similitude transformation and parallel displacement (A± = rA0+b, r ^ 0, b is a vector).

Inequality (2.98) was proved by Brunno (in 1887); the case of equality was proved by G. Minkovskii (in 1891).

Inequality (2.98) implies that V nC *) is a concave function for any t in [0, 1].

The theorem still holds for non-convex bodies, and in general for any sets, if Jn(A) is understood as the exterior /t-dimensional measure of the set A or the measure when it is measurable (see ref. 7).

The Brunno-Minkovskii theorem is employed in proving the iso-perimetric and numerous other geometric properties of convex bodies.

CHAPTER ΙΠ

SERIES

Introduction

Infinite sequences and their limits have been discussed in Chapter I. Infinite series, or simply series, are closely connected with sequences, e.g.

oo

*! + <%+ . . . +an+ . . . = £ an (3.1) 7l« l

is "the sum of an infinite number of terms". Series are commonly employed in the most widely spread branches

of mathematical analysis and in the solution of applied problems, since they represent one of the most universal and effective means both of investigation and computation.

Examples of series (infinite geometrical progressions) were known to the mathematicians of antiquity. In the process of developing the analysis of infinitesimals, series made their appearance, with the aid of which the values of various functions could be computed: Mercator's series for the logarithm, Newton's series for sin x, cos x, arc sin *, arc cos xy (1 -fjc)a, etc. Series were widely used by Euler for representing functions in various works; in addition to power series, trigonometric series were employed by Euler. He also made use of divergent series, and seems to have been the first to be concerned with improving the convergence of series. The vast amount of factual material on series that had accumulated by the beginning of the nineteenth century posed to scientists the problem of finding a strict basis for the theory of series. The investigations of Abel, Gauss, Cauchy and others in this direction played an important part in laying the foundations of mathematical analysis as a whole.

The present chapter is devoted to the basic theory and practice of evaluating series. Numerical series are considered in § 1, functional in § 2, and the various methods of computing them in § 3. (See Chapter IV for vector series.)

85


1. Basic concepts

DEFINITION: A series is defined as in (3.1), composed of the terms of an infinite sequence {an} (n = 1, 2, 3, . . . ) . For every series there is a corresponding sequence {sn} (n = 1, 2, 3, . . . ) of partial sums, where

n sx = al9 s2 = αχ + α2, . . . , sn = X ah9 . . . . (3.2)

EXAMPLE 1. Given the series

1 + T + y + i 7 + · · · = „ ! α " ' · · · we have

_ i - 4 - 3 Λ 1

sx — i , $2 — "5~ > · · · > ^n — T I 3^

The following two cases are possible when considering the sequence of partial sums.

Case 1. The sequence {sn} has a definite finite limit

S = lim sn. Ϊ Ι ->·οο

This limit is called the sum of series (3.1) and is written as

S= f,an. (3.3)

In Example 1,

-j,Gr^· Case 2. The sequence {sn} has no finite limit. EXAMPLE 2.

oo

1 + 2 + 3+ . . . +H+ . . . = £ n;

_ 1 _ Q _ ( 1 + Π ) Λ ^1 — A, Λ"2 — J, . . . , Sn — - ,

As n -+ oo, the partial sum sn — oo.

SERIES 87

EXAMPLE 3.

1-1 + 1 - 1 + . . . = Σ ί - 1 ) 7 1 - 1

71=1

ί θ when n is even

1 when n is odd.

EXAMPLE 4.

1 - 2 + 4 - 8 + . . . = Σ ί - 2 ) * " 1 ; n* l

_ l - ( - 2 ) n

j* ! = 1, 5"2 = — 1, 5*3 = J , ^4 = — J, . . . , Sn — - .

In Case 1 the series is said to be convergent, whereas it is divergent in Case 2 (Examples 2, 3, 4).

Example 2 illustrates an important particular case of a divergent series, when ^n — oo (or ί η - ^ -οο ) ; Example 3 illustrates the case when sn oscillates while remaining bounded, and Example 4 the case when sn oscillates but is unbounded.

We shall only discuss convergent series in this section. In the case of a convergent series (3.1), the sum S can be written as

S=sn + Rn, (3.4) where

oo

^ n = 0n+l + 0n+2+ · · · = Σ ah ( 3 · 5) ft=*n+l

is called the remainder or remainder term of the series. Evaluation (or summation) of a series implies finding its sum S. We must therefore verify the convergence of a series before finding

its sum. Strict summation (some methods of which will be given below)

is only possible for a narrow class of series. For the majority, the sum is found approximately. In approximate summations, the sum S is replaced by the partial sum sn.

It follows from the definition of the sum of a convergent series that

lim Rn = lim (S-sn) = 0. (3.6)

This indicates that, given a sufficiently large n, the absolute value of the remainder term i ^ can be made as small as desired.

To ensure the required accuracy when replacing S by sn, it becomes necessary to estimate the difference

S—sn = Rn, i.e. the remainder term.

In addition, practical convenience in computations requires that the number n of terms of the partial sum sn, required for achieving the necessary accuracy, be not too large. Reasonably accurate summation of certain series would require taking thousands, or even tens of thousands, of terms. Such series are said to be slowly convergent.

EXAMPLE 5. To compute the sum of the series 1 1 1 °° 1

, 44-Τ + · · · · -£<-ΉΓ to an accuracy of 0-001, the summation of a thousand terms is needed.

The question thus arises of improving the convergence of a series, i.e. transforming it in such a way that summation of the transformed series with the same accuracy requires a smaller number of terms in the partial sum sn. The fundamental questions for the summation of series are therefore: convergence, estimation of the remainder term, improvement of the convergence.

2. Some convergence tests for series

Cauchy's test for the convergence of a sequence (3.2) takes the following form as applied to series.

CAUCHY'S TEST. The necessary and sufficient condition for convergence of the series (3.1) is that, given anye^O, there exists an ne such that

|on+iH-ûn+2+ * · · + ö n + ml < ε

for all n>nt and any integer m. An important corollary follows from this test with m = 1. If series (3.1) is convergent, its general term an tends to zero as n

increases: an -+ 0 as n -* «>.

SERIES 89

This condition is necessary but not sufficient for convergence. EXAMPLE 6. The harmonic series

1 1 1 S I 1 + 7 + Τ + · · · = , Σ 7

is divergent, although an = l/n — 0 as n — <». Cauchy's test is the most general necessary and sufficient criterion

for the convergence or divergence of a series. Direct application of it is difficult, however. The theory of series

includes a whole range of sufficient tests for convergence and divergence of varying degrees of generality, and widely employed. Historically, the first of these is

D'ALEMBERT'S TEST. If the limit

hm - r — — = q

exists for series (3.1), the series is convergent when q< 1 and divergent q > 1. (The test is inconclusive when q = Ì).

A more general test is THE CAUCHY-HADAMARD TEST. We write

q = lim Vi «n I -n—> »

Series (3.1) is now convergent when q<\ and divergent when # > 1 . (The test is inconclusive when q = \.)

THE PRINCIPLE OF MAJORANT SERIES. The series

Μ Λ + . . · = ΣΚ (3.7) 7 1 - 1

with non-negative terms bn is said to be majorant for series (3.1) if, given any n,

Kl ^K as from some n N.

The convergence of series (3.7) implies the convergence of series (3.1).

Numerous convergence tests are based on the principle of majorant series, including in particular the d'Alembert and Cauchy-Hadamard tests.


§ 1. Numerical series

1. Alternating series and series of constant sign

If all the terms of a series are of the same sign, the series is said to be of constant sign. Such series include those, all the terms of which are positive (positive series), and those, all the terms of which are negative (negative series).

If all the terms of a series are not of the same sign, the series is described as alternating. When there is a finite number of terms of the same sign, they can be neglected when investigating the convergence and attention paid only to the remaining series of constant sign. The theory of series with an infinite number of positive and negative terms has certain differences in principle from the theory of series of constant sign.

The following is a particular case of the principle of majorant series.

THEOREM 1. The series

ax + a2+ ... +an+ . . . = £ an (3.8)

with terms of arbitrary sign is convergent if the series

Ι * ΐ Ι + Ι * 2 Ι + . . · + Ι * η Ι + . . . = Σ \°n\ (3.8*)

formed from the absolute values of the terms of series (3.8), is convergent. In this case series (3.8) is described as absolutely convergent. All

series of constant sign are absolutely convergent. Cases are possible when series (3.8) is convergent, and series (3.8*)

divergent. Series (3.8) is then described as non-absolutely, or conditionally convergent.

EXAMPLE 7. The series 1 1 ^ ( » l ) n - 1

■-T+T---Ä » is convergent and its sum is S = In 2.

On the other hand, the series of the absolute values of the terms is the divergent harmonic series.

SERIES 91

2. Properties of convergent series. The associative property

If the terms of the convergent series oo

0i + tf2+ . . . +an+ . . . = Y an

are grouped in any manner, without changing their order:

aL+ . . . +ani> tfni+i + · · · + tfnr απ2+ι + · · · + #π3> · · ·

• · · y ank—i+l + · · · + Gnk> · · . j

where {nk} is a partial increasing sequence of numbers of the natural series, the series of the sums of the terms of these groups

(aL+ . . . +αηι) + (αηι+1+ . . . + tfn2) + . . . is always convergent and has the same sum as the initial series. This property can be utilized for improving the convergence of series.

The rearrangement property of absolutely convergent series. If the oo

series ]£ an is absolutely convergent, the series obtained from it 7 1 - 1

by any rearrangement (commutation) of the terms is also convergent and has the same sum as the initial series.

Non-absolutely convergent series do not have the commutative property. We have, furthermore,

THEOREM 2 (Riemann). The terms of a non-absolutely convergent series can be rearranged in such a way that the transformed series has a sum equal to any previously assigned number, or becomes divergent.

3. General tests for the convergence of series of positive terms

If we want to determine the convergence of the series with positive terms

Σ «» (3·9) Ä » l

we choose at discretion another series with positive terms

Σ h (3.10) Ä - 1

such that its convergence and sum are already known to us. We introduce the notation:

*m = Σ **; (3·11) fc-»m

Rm is the remainder term of the series, which is equal to a finite positive number if series (3.9) is convergent, and to e», if series (3.9) is divergent;

Λ(/) = - ^ , (3.12)

where / is a parameter taking integral values such that m + / ^ 0, m is a fixed positive integer;

lim Ah(l) = AQ)9 (3.13)

Km Ak(l) = 1(1). (3·14) Let

Hm Σ Z>* = Bm, (3.15)

ita £ Z>* = ï m ; (3.16) n—► « fe«m

in the case when the lower and upper limits coincide, i.e. when the limit exists, we introduce the notation:

lim Ah(I) = A(l\ (3.17)

[lim £ bk=Bm. (3.18)

We have the following sufficient tests for convergence of series (3.9).

Test I. IfAIJ) > 0 and i?m -< + «>, series' (3.9) is convergent, whereas if _ » <Λ(/) ^ 0 α/îrf 2?m = — «>, /Ae 5ene5 w divergent.

Test IL îfA(l)^0 andBm> - « , /Ae seri&y w convergent, whereas ifO^ A(J) < + » awrf 2?m = + ©o, *Ae jer/ej w divergent.

SERIES 93

Test III. If A(J) > 0 and £ m < + ^ or A(l)<0 andBm> - <~, the series is convergent; ifO ^A(l) < -f- oo andBm= t ooor-<»< A{T) ^ 0 and Bm= — o°9 the series is divergent.

By a suitable choice of auxiliary series (3.10), we can obtain as particular cases from Tests I, II and III either new or already famihar sufficient tests for the convergence of tests with positive terms (see sec. 5). These tests include, in particular, the familiar necessary condition for convergence which says that, if series (3.9) is convergent, lim an = 0.

4. Remainder term estimates corresponding to the various convergence tests

When one of the convergence Tests I, II, III is satisfied for series (3.9), a corresponding estimate of the remainder term can be given.

(1) For Test I:

Bm ^ B m + l ^ *mA m , (3.19) inf Ak(l) - m+L - sup Ah(l) '

where m is such that all the Ak(l)^0; l is an integral parameter, satisfying the condition m+/^ 0.

(2) For Test II:

where m is such that all the Ak(l) < 0 and m-f / ^ 0. (3) For Test III, if Λ(/)>0, inequalities (3.19) hold, while (3.20)

hold if A(I) < 0. For Test III, it becomes convenient to utilize the following par

ticular cases of (3.19) and (3.20): (a) When A(J) >■ 0 and the sequence {Ak(l)} is monotonically in

creasing for k ^ m, we have

W)-Rm+l-^U)' (121)

(b) When A(l) < 0 and the sequence {Ak(l)} is monotonically decreasing for k m, we have

Ä^ij-Rm«-W)' ( 3 · 2 2 )

(c) When A(T) < 0 and the sequence {Ak(I)} is monotonically increasing for k^ m, (3.21) holds.

(d) When A(J) > 0 and sequence {Ak(l)} is monotonically decreasing for k m, (3.22) holds.

EXAMPLE 8. Let us estimate the remainder term of the series oe 2

n « l "

On putting, say, bn = rn-1—2n, where zn = —1/2«2, we have, provided that w > 1 and / 0:

(2m + l)(/w + /)3

4»(0 = 2m2(m+l)2

Λ(/) = lim A L = i, Bm = Urn ( r n + 1 - z j = ^ .

It is easily verified that the sequence { 4m(0)} (m = 1, 2, 3, . . . ) is monotonically increasing, while the sequence {Am(l)} (m = 1, 2, 3, ) is monotonically decreasing.

Consequently, when / = 0 and m ^ 1, we have by (3.21):

2w2 m (2rfJ+l)m3 '

while with / = 1, by (3.22), 1 _ 1

R. (2m+l)(m+l) m + 1 2m2' We obtain from the last two inequalities, in choosing the greatest lower and least upper values,

1 ^ ^ ( » + » (2m-l)w m (2m+l)m3'

This example shows that, given the same choice of sequence {bn} and different values of the index /, estimates can be obtained that do not overlap each other.

SERIES 95

5. Special tests for the convergence of series of positive terms. Estimates of the remainder term

Numerous familiar classical tests for the convergence of series with positive terms follow from the general convergence Tests I, II and III (sec. 3). We shall give here certain of these and the corresponding remainder term estimates (see ref. 9).

Γ. THE GENERALIZED D'ALEMBERT TEST. The following two tests can be regarded as a generalization of the d'Alembert test mentioned above.

Test A. If ; for a fixed /,

1(1) = fim Û W l ~ a w < 0, (3.23) oo

the series ]T an is convergent, whereas the series is divergent if

A(f) = lim SztlZf» > o, (3.24) n—~ an+l

Test B. If, for a fixed /,

A(l)= lim *"+*"-*" η _ * β ο an+l

is negative, the series is convergent; whereas if the limit is positive^ the series is divergent.

In particular, when / = 0 Tests A and B yield d'Alembert's test (Introduction, sec. 2).

oo

If the convergence of the series £ an is established with the n - l

aid of Test A or B, we have on the basis of (3.20):

=^2 ^ Rm+l ^ Z^H . (3.25) i n f *fe+i-*fe s u p *k+i-"k

k^m ak+l k^m ah+i

If the sequence {Ah(l) = (^k+i—ah)Iak+i} ™ monotonically increasing for k ^ m, we have

~üm S Rm+i m ~am „ . (3.26) g m + l ~ " g m j . ak+l — ak am+l A-*«» ak+i

If the sequence {Ah(l)} is monotonically decreasing for k^m, we have

-an lim "h+i-'h

^ Rm+l — an

*m + l "~um -*m+i

EXAMPLE 9.

(3.27)

(3.28)

It is easily verified that, if /+ 1 < 0, the sequence {Ak(l) = (<ζ&+ι— —ak)/ah+i} f°r s e r i e s (3.28) is negative and monotonically decreasing as from a certain fc JV(/). If /+1 > 0, the sequence {Ak(l)} is negative and monotonically increasing.

We therefore have, on the basis of (3.27) with /-f 1 ^ 0, as from a certain m ^ N(l) (say as from m ^ 1 when / = —1):

1 - (m-M)2

2 m + i - l m 2 ^ *m+l ^ 2^H-i(w + / ) 2 ( m 2 + 4 m + 2 )

When / -f 1 > 0 we have to reverse the inequality signs in the above. For instance, on putting m + / = 6, and accordingly assigning the

following values to m and /:

m

I

9

- 3

8

- 2

7

- 1

6

0

5

1

4

2

it may be seen that the best upper estimate is obtained with / = — 1, and the best lower estimate with/ = 0, so that in this case 0-000686 ^R^ 0-000703.

2°. THE GENERALIZED CAUCHY TEST. On putting bn = ρη and applying Tests, I, II (sec. 3), we get the following convergence test, which may be regarded as a generalization of the Cauchy-Hadamard test.

Test C. If for a fixed I and ρ < 1,

lim 0, *n+l

SERIES 97

the series £ an is convergent; whereas the series is divergent if,

for a given I and ρ > 1,

0 ^ lim Qn

n^oo an+l

When / = 0, this test yields the familiar Cauchy-Hadamard test (Introduction, sec. 2).

If the convergence of the series can be established with the aid of Test C, we have by (3.19), for any m+/>0 ,

om nm — -jr =5 Rm+l S -jr . (3.29) (1 — ρ) sup —— (1 — ρ) inf ——

Here, ρ < 1 is an arbitrary number, for which lim (Qn/ak+ì) > 0.

If the sequence {Ah(l) = Qklak+X} is monotonically increasing for k ^ m, we have

s i ^ i ^ . (3.30) ( 1 _ ρ ) Η ι η - £ _ ?

Whereas, if the sequence {Ak(l)} is monotonically decreasing for k ^ w, we have

f ^ - 2 W ^ ^ — j r . (3.31) * ( l - ρ ) lim -2—

EXAMPLE 10.

Σ Α (0^*<l). (3.32)

It is easily shown that estimate (3.31) can be applied to series (3.32) on condition that m+l ^ 3.

On taking ρ = x, we have

3°. RAABE'S TEST. A particular case of the general Test III (sec. 3) is


Test D. The series £ an is convergent if, for any integral / ^ 0, n « l

AC) = lim " W f r - I K ^ 0) (3 33) n->oo an+l

and is divergent if A(l) > 0.

When / = 0, we get oo

RAABE'S TEST. The series ][] an is convergent if

and is divergent if

for all n^ n0 say. EXAMPLE 11.

Km „A_fa±A^if

7 i [ l - * 2 ± l | ^ L

1 nfrilK"+l)(« + 2)· ( 3 3 4 )

The convergence of this series is readily established by applying Test D. We can also make use here of the estimates for the remainder term of sec. 4, corresponding to cases (3), (c) and (3), (d).

On applying estimate (3), (c) (sec. 4) with m ^ 9, and 1 2, we get

( w _ l ) ( w + 3) m _ i (2w-3)(w + /)(w + /+l)(m + /+2) ~ m+t ~ (m+l)(w + 2) '

(3.35) On applying estimate (3), (d) (sec. 4) with m-f-/>0, / ^ l and w > 4//(2/-3), we have

(m-l)(m + 3) m-1 (2m-3)(m+/)(m + /+l)(m + /+2) " m+I " 2m(m+1)(m + 2) '

(3.36) On comparing inequalities (3.35) and (3.36), say with m+/ = 12, it is easily seen that

0-00315 ^ Rj2 ^ 0-00337, he true value here being R12 = 0*00320.

SERIES 99

4°. GAUSS'S TEST. If lim an+1/an = 1, d'Alembert's test gives n - > o o

no answer as regards the convergence of the series, as mentioned above. If the ratio an+1/an has the form

ηλ+ρηλ~1 + φ(η) an ~~ nx + qnk"1'\-y)(n) '

"n+1

where <p(n) and ψ(ή) have lower orders than Ηλ-1, the convergence or divergence of the series may be established with the aid of the following test.

oo

GAUSS'S TEST. The series ]T an is convergent when q—p> 1, 71—1

and divergent when q—p^ 1. EXAMPLE 12. When x = 1, the hypergeometric series (ref. 7):

ccß ì α(α+ ϊ)β(β+1)+ *(*+1)(* + 2)β(β+1)(β + 2) + _ (3>37)

l .y 1.2.y(y+l) 1.2.3.y(y+l)(y+2)

is convergent provided that

y > α + /?.

Notice that, from a certain H onwards, series (3.37) is of constant sign for any real α, β, y (it is assumed that none of the numbers a, /?, y is negative).

We have in the present case, say with / = 0,

y-tn

and A(0) = - ( y - a - j S ) < 0.

On observing that, for all n > — y, the sequence {An(0)} will be mono tonically increasing or monotonically decreasing, and making use of (3.26) and (3.27), we get

{m + y)mam ^ ^ ττ^ *„ -*m

(ηι+γ)(γ-χ-β)-(γ-*)(γ-β) s m ë y - a - / î ' (wi> - y ) ,

where α(α+ 1) . . . (α + m - ϊ)β(β+ 1) . . . tf + m - 1)

m ! y ( y + l ) . . . ( y + m - l ) 5°. CAUCHY'S INTEGRAL TEST. Let the general term of the series

oo

Σ an be monotonically decreasing with increasing n. Obviously, n - l it is now possible to choose an infinite set of positive functions a(x)9

monotonically decreasing and continuous for x > 0, and such that

a(n) = an. Let a(x) be one of these functions. The series £ an is now converg-

n«-l ent or divergent, according as the improper integral

i: a(x) dx, (3.38)

where a ^ 1, is convergent or divergent. This test follows as a particular case from general Tests I and II

(sec. 3), if we put say fn+l

bn = I a(x) dx and / = 0.

EXAMPLE 13. We are given the series ». i

n?2 «ln1 + < rn ^ We choose as a(x) the function l/(x ln1+<r J:) and put a = 2, so that

0). (3.39)

i; 1 ;cln1+<rx

dx = — ■ 1 σ1η σ χ

1 σ1ησ2

the series is therefore convergent. EXAMPLE 14. We are given the series

^ 1 n - 3 « In Λ In In ?2 '

We choose a(x) = 1/(χ1ηjclnlnjc) and put a = 3:

1

(3.40)

x In x In In x dx = In In In x

so that the series is divergent.

SERIES 101


n - i » ' (3.41)

we choose a(x) = l/x" and put « = 1. On recalling that

= ) T=7 for σ * U ί dx_ X?

Λ-cr

a lnx

we get: dx IF

1 a-\

for

for

for

for

< r = l ,

σ > 1,

o ^ 1. The series is thus convergent for cr> 1 and divergent for a ^ 1. Notice the following points:

(1) In view of the fact that the choice of the function a(x) is to a large extent arbitrary, an infinite set of estimates can in general be quoted, corresponding to the test.

(2) When the sequence {an} is monotonie, estimates corresponding to Cauchy's integral test can be used, independently of the

particular test used to prove the convergence of the series £ an. n - l

EXAMPLE 16.

Σ — (a > 1).

Let a(x) = x/fl*, say. The function a(x) is monotonically decreasing for x > 1/loga. Therefore, on evaluating

ak fÄ+1 x , α - 1 α - 1 η α - 1

Ak = T" x a dx = " Ί — + y i 2 k J ft a In a Ara In2 a

and making use of estimate (3.26), we have

m(m ln<z+ 1)

a m - i ( e - l) /"w in e +1 - ^ Λ ^ * m ^

w In a + 1

(m ^ 1).


6°. N. V. BUGAEV'S THEOREM; V. P. ERMAKOV'S TEST. Let a(x) be the function introduced in the statement of Cauchy's integral test, and δ(χ) some positive differentiable function, increasing as x increases and such that lim 1/δ(χ) = 0. The following theorem can now be

obtained on the basis of Cauchy's integral test. THEOREM 3 (N. V. Bugaev). If the function δ\χ) a [δ(χ)] is mono-

©o oo

tonic for sufficiently large x, the series £ an and ][] δ'(η)α [δ(η)] n~l n - i

are simultaneously convergent or divergent. oo

Thus any convergent test applied to the series £ δ'(η)α[δ(ή)] π«*1 oo

will at the same time yield a convergence test for the series ]T an. 7 1 - 1

In particular, we have the following corollary of N. V. Bugaev's Theorem. V. P. ERMAKOV'S TEST. If

ema(em) t hm — T ^ - C 1, m-+oo a(m) oo

the series ]T an is convergent; whereas if n - l

ema(em) hm — 7 ^ ~ > 1,

τη- βο a(m) the series is divergent.

rn+i On putting say bn = δ\χ)α [δ(χ)] dx and using the general

tests of sec. 4, an infinite set of estimates can be obtained (for the

remainder term of the series £ ύ^), corresponding to different n - l

methods of choosing the functions δ(χ). 7°. LOBACHEVSKII'S TEST. If the terms of the series

n - l

are monotonically decreasing, then the series is convergent or divergent at the same time as the series

m « l

SERIES 103

where pm is given by a(pm)^2-m, a(pm + l)^2-m.

We can also define pm from the equation aipj = 2-»\

if the function a(x) is monotonie and defined for any value of x. EXAMPLE 17.

The equation 1/P2m = 2 " ^

i_m

gives us pm == 22 ; we form the series oo oo . 2 5

EAn2-m= Σ 2 2· m=»l m—1

This is a convergent geometrical progression, so that the series 1

Σ, „2 s also convergent.

6. The convergence of alternating series

Given the alternating series (3.8), we form series (3.8*) from the absolute values of its terms.

To establish the absolute convergence of series (3.8), we can apply to the positive series (3.8*) all the tests described above for series ofconstant sign. As already mentioned, however, tests for the convergence of series (3.8*) may not be valid for series (3.8).

Alternating numerical series. Alternating series are those, the terms of which are alternately positive and negative. The following holds for such series:

oo

THEOREM 4 (Leibniz). If the terms of an alternating series £ an n - l

are monotonically decreasing in absolute value, \an+1\ ^ \an\ (n = 1, 2, 3, . . . )

and the n-th term tends to zero, lim an = 0,

n—►<»

the series is convergent. Estimation of the remainder term of an alternating series is simple

and effective. The remainder term of an alternating series that satisfies the con

ditions of Leibniz's theorem,

&η ~ β η+1 + ΰ η + 2 + · · · >

has the sign of its first term ûn + 1 and is less than it in absolute value:

l-Rnl = an+l + an+2 +

The criteria for convergence of Abel and Dirichlet are more general than Leibniz's test.

Suppose we have the series

Σ * A = αΑ + βΑ + - - - +anbn+ . . . , (3.42)

where {an} and {Z>n} are sequences of real numbers. ABEL'S TEST. Series (3.42) is convergent if the series

Σ*η = *1 + * 2 + · . . + * η + · · · (3.43)

is convergent, while the numbers an form a monotonie and bounded sequence:

\an\^K (TI= 1,2, 3, . . . ) .

EXAMPLE 18.

DIRICHLET'S TEST. Series (3.42) is convergent if the partial sums of series (3.43) are bounded in aggregate:

\sn\*M ( T I = 1,2, 3, . . . ) , (3.44)

while the numbers an form a monotonie sequence tending to zero: lim an = 0.

SERIES 105

Assumption (3.44) is more general than the assumption of the convergence of series (3.43), so that Abel's test is a consequence of Dirichlet's test.

7. Infinite products and their convergence

Suppose we have a sequence of numbers (or functions)

A» A i A* · · · > A i , - - . - (3.45) The symbol

Π Α ι = Α Λ Α · · · Α ι · · · (3·46)

s known as an infinite product. The product of a finite number of consecutive terms

Λ = At Λ = AA> Λ = AAA> -. - , Λ ι = A A · · · A i - i A (3·47)

is called a partial product. The sequence of partial products will be denoted by {Pn}.

EXAMPLE 19.

~ K(ft + 3) 1.4 2.5 3.6 JJ(/i + l)(* + 2) - 2.3 3.4 4.5 " "

Tfte convergence of infinite products. Four fundamental cases may be distinguished. (1) The sequence of partial products {Pn} has a finite limit different

from 0: lim Pn = P. (3.48)

This limit is called the value of the product and is written as

P = Π Λι- (3.49) n-i

The product itself is said to be convergent in this case. In Example 19,

P = hm - r = —. n-^eo 3 «-h i 3

(2) The sequence {Pn} tends either to + °°, or to — <».

106 MATHEMATICAL ANALISYS

EXAMPLE 20.

oo"1

Π « = 1 . 2 . 3 . . . . . . . ; Pn = n\; lim ni = oo.

(3) The sequence {Pn} has the limit 0. EXAMPLE 21.

T for any n, n, Pn = — , lim —

(4) The sequence {Pn} has no limit (e.g. oscillates). EXAMPLE 22.

Π(-1)* "(Wf3) ° i-V ' (n+l)(« + 2)

" 2 . 3 ^ 3 .4^/4.5^ 5.6J·"9 3 ^ r " < 3 *

In the last three cases the sequence is said to be divergent. Case (3) / lim Pn = 0\ implies a different classification to that accepted

for infinite series. However, this is convenient for the statement of many theorems on infinite products.

An infinite product can be written in the form

oo

Π Pn = Pm*m> (3.50) n - l

where Λη = PlPf-Pm

is the partial product of the first m terms, and

oo

nm = Pm+lPm+2 · · · = Π Pn (3.51) n-m+l

is called the remainder product, this being analogous to the remainder term of a series.

THEOREM 5. If product (3.46) is convergent, the remainder product (3.51) is also convergent for any m (if all the pn ^ 0) ; the convergence of the remainder product implies the convergence of the original product.

SERIES 107

Thus discarding a finite number of initial factors or combining at the start a finite number of factors has no effect on the convergence of an infinite product.

If an infinite product is convergent, then lim nm= 1. (3.52)

This follows from (3.50); _ _P_

ητη — p

and from the fact that P ja 0. If an infinite product is convergent, then

lim /?n = 1. (3.53)

This follows from the chain of equations:

n - ~ l imPn. t P

In the case of a convergent product, pn > 0 as from a certain «. This follows from (3.53). In view of Theorem 5, we can assume without loss of generality that all the pn > 0.

There is a connection between the convergence of infinite products and that of series.

A necessary and sufficient condition for convergence of the infinite product (3.46) is that the series

• convergent. Σ lnPn

If S is the sum of series (3.54), we have

On writing ^n

P = es. for the partial sum of series

sn = lnPn, Pn = e8» (3.54), we have

(3.54)

(3.55)

(3.56) It follows from the continuity of the logarithmic and exponential

functions that, when Pn tends to a finite positive limit P9 the partial sum £η tends to In P, and conversely, if the finite limit S exists, the limit for P is equal to es.

The «th term of an infinite product may be conveniently written as

and product (3.46) written as

Π(1 + *η)> (3.57)

while series (3.54) is written as

£ l n ( l + e„). (3·58) n-»l

THEOREM 6. If for sufficiently large n9 all the an> 0 (or an< 0), the necessary and sufficient condition for convergence of product (3.57) is that the series

Σ «»· (3-59) n=l

be convergent. A necessary condition for convergence of (3.57) and (3.58) is that

lim an = 0, whence lim X^±^ = 1.

In the general case an % 0, the infinite product (3.57) is convergent if the series

Σ <*· (3·60)

is convergent along with series (3.59). The product of Example 19 can be written as

Π "(w+3) = ft (i - Ì · n

1J1(»+l)(» + 2) JLU (" + 1)(" + 2)y Here - 2 On = n (Λ+1)(Λ + 2)

and lim an = 0.

n-

SERIES 109

The following will make it clear why, in the case

P = 0,

the infinite product is classified as divergent. The necessary and sufficient condition for the infinite product to have

a zero value is that series (3.54) or (3.58) have the sum — «>. This is the case for example when an < 0 and series (3.59) is diver

gent, or, when series (3.59) is convergent, but series (3.60) is divergent.

Product (3.46) is said to be absolutely convergent when series (3.54) of the logarithms of its factors is absolutely convergent.

An absolutely convergent product has the commutative property. The necessary and sufficient condition for absolute convergence of product (3.57) is that series (3.59) be absolutely convergent.

We have, for Example 19:

2 1 J_ | α η Ι " (Π + /)(Λ + 2) - 2 ( Λ + 1) (* + 2) < 2 , ζ 2 ·

oo

But £ I/**2 is a convergent series (see example 15), so that the 71=1

product is absolutely convergent. It may be remarked that the theory of functional products bears

the same relationship to the theory of numerical products as the theory of functional series to that of numerical series. Let us give an example of a functional product.

EXAMPLE 23. The product

"â('-w) represents sin* for any x.

8. Double series. Fundamental concepts and definitions

In addition to ordinary (simple) infinite series, multiple series are employed in analysis and applied mathematics, e.g. double, triple, etc., series. We shall confine ourselves here to a brief outline of the theory of double series.

DEFINITION. A double series is defined by the symbol

Σ Σ*« (or Σ *«Y (3.61) fe«=0 l»0 \ k, 1-0 /

#wd is made up of the terms of the double sequence {ahl} (k = 0, 1, 2, 3, . . . ; / = 0, 1, 2, 3, . . . ) ; corresponding to it, we have a double sequence {smn} of partial sums

Ä«m-1, l«*n—1 Smn = Σ *«· (3.62)

If a finite limit exists on simultaneous and independent increase of the indices m and n,

oo

S = lim smn = Σ *«> (3.63)

the limit is called the sum of the double series (3.61), and the series is said to be convergent in this case; otherwise it is divergent.

If the terms au are summed consecutively, first over one index, then over the other, the double sum will be described as an iterated series. Obviously, the following two cases are possible here:

or

oo / oo \

Σ ( Σ <*)

1(1««). Ä-0 \ I - 0 J fe

derated series (3.64) is said to be convergent if the series

(3.64)

(3.65)

Ai= Σα* (3.66) fc*=o

(for any fixed subscript /) and

Σ Ai (3-67) i«o

are convergent. Similarly, series (3.65) is said to be convergent if the series

Bh = Σ «« (3.68)

SERIES 111

(with any fixed subscript k) and

Σ *h (3.69)

are convergent. Let us introduce the notation

A = lim ( lim ^ m n ì . (3.70)

Similarly, B = lim ( lim ^ ,Λ . (3.71)

A series for which all the ahl ^ 0 is called a double series with positive terms.

oc oo

If the double series £ akl and £ \akl\ are simultaneously ft, 1=0 fc, 1=0

convergent, the double series is said to be absolutely convergent. oo oo

Whereas if £ akl is convergent, but £ |af t l | is divergent, the k, 1=0 k, i«0

oo

double series ]T akl is non-absolutely (or conditionally) comer-K i -o

9. Some properties of double series

I. //* double series (3.61) awrf rer/ej (3.66), (3.67) are convergent, the iterated series (3.64) is also convergent and has the same as the double series.

Similarly, the following property holds: II. If double series (3.61) and series (3.68), (3.69) are convergent,

iterated series (3.65) is also convergent, and has the same sum as the double series.

REMARK. Generally speaking, the convergence of series (3.66) and (3.68) does not follow from the convergence of double series (3.61)

III. The necessary and sufficient condition for the convergence of the double series (3.61) with positive terms is that its partial sums be bounded.

IV. If one of the three series (3.61), (3.64) and (3.65) with positive

gent.


terms is convergent, the other two are convergent and have the same sum.

A one-to-one correspondence can be established between pairs of equal terms of the double sequence {akl} and the ordinary sequence {bs}. We shall say for brevity in this case that the double series

oo oo

£ akl and the series £ bs consist of the same terms. k, 1=0 s « 0

oo oo

V. If the double series ]T akl and the series £ bs consist of

the same terms, the convergence of one implies the convergence of the other, and they have the same sum.

oo oo

VI. If the double series £ \akl\ is convergent, the series ]£ akl

is convergent (but the converse does not hold). oo oo

VII. If the double series ^ aM and the series ^ bs consist k, l*=0 8*0

of the same terms, the absolute convergence of one implies the absolute convergence of the other. Both series now have the same sum.

VIII. The terms of an absolutely convergent series can be rearranged in any manner without changing the sum.

The above properties can be useful in proving the convergence, and computing the sum of a double series.

EXAMPLE 24. It is easily shown that the double series

is convergent for a > 2 and divergent for α ^ 2. For, on putting k+l = m, we find that m— 1 terms of series (3.72)

are equal to l/ma. Series (3.72) can therefore be written as the simple series

m « 2 m m = 2 "* m = 2 m

Both the series on the right-hand side of (3.73) are convergent for a > 2 (Example 15), so that the series on the left-hand side is convergent. By Property V of double series, the double series (3.72) is also convergent, and its sum is equal to the sum of the simple series (3.73).

SERIES 113

10. Some convergence tests for double series of positive terms. Estimates of remainder term

A necessary condition for the convergence of any double series £ akl is that

lim akl = 0, (3.74)

when k and I tend to infinity independently of each other. A stricter necessary condition than (3.74) can be given for the con

vergence of double series with positive terms. For a double series with positive terms to be convergent, it is necessary

that

and

lim £ * w = 0 (3.75)

lim £ a H = 0, (3.76) t - χ » ft-0

so that all the more, the following conditions must be satisfied:

lim Ì>f t I = 0 (3.77)

and

lim J>w = 0· (3-78) l ->oo fc«0

From (3.77) and (3.78), we see that the following conditions must also be satisfied:

lim akl = 0, \ k-

lim ahl = 0 for any /,

lim akl = 0 for any lc.

(3.79)

As in the case of ordinary series with positive terms (see § 1, sec. 3-5), comparison of the terms of two series leads to a number of general tests for the convergence of double series with positive terms, and corresponding remainder term estimates can be given.

Suppose we have the double series

Σ *« (3-80)

with positive terms, and some other series

Σ hi- (3.81) k, 1-0

We assume that the convergence or the sum of the series (3.81) is known to us in advance. We introduce the notation: k+l = n9

lim 5« = i4, (3.82) T l ~ ~ akl -

hin ^ = 1. (3.83)

In the case when A = Aywc have

lim Ï£ = A. (3.84)

The remainder term of double series (3.80) will be defined by

Rmm = rm0 + r0m + rmm» (3 .85)

where

and

ft=00, l = m - l Ä«m-l , l=eo rmo = Σ akb r0m = Σ akl

k, l—m

We shall write Bmm for expression (3.85), corresponding to the remainder term of series (3.81).

Using the above notation, the following general sufficiency tests can be stated for the convergence of the double series (3.80):

1 °. IfO Bmm< + °°/or allm>N and A_ > 0, the series is convergent. 2°. If — oo ^ Bmm ^ Ofor allm^N and A^09 the series is converg

ent. 3°. If with finite A and A of the same sign, Bmm = ± <», the series

is divergent.

SERIES 115

4°. If 0 < Bmm < +00 for all m > N, and the conditions — oo < A < +0O and A = + oo are satisfied, the series is convergent.

5°. If — oo < i?mm < 0/or m> N and the conditions — <*> < ^ < 0 and i = - oo are satisfied, the series is convergent.

When series (3.80) is convergent, the following estimates hold for the remainder term:

Bmm — r> ^ Bmm /^ Q/~v Γ - < ^mm _ IT - (J.OÔJ

. e. Ohi ^ ^ ^κι

inf — sup —'

When y4 = + oo or ^ = — oo (or A = ± ~ ) ,

I A, ^ n m ^ •'mm I

inf (3.87)

Further, on choosing the bkl by different methods, we can obtain from the above general convergence tests various practical tests which are sufficient for the convergence of a double series with positive terms. Inequalities (3.86) and (3.87) enable us to give remainder term estimates corresponding to the convergence test chosen.

We shall confine ourselves here to one concrete case, e.g. by putting

hi = ak+i, i+i - ak+i, ι - ακ i+i + *«· ( 3 · 8 8 )

It now follows from the above general tests that the double series

£ akl is convergent if:

(1) Um app = 0; (3.89)

(2) A = lim 5« > 0; (3.90)

and is divergent if:

J = Um ^ l < 0 . (3.91) fe+l-> oo tffcl

The remainder term estimate corresponding to this test has the form

7 S -i mrn s Γ 9 yl.yZ) sup 2« inf 5«

where bkl is given by equation (3.88). Fulfilment of condition (3.90) only is not sufficient for convergence

of the double series. EXAMPLE 25. The series

3 W 6 Y :*%£.( W is obviously seen to be divergent; at the same time, the condition A = 0-1 > 0 is satisfied although lim app = <».

EXAMPLE 26. Let us take the double series

5 = kic <*+l)!(/+l)! * ( 1 9 3 )

Condition (3.89) is satisfied here, since

^-^",Ϊ.ΚίΤϋϊΡ"0· We have by (3.90):

lim — = lim gfe+i''+i~g»-'+i~gfc+:i,i + flfei _ fc+l-»-~ Oft; fc+I-s-οβ ûftl I ft+I-s-o» ûftl

/ / + ! /+2 * + l

" k + S - (* + 2)(/+2) _<{

for fixed /;

for fixed &; Ä: + 2

1, if simultaneously k and / — ».

But obviously,-J-S(/+l)/(/+2)S 1 and-J-S (À:+l)/(A:+2) S 1,

i.e. A = —, for all k, / ^ 0. Series (3.93) is thus convergent.

To estimate R^^ in accordance with (3.92), we first observe that, for the function F(x, y) = (*+1)(J+1)/(;CH-2)0>-{-2), necessary

SERIES 117

conditions for an extremum: dF/dx = dF/dy = 0, are not satisfied for any finite positive x and y, i.e. the greatest and least values of the function can only be attained on the boundary of the domain m ^ x, y < <».

It may easily be shown by using this fact that

. f bkl (* + !)(/ + !) w + 1 1 ini = Im 77 χτ-τζ -rr = — · ττ-

fe+l^m^ì k+lZzm (* + 2) (1+2) m + 2 2 and

(fc+!)(/+!) _ * i r |m(* + 2)(/ + 2 ) " le

On substituting the values obtained in inequality (3.92), we get

2 1 D 4(m + 2) 2(m + 2) (m+1)! |(w-fl)! |2~ m m~(w-f l ) (m4-l) ! (m+l)[(m + l)!]2 '

For instance, we obtain with m = 4:

0-0159 ^ i?mm ^ 0-0382.

§ 2. Series of functions

1. Fundamental properties and convergence tests

We shall discuss functional series in this article, i.e. series whose terms are functions. We shall confine ourselves simplicity to the series

ιφ) + ιφ)+ . . . + "»(*) + . . . = Σ wn(*), (3.94)

the terms of which are functions of a single variable. Immediate generalization is possible to the case of two or more variables.

DEFINITION. The expression (3.94) is called a functional series; it is formed from the terms of the function sequence {un(x)}, defined on the set X = {x} of the numerical axis El9 and has a corresponding sequence {s^ :

n Sn = Σ uk(x)

fc-1

of partial sums.

There are different types of convergence for the function sn(x): uniform, non-uniform, in the mean, etc. Corresponding to these we have different forms of convergence of the functional series.

Let the sequence {sn(x)} be uniformly (non-uniformly) convergent on X to the function S(x). Series (3,94) is then said to be uniformly (non-uniformly) convergent, and S(x) is its sum:

A Condition for Uniform Convergence of a Series: The necessary and sufficient condition*, for series (3-94) to be uniformly convergent on X is that, given any ε >- 0, there exists an N independent of x such that, given any n>N and any m = 1, 2, 3 , . . . , the inequality

I n+m I Σ uh (*) < ε

I fc -n+l I holds for all x € X.

If all the terms of series (3.94), uniformly convergent on the set X, are multiplied by the same function v(*), bounded on X,

\v(x)\*Mt

the uniform convergence is preserved. Tests for uniform convergence of series. Γ. WEIERSTRASS'S TEST. If the terms of series (3.94) satisfy on the

set X the inequalities

k W I S c n ( « = 1, 2, 3, . . . ) , (3.96)

where cn are the terms of a convergent numerical series, (3.94) is uniformly convergent on X.

When (3.96) holds, the series

Σ cn π-α

is described as majorant for series (3.94). A functional series satisfying Weierstrass's test is absolutely convergent, and moreover, the series

Σ l«n(*)l· is uniformly convergent.

SERIES 119

Cases are possible when series (3.94) is uniformly but not absolutely convergent.

The following tests hold for functional series of the form

Σ an(x)t>nW = afrMx) + . . - + **(*)*»(*) + · . . . (3.97) n = l

2°. ABEL'S TEST. Series (3.97) is uniformly convergent on a set X if the series

£ *»(*) = * i ( * ) + W * ) + · · · <3·98)

is uniformly convergent on the set X9 while the functions an(x) form a monotonie sequence for any x, and are bounded for any x and n:

K(*)l ^K.

3°. DIRICHLET'S TEST. Series (3.97) is uniformly convergent on the set X if the partial sums sn(x) of series (3.98) are bounded for all x and n:

\sn{x)\*M,

while the functions an(x) form, for any x, a monotonie sequence, uniformly convergent to zero on the set X.

Let us also note some properties of the sum of a functional series. If the functions un(x) of series (3.94) are defined in the interval X = (a, b) and are all continuous at the point x = *0 of this interval, and in addition, series (3.94) is uniformly convergent, the sum S(x) of the series is also continuous at the point x = xQ.

The following proposition is a consequence of this: if the functions un(x) are continuous throughout the interval X = (a, b) and if the series is uniformly convergent there, the sum S(x) of the series is continuous throughout the interval. The uniform convergence in these propositions is a sufficient but not a necessary condition, since there are series, non-uniformly convergent in an interval, having a sum continuous in the interval. We shall not state here necessary and sufficient conditions for continuity of the sum of a series, and shall only mention that uniform convergence is necessary and suffi-icent provided the terms of the series are in addition positive in the interval.


Term by term integration of series. If the terms un(x) of series (3.94) are continuous in the interval X = (a, b) and the series is uniformly convergent in this interval, the integral of the sum S(x) of the series is equal to the sum of the integrals of the terms:

F S(x)dx= g Î\(x)dx. (3.99) Ja *-l J a

In other words, term by term integration of the series is permissible under these conditions.

The following is a generalization of the above theorem: if the terms un(x) of the series are integrable (in Riemann's sense) in the interval X = {a, b) while the series is uniformly convergent, the sum S{x) of the series will also be integrable and (3.99) will hold.

Term by term differentiation of series. If the terms un(x) of series (3.84) are defined in an interval X = (a, b) and have continuous derivatives un{x) in X, and series (3.94) is convergent in X, while in addition the series formed from the derivatives

Σ «nW β!«ΐ(*) + (*)+ · · · +««(*) + n«=l

is uniformly convergent, the sum S(x) of series (3.94) has a derivative in X, and

S'(x) = f un{x\ (3.100) n « l

in other words, the derivative of the sum is equal to the sum of the series formed from the derivatives, i.e. term by term differentiation of series (3.94) is permissible. It may be mentioned that the restrictions of this theorem can be somewhat relaxed (see ref. 11, vol. II, § 407).

2. Power series

This branch of the theory of functional series is particularly important.

A series of the form oo

Σ an(x - x0)n = a0+a^x - x0) + ... + α„(χ - x0)n + ... (3.101)

SERIES 121

is called a power series. On substituting x± = x—x0 (we shall omit the subscript 1), it reduces to the form

oo

£ anxn = ao + x + x2^ . . . +anxn+ . . . , (3.102)

with which we shall be concerned in what follows. There are three possible cases. Case 1. Series (3.102) is convergent for any real x, i.e. throughout

the axis Ev In this case the series is said to be convergent everywhere. Case 2. Series (3.102) is convergent only for x = 0. The series is

now said to be divergent everywhere. Case 3. The series is convergent in some interval (—R, +R),

0<Ä< + oo; the number R is called the radius of convergence of the series.

We shall accept the convention that Case 1 corresponds to a radius of convergence R = o°, and Case 2 to a radius of convergence R = 0.

EXAMPLE 27. The series

^ xn x x2 x3

η?1^Γ=1+π+2Γ+ΐΓ+··· is convergent for any x (R = «>).


f n\xn = x-f-2f;c2 + 3!;c3+ . . .

is convergent only for x = 0 (R = 0). EXAMPLE 29. The series

Σ *n = l+jc + ;c2-h . . . 7 1 - 1

is convergent in the interval ( - 1 , -hi) (R = 1). EXAMPLE 30. The series

~ xn x2 x3

is convergent in the semi-interval (— 1, +1) (R = 1).

~ xn x2 x3

is convergent in the segment [—1, 4-1] (R = 1). If the limit

lim *n+i

exists, (see d'Alembert's convergence test) then

* = —. (3.103) e

(R = co for ρ = 0, R = 0 for ρ = «>).

On starting from Cauchy's test, we get

* = - = — * (3.104) ρ Um VI η I

If series (3.102) has a radius of convergence R > 0, no matter what the positive number r< iÊ, series (3.102) is convergent uniformly with respect to x in the segment [—r, r],

The sum £(;*:) of series (3.102) is a continuous function of x for all x between — R and R.

THEOREM 7 (on the identity of power series). If two power series

oo oo

£ flnxn uTzrf X bnxn

Aave /Ae same sum in the neighbourhood of the point x = 0, iAe 5eri££ are identical, i.e. their corresponding coefficients are equal: an = bn.

The following propositions concern the behaviour of a power series at the ends of the interval of convergence. If power series (3.102) is divergent at an end of the interval of convergence, the convergence cannot be uniform in the semi-interval (0, R).

SERIES 123

The converse is: if the power series (3.102) is convergent for x=R (x = — R) (even non-absolutelyj the series is uniformly convergent throughout the segment [0, R] ( [-R, 0]).

THEOREM 8 (Abel). If series (3.102) is convergent for x = R9 its sum remains continuous (from the left) for this value of x9 Le.

lim S(x) = S(R).

See Example 30 for x = — R = —1 and Example 31 forx== — R= = - 1 , x = R= 1.

3. Operations on power series. Taylor series. Integration and differentiation of power series

(a) Power series (3.102) can be integrated term by term in the segment [0, x], where |* | < R:

S(x)dx = Y anx*dx= Σαη^—τ\ (3.105) Jo " -Ojo n«0 " + *

x may reach the end of the segment at which series (3.102) is convergent.

(b) Power series (3.102 ) can be differentiated term by term as many times as desired inside its segment of convergence :

S'(x) = g nanxn-K

I S"(x)= g n(n-l)anx"-\ \ n~2

S<m>(χ) = £ n(n - 1) . . . (n - m + l)ûnxn-m;

this also holds for an end point of the segment at which the series is convergent.

Taylor series. Different forms of the remainder term. A function expressible as a power series in its interval of convergence has derivatives of all orders inside the interval. The series itself is the Taylor series of the function.

If the function is expanded as a Taylor series in the neighbourhood of the point x0,

/ ( * )= Σ*η(*-*ο)Λ (3.106) n«o

in this neighbourhood, where

„ / ( n W 72!

(0! = 1). (3.107)

The particular case of a Taylor series when x0 = 0 is sometimes called a Maclaurin series. As follows from (3.106) and (3.107), it has the form

/(*) = Σ *n*n> (3.108)

where /<n>(0)

fl = r — . n\

On confining ourselves to a finite number of terms in a Taylor series, a function f{x) can be written as a partial sum sn of the series plus a remainder term rn:

/ (* )= Σ %(*-*<>)*+ Σ β*(*-*ο) = *η(*) + ^ι(*). (3.109)

When * -* x0, the remainder term rn(x) is an infinitesimal of order higher than n (compared with x—x0). There are various famihar forms of the remainder term of a Taylor series.

The form of Schlömilch and Roche:

nip

Here /?>0, 0 < 0 < 1 On assigning concrete values to ps we get more specialized forms

of the remainder term: Lagrange's form (p = 72+1):

*»(*) = / < η + 1 ) ^ + ^ ~ Χ ο ) ] (*-s0)"+i (O<0<1) . (3.111)

r (je) = i V*T"\* ^oyj ( 1 _ 0 ) η + ι - ρ ( χ _ j ^ n + l . ( 3 . 1 1 0 ) *7 I f

SERIES 125

Cauchy's form (p = 1):

{χ) = /(η+1)[Χο+θ(χ-Χο)] (1 _ 0 ) n (^_Xo)n+i (0 < 0 < 1).

(3.112) In spite of the indeterminate size of 0, these forms of the remainder

term enable us to estimate the accuracy of replacing/(JC) by the nth degree polynomial sn(x). If the (>2+l)th derivative is boimded in absolute value in the interval between x and x0 by the number M9 we have from (3.111):

I r (χ)\ -s m ' x x° '

EXAMPLE 32. The Taylor expansion of the function f(x) = e* in the neighbourhood of the point JC0 = 0 is

where

Γη(*} = (ίΓΠ^+^Τ2)Γ+

By (3.111): ρθΧ

Here (when jc>0): v n + l

(n + 1)!· For instance, when x = 1 :

ir«(i) i<(«-+V The remainder term can also be expressed in the integral form,

which contains no indeterminate numbers:

rn(x) = ZJ n\ f<-n+1Kt)(x-t)ndt. (3.113) * 0

We can write, in the above example:

Φ ) = „f é(x-t)ndt.

Notice that a Taylor series may be convergent, and not represent its generating function /(*).

EXAMPLE 33 (Cauchy). Let φ(χ) be a given function, expressible in the neighbourhood of x = 0 by the series

Ψ(χ) = <ρ(0) + ψχ+'ψχ>+.... (3.114)

We add to it the function ip(x) = e~llxZ. If we complete the definition of φ(χ) by putting ψ(0) = 0, both y>(0) and all its derivatives ψ{η) (g) vanish, so that the Taylor expansion of ψ(χ) in the neighbourhood of x = 0 is identically zero. But now:

/(*) = ΨΜ+ΨΜ = φ(θ)+^χ+^*+ . . . . (3.115)

It is clear from this that equality of the expansions of the functions (3.114) and (3.115) does not allow us to deduce the equality of the left-hand sides: f(x) ?* <p(x).

In general, if two functions f(x) and φ(χ) are equal at x = x0 and all their derivatives are equal, while at the same time the functions are not identically equal, their Taylor expansions are nevertheless the same in the neighbourhood of x = x0.

Various operations on power series. Substitution of a series into a series. Suppose we have the series

z=/0>) = 00 + 00 + 02?+ · · · = Σ flnJn5 (3.116)

n-o

convergent in the interval (—R9 R). The variable y is expressible in turn as a power series in x:

y = <p(x) = b0 + b1x + b2xi+ . . . = £ 6«*n, (3.Π7) n-o

convergent in the interval (—r, r). Say we want to express z as a power series in x and to find the interval of convergence of the series.

SERIES 127

On formally substituting series (3.117) into series (3.116), we get

z= ΣΑηχ\ (3.118)

where

Λ = «0 + *1*0+ fl2è0 + · · · >

A2 = a16a + fla(*? +2*0*2)+ 2Α,(*Ο6Ϊ + Α86^)+ · · ·

The question of the convergence of series (3.118) is answered by THEOREM 9. 1°. If

\b0\>R,

series (3.118) is divergent, whereas, if

the series is convergent in the interval (— Rl9 R^)9 where

Rl~ M + R-\b0\> ( 1 1 1 9 )

ρ being an arbitrary positive number that satisfies the condition ρ < r and can be taken as close as desired to r, while M is the least upper bound of the numbers \ bm \ Qm (m = 1, 2, 3, . . . ) , so that \ bm \ Qm ^ M or all m.

2°. Ifb0 = 09 series (3.118) will be convergent in an interval (—Rl9 R^9 where

3°. If the series z = £ α„γη is convergent for ally, Le. in the interval (— oo, -j- oo), series (3.118) will be convergent for \x\ < r, i.e. in the interval (—r, r).

Multiplication and division of power series. THEOREM 10. The product of the series

oo oo

f{x) = £ anxn and φ(χ) = £ bnxn9

convergent respectively in the intervals

h = (-*!, *i) and h = ( - * * *2), is given by the equation

n ~ 0

w/zere c0 = a060> 1 ■= α0^+αφθ9 c2 = Û0^2+ÛA+02&O , £*C·» #w<i ir convergent in the smaller of the intervals Il9 I2.

00

The quotient of the division of 1 by the power series 1+ ]T bnxn: n « l

/(*) = = . n-1

convergent in an interval (—r, r), is given by the series

/(*) = 1 +A±x + At*+ . . . = 1 + g Anx\ (3.120) n - l

where A1 = —èl5 Λ(2 = b\—b2, Az = —èj +20x02—63, etc. THEOREM 11. Series (3.120) is convergent in the interval (—ρ/ [ΑΓ+1],

ρ/ [M-f-1]), where 0 «<= ρ < r and ρ CÛ/Î be taken as close as desired to r, while M is the least upper bound of the numbers \ bm \qm{m = 1, 2, 3, . . . ) .

We can easily pass from this to the division of one power series by another:

00

Y a xn

ψ(χ) _ ao + a1x+a2x2+ . . . _ n-0 n

ψ(χ) bQ+b1x + b2x* + ... Ä b ^ *

We have

J g . = J-(e0 +α1χ+β2χ2+ . . . ) (1 +^1x + ^2x2+ . . . ) ,

where

ι+Σ4^ = r1

n « i - 1 ι+τ-Σ*»*" "0 1

SERIES 129

is obtained as indicated above. The expansion

is now convergent in an interval determined by the above theorems»

4. Complex series

A numerical (or functional) complex series

Σ c » (3·121) 7l» l

is one in which the terms are complex numbers (or functions):

£η = <*πΤ#η· (3.122)

The convergence of complex series (3.121) to the sum S = A+iB is equivalent to the convergence of two real series

Σ "n> (3.123) n - l

Σ *». (3.124)

to the sums A and B respectively. We have: THEOREM 12. If the positive series

Σ k™l= Σ Λ Κ + ^ . (3-125)

composed of the moduli of the terms of series (3.121), is convergent, series (3.121) is also convergent.

In this case series (3.121) is said to be absolutely convergent. D'Alem-bert's and Cauchy's tests retain their validity for complex series. The theorem on rearrangement of the terms of a series and the rule for term by term multiplication of series can be carried over of absolutely convergent complex series. All the theorems on absolutely convergent real series retain their validity for absolutely convergent complex series.


Functions of a complex variable. If, for every value of a complex variable z from a domain Z in the complex plane, there is a corresponding single value of another complex variable H>=M+iv, W is called a complex function ofz in the domain Z and we write

w=/(z) . (3.126)

If the function/(z) is expressible by a Taylor series in the neighbourhood of the point z0, i.e. can be expanded as a convergent power series in powers of (z—z0)> it is called an analytic function ofz at the point z0. If/(z) is expressible as a power series in the neighbourhood of any point of the domain Z (open or closed), it is said to be an analytic function ofz in the domain Z. Functions having this property are of the greatest interest in applied mathematics.

The complex power series

f cn(z-zor (3.127)

has a number of properties similar to those of real power series. We can consider in future, without loss of generality, the series

Σ V n · (3.128)

Since this series has a circle of convergence instead of an interval of convergence, the term radius of convergence acquires a strict meaning here.

If the coefficients cn of power series (3.127) are real numbers, the radius R of the circle of convergence coincides with the previous radius of convergence.

The following proposition holds: if the series (3.128) is convergent at some point z0 of the circumference |Z|=JÊ, then as the point z approaches the point z0 from inside the circle along a radius, we have

OO CO

lim £ cnzn = £ cnzl.

A complex power series can be differentiated term by term inside its circle of convergence.

If a function is expanded, as a series in powers of z, the distance

SERIES 131

from the origin (z = 0) to the nearest singulart point of the function is equal to the radius of convergence of the sum of the series.

EXAMPLE 34.

_ L _ = 1_ 2 2 + Z 4_ Z 6 + mmm + ( _ 1 ) n z 2 n + . . . = f (-1)»Z2*. l-f-z n = 0

(3.129) If we take the real axis z = x, the expansion

—i-y=: l - x 2 + jc4-x e+ . . . (3.130) 1+jc

has a radius of convergence R = 1, although the function 1/(1 +*2) and its derivatives have no discontinuities on passing through the points ±1. On returning to expansion (3.129) in the complex plane, we see that the function 1/(1 + z2) has discontinuities at the points ±/. This is in fact the reason for the divergence of expansion (3.130) for |JC|» 1.

If we introduce the notation cn = an+ibn9 (an9 bn are real numbers), \z\ = r, arg z = 0, series (3.128) can be written in the form:

= ]£ rn(örn + i6n)(cos/ÎÔ-f / sin/2Ö) = n « 0

= · £ rn(an cos ηθ - 6n sin ηθ) + i £ rn(on cos ηθ + αη sin «0). n-o π«ο

Further, we have, on writing anrn = An9 —bnrn = i?n: oo oo oo

£ cnzn = J] (AnCQsnd + BnûnnQ) + i £ ( — 2?ncosH0-Masinw0).

The real and imaginary parts of the series are thus written as trigonometric series.

t A singular point is one in the neighbourhood of which a function is not expressible as a power series.

EXAMPLE 35.

T l 7 = f z» (r=|z |<l ) ; A " ~ Z 71*0

1 1 1 - r cos 0 + 1-z 1 — (r cos 0-f/r sin0) 1 — 2rcos0+r2

+ *1 9 ^ β - ^ = Σ ** COS W + / £ 1*8111*0, 1 - 2r cos 0 + r k% fc-i

from which it follows that

1 —rcos0 ^ . ΤΛ l - 2 r cos0+r 2

fctr0

r sin , ' ;—2 = y r* sin *e· l -2 rcos0 + r2 jf t

5. Trigonometric Fourier series

A function/(x) having a periodic (f(x)=f(x+2nn), n= ± 1, ±2,...) can be expressed, under certain restrictions to be described below, as an infinite trigonometric Fourier series

Û °° /(*) ~ -T + Σ (an c o s nx + bn sin nx). (3.131)

The Euler-Fourier formulae are used to find the coefficients of series (3.131): ,

1 f* π J-« :) cos «* dx (Λ = 0, 1,2, ...)>

/(*) sin nx dx (n = 1, 2, 3, ♦..). (3.132)

In order that the Fourier series (3.131) be convergent to the given function f(x\ i.e. for its sum to be equal, at every point x0 of the interval (0, 2π), to the value f(x0) at this point, the function f(x) must satisfy certain conditions (see below, and Chapter IV, § 2, sec. 5).

t If the period of/(*) is equal to 2L, it can be transformed to the period 2π by bringing in the variable y=nx/L, so that all subsequent discussions retain their generality

SERIES 133

THEOREM 13 (Localization theorem). The behaviour (i.e. convergence or divergence) of the Fourier series of the functionf(x) at somepoint Xo depends exclusively on the values taken by the function in theimmediate vicinity of the point xo.

The sum So of the series (3.131) for f(x) at Xo is determined asfollows:

(a) when f(x) is continuous at Xo, we have

So =f(xo);

(b) when f(x) has a disf?ontinuity of the first kind at Xo (so thatthe limits f(xo+O) and f(xo-O) exist), we have

So =!(xo+O)+!(xo-O).2

We say that I(x) is smooth in the segment [a, b] if it has a continuous derivative in this segment.

A continuous function f(x) is described as piecewise smooth inthe segment [a, b] if the segment can be split into a finite number'of subsegments, on each of which I(x) is smooth.

A discontinuous f(x) is described as piecewise smooth on the segment [a, b] if: (1) it has only points of discontinuity of the first kind(and only a finite number of these) on the segment, (2) on each of thesubsegments [ex, ,8] into which the original segment is split by thepoints of discontinuity, the continuous function

{

!«(%+O)g(x) = [(x)

f({3-0)

for x = (%,for (% -< x <: {3,for x = {3

is piecewise smooth.A convergence test for Fourier series. The Fourier series of a piece

wise smooth (continuous or discontinuous) function/(x) ofperiod 2nis convergent for all values of Xo, its sum being equal to So.

If a piecewise smooth function I(x) is continuous everywhere, itsFourier series is absolutely and uniformly convergent.

THEOREM 14. Let I(x) be an absolutely integrable function ofperiod27&, continuous and possessing an absolutely integrable derivative insome segment [a, b] {the derivative may cease to exist at individual

points). The Fourier series is then uniformly convergent to f(x) in every segment [α+δ, ό— δ] (δ > 0).

This theorem holds, in particular, for an absolutely integrable f(x) of period 2π, continuous and piecewise smooth on the segment

EXAMPLE 36. The function/(x) = - I n 2 sin — x 2

, which becomes

infinite for x =2far (k = 0, ± 1 , ± 2 , . . .)> has period 2π. Its Fourier series is

cos 2x cos 3x = COS* H - 1 r h . . . . - I n 2 sin—-

2 THE DIRICHLET-JORDAN TEST. The Fourier series of a function

f(x) is convergent at x0 to the sum S0 if the function is of bounded variation in some segment [x0— δ, χ 0 + δ],

A more specialized statement is: DIRICHLET'S TEST. Iff(x) of period 2π is piecewise monotonie

on a segment [—π, π] and has a finite number of discontinuities there, then its Fourier series is convergent to the sum f(x0) at every point

of continuity and to the sum S0 = — [f(xo+0)+f(x0—0)] at every

point of discontinuity (see Chapter IV, § 2, sec. 5). DINI'S TEST. The Fourier series oj a function f(x) is convergent at

a point x0 to the sum S0 if given some h > 0, the integral

/ •dt

exists, where <K0 = /(*o + 0 +/(*o - 0 - 250.

The following is a particular case of Dini's test: LIPSCHITZ'S TEST. The Fourier series of a function f(x) is conver

gent at a point x0 which it is continuous to the sum S0 = f(x0) if given sufficiently small t>~ 0,

where L and a are positive constants (a ^ 1). In particular, piecewise differentiable functions are suited to this

test.

SERIES 135

The Dini and Dirichlet-Jordan tests are not consequences of one another.

In the case of a function /(*), only specified on the semi-interval (—7T, π), or defined outside it but non-periodic, application of the above theory is made possible by replacing f(x) by an auxiliary funct ion/*^) , having the following properties:

/ * ( - π ) = / * ( π ) ,

while f*(x) is extended to the remaining real values of x in accordance with the law of periodicity.

All the above theorems and propositions are applicable to the function f*(x). The series represents the function/(x) inside the interval.

This construction is not required if /(—τι) = /(π). In this case the Fourier series is convergent to f(x) throughout the open interval (excluding the ends). But outside the interval, the sum of the series in general no longer coincides with/(;c), assuming the latter is defined throughout the real axis.

On bearing in mind the footnote at the start of the present section, we can say that a function specified in any manner in any interval can be expanded as a trigonometric series in a very wide class of cases (this includes piecewise differentiable and piecewise monotonie functions) (see also Chapter IV, § 2, sec. 5).

Expansions in sines only or cosines only. The Fourier series of an even function /(*) = (/(—*)) contains cosines only:

a. °° /(*) ~ "T-+ Σ ancosnx.

^ n= l

The Fourier series of an odd function (fix) = —f(—x)) contains sines only:

oo

/(*) ~ Σ bn sin nx-71« 1

Every function/(x), specified in the segment [—π, π], can be written as the sum of an even function fx(x) and an odd function f2(x):

/ W = / i W f / 2 W ,

where the component functions are

If a function is only specified in the segment [0, π], we can define it arbitrarily in the interval [—π, 0] and thus obtain different Fourier series, which will have a sum S0 at a point x0 between 0 and π, the sum

being convergent either to/(x0) or to —[/(x0+0)+/(;c0--0)] (in the

case of a discontinuity). If we complete the definition of the function in (—π, 0) as an even

function, i.e. put/(—x) =/(*), we get an expansion in cosines only; and if we complete it as an odd function (/(—x) = —/(*)), we get an expansion in sines only.

EXAMPLE 37. f(x) = x (0 ^ x ^ π). (1) Let f(x) = / ( - * ) = - x (-π*£ x < 0); then

~ x n 4 / cos3x cos5x \

(2) Let /(JC) = - / ( - * ) = JC ( -π < x < 0); then

~ N „/sin* sin2x sin3x \ f(x) = 2(— — + — . . . J . \ T ~ 2 + ~ T

The order of the Fourier coefficients. If a periodic function /(*) has a finite number of discontinuities of the first kind (in a period), its Fourier coefficients will be infinitesimals of the form 0(1 jn) as n — oo (see Chapter 1, § 3, sec. 8 and 15).

This means that

n\an\^M, n\bn\^M,

where AT is a constant positive number independent of n. If the periodic function f(x) is continuous everywhere, its Fourier

coefficients will be infinitesimals of higher order than 1/« as n -*· oo, i.e. of the form o(l/n) (see Chapter I, § 3, sec. 8 and 15).

This means that lim nan = lim nbn = 0.

SERIES 137

If a periodic continuous function has continuous derivatives everywhere up to and including the (m— l)th order, its Fourier coefficients an and bn will be of higher order than l/nm as n -* «>, i.e.

lim nman = lim nmbn = 0.

In particular, if a periodic function f(x) has continuous derivatives of any order everywhere, its Fourier coefficients an and bn satisfy the condition

nman -* 0, nmbn »* 0

as n -+ », independently of m. Integration of Fourier series. Let us first consider a continuous

function /(JC), specified everywhere on Ex. THEOREM 15. If an absolutely integrable function f(x) is specified

by its Fourier series:

/(*) ~ττ+Σ (°n c o s nx + bn sin nx\ (3.133) 2 n - 1 fb

we can find f(x)dx by term by term integration of series (3.133),

independently of whether the latter is convergent or not, i.e. i rt \ j ao /!_ Λ ^ an(sin nb — sin na) — bJcos nb — cos na) f(x)dx = -f(b-a)+ £ -2i ί ^ \

(3.134)

THEOREM 16. Lei a/z absolutely integrable function be specified by its Fourier series (3.133) (convergent or not). The following Fourier expansion now holds for its integral:

i: f{x) dx= f 2- + V - K cos nx + fa + ( - l)n+^0) sin nx n - l Λ n s l 72

( - π < χ < π ) . (3.135)

A particular case of this theorem is when a0 = 0 (the other conditions are retained); we now have for all x:

ί f(x)dx = Σ - f Σ -*" c o s ** + a " s i n n * . (3.136) 0 n « l « n= l Λ


Differentiation of Fourier series.

THEOREM 17. Let f(x) be a continuous function of period 2π, possessing an absolutely integrable derivative (which may not exist at a finite number of points). The Fourier series for f\x) can now be obtained from the Fourier series (3-133) forf{x) by term by term differentiation:

oo

/ '(*) ~ Σ n{bn cos nx—On un nx). (3.137) n«=i

THEOREM 18. Let f(x) be a continuous function of period 2π, possessing m derivatives, the first m—\ derivatives being continuous, and the m-th absolutely integrable (the m-th m-ay cease to exist at a finite number of points). Now: (1) The Fourier series of all m derivatives can be obtained by term by term differentiation of the Fourier series for f(x), all these series, except possibly the last, being convergent to the corresponding derivative; (2) the following relationships hold for the Fourier coefficients of f{x) (see above):

lim nman = lim nmbn = 0. (3.138)

The series for f(x) and all the series obtained from it by term by term differentiation (except possibly the last) are uniformly convergent in this case.

This theorem has the following converse. THEOREM 19. Given the trigonometric series

a °° ~T + Σ (an c o s nx + bn sin nx), (3.139)

if the following relationships hold for the coefficients an and bn, I nman | ^ M, \ nmbn | ^ M(m ^ 2, M = const),

then the sum of the series is a continuous function of period 2π, possessing m — 2 continuous derivatives, which may be obtained by term by term differentiation.

THEOREM 20. Let f(x) be a continuous function specified on the segment [—π, π] and having an absolutely integrable derivative (which may cease Jo exist at a finite number of points). Then

fix) ~ 4 + Σ \inbn + ( "" ^^) C0S nX "" mn sin ηχί> (3· 1 4°) 2 n=l

SERIES 139

where an and bn are the Fourier coefficients off(x), while the constant c is given by the equation

* = ^ [ Λ * ) - Λ - * ) ] . (3·141)

THEOREM 21. Given the series a °° "T + Σ (ön c o s nx + n s i n nx)> (3.142)

// iAe jene,? c °°

T + Σ ΚΛ*η + ( — 1)W0 c o s w* — wan SÌn Λ*1> (3-143) 2 n- l wAere

c = li

m [(-l)n+1nb

n], (3.144)

w the Fourier series of an absolutely integrable function φ(χ)\,

then the series (3.142) is the Fourier series of f{x) = φ{χ)άχ ·+■ J oo Jo H α0+ ]Γ αη, continuous for —π < Λ: < π, aw/ w convergent 2 nml

io /Aw function, while obviously f\x) = φ(χ) at all the points of continuity of y{x).

THEOREM 22. Given the series

T + Σ ( - Dn («n c o s "x + bn s i n **λ (3·145) ^ n - l

wAere an uf/7i/ όΛ are positive, if nan, nbn do not increase (as from a certain n) and tend to zero as n -+■ °°, the series is convergent for —π < x < π and has a differentiable sum f(x), where

oo

/ ' (*) = Σ ( - 1)ηΦη c o s nx-an sin nx), n=»l

/.e. jer/e? (3.145) can be differentiated term by term. Similar theorems can be stated for functions specified on the seg

ment [0, π] (see for example ref. 10). Further information on trigonometric series will be found in

Chapter IV.

t We do not assume the convergence of series (3.143).


6. Asymptotic series

If a function F(x), defined for x ^ x0, is to be considered for large values of the argument x, it is sometimes useful to find a function S(x) of simple structure such that

R(x) = F(x)-S(x) - 0 as x -* «>.

In this case F(x) can be replaced by S(x) for large x. If i^*) admits of the expansion (for x > x0) :

^ = Λ - ^ + . . . 4 Φ + . . . = Σ 4 > (3.146) we find, on putting

and

that

fc«=0 Λ Λ Λ

Mx) = F(x)-sn(x) = Σ 9 = ΐ & ΐ + · · · . (3·148)

lim x"Rn(x) = 0, or *„(*) = o ( -3 ] , (3.149) :*> = ' ( ? ) . i.e. Rn(x) is an infinitesimal of higher order than n.

If F(x) does not admit of an expansion (3.146), it is still sometimes possible to choose a series of the form (3.146) such that condition (3.149) is satisfied for any fixed n.

Series (3.147) is said to be asymptotic for F(x\ and we write

*"(*)- Ë ê · (3-150)

Series (3.147) can be divergent, but it is still useful since it yields the approximate formulae

F(x)^A0+^+...+^,

the degree of the approximation being indicated by (3.149). EXAMPLE 3S. Let us consider the function

™=1>'τ·

SERIES 141

Repeated integration by parts gives: 1 1 V in— IV

F(jc) = Ί-ϊ?+Ί?~ · · · + ( - 1 ) n - 1 ^ L + * n W · where

i; Rn(X) = (-\)nn\[~Ç^dt9

e*-1 . e*-* dt = -/ n + 1 ί η + 1 - (Λ + 1)

| ί - χ

K-t

ί η + 2 Ä *·η+1>

consequently,

Ì*n(*)l r n + l

and condition (3.149) is satisfied· Thus

f W „ I _ ^ + «_.. .+ (_,r-ifc=W+.. . . jc x2 ΛΤ * Λ

This series is divergent, since

hm 0 n = hm ( - ì y i L ' - = «>. 71—> oo 71 —> oo X

If F(x) admits of an asymptotic expansion, this latter is unique. The asymptotic expansion of the product of two functions F{x)

and G(x), each of which has an asymptotic expansion

F(x) ~ Σ Akx~\ G(x) ~ X B&r\ fe=0 fc«0

(3.151)

is formally equal to the product of asymptotic expansions (3.151):

F(x)G(x) ~ Σ Λ * ~ * Σ 5ft*~" = Σ CkX-k, (3.152)

where

Ä - 0 (3.153)

If the asymptotic expansion (3.150) of F(x) starts with the term A2x~2

9 it can be formally integrated term by term in the interval from x to + » , i.e.


Formal differentiation of asymptotic expansion is not permissible. If F(x) has an asymptotic expansion with no free term (A0 = 0 and

F(x) — 0 as x -» oo), this expansion can be exponentiated

eF(x)= 1+ Σ i [ F M ] m ~ m=l ml

~ , + # + · · · + [ 5 + · · · + τ ΐ · ] ? + · - (3·154> The asymptotic expansion is thus a source of approximate formulae

for computing a function for large values of its argument, the accuracy of formula (3.147) being the greater, the greater the value of the argument.

There exist functions, not identically zero, for which all the coefficients Ak in the asymptotic form (3450) are equal to zero. Such functions are described as asymptotic zeros. An asymptotic zero is any function F(x), for which

for any n. For example, F(x) = e~x is an asymptotic zero. Addition of this function to the left-hand side of (3.150) does not change its right-hand side.

Not every function F(x), defined in the semi-interval [a, + «>], admits of an asymptotic expansion (3.150), but if such an expansion exists, it is unique for the given F(x) (the coefficients are uniquely defined). On the other hand, there exists for any sequence of numbers {Ak} a function F(x), for which the Ak are the coefficients of an asymptotic expansion (3.150). This F(x) is not uniquely defined, however (to an accuracy of an asymptotic zero).

7. Some methods of generalized summation of divergent series

Certain classes of divergent series, i.e. series lacking a sum in the ordinary sense of the word, are capable of generalized summation.

The definition of generalized sum usually satisfies two conditions:

1°. The linearity condition. If the generalized sum A corresponds to the series ][]tfn, and the generalized sum B to £ £ n , the series

SERIES 143

Σ(ρ0η+4*η)> w^ere p and q are arbitrary constants, must have a generalized sum equal to pA + qB.

2° . The permanence condition. If a series is convergent in the ordinary sense to the sum A, it must also have a generalized sum, equal to A.

The general plan for constructing linear permanent methods of summation is as follows.

Let ?<>(*). 7i(x)> · · · > 7η(*)> · · · (3.155)

be a sequence of functions given in a domain X of variation of the parameter x, and let X have a point of condensation — the finite or improper number ω. We construct, in accordance with the numerical series

Σ <*n, (3.156)

the functional series

f an7n(xl * (3.157)

If this series is convergent, at least for x sufficiently close to ω, and its sum S(x) tends to a limit A as x -*· ω, this number A is in fact taken as the generalized sum of series (3.156). To ensure that the method has permanence, two conditions are imposed on the functions γη(χ):

(1) lim γη(χ) = 1 (n = 0, 1, 2, . . . ); (3.158)

(2) for all x g X:

I JO(*)|+ Σ I ?η(*)-7η-ι(*) I ^ Ü:< oo (* = const). (3.159) n - l

We shall next consider two methods of generalized summation, embraced by the general plan just described.

1°. The method of power series (Poisson). In accordance with the given numerical series (3.156), we form the power series

Σ anxn = fl0 + öi*+ . · - +αηΧη+ ~l (3.160)

if this series is convergent in the interval 0 < x < 1 and its sum S(x) has the limit A as x -*- 1 :

lim S(x) = A, n - * l - 0

the number A is called the generalized sum (in Poisson's sense) of the given series (3.156).

This method follows from the general scheme if we put

[X = (0, 1), ω = 1, γη(χ) = xn (n = 0, 1 ,2 , . . . ) . ;

EXAMPLE 39. Let us take the series discussed by Euler:

1-1 + 1 - 1 4 - 1 - 1 + (3.161) It has no sum in the ordinary sense, since sn oscillates between 0 and + 1. The corresponding power series (3.160) has the form

71-0

For 0 < x < 1, its sum (as the sum of an infinite geometrical progression) is equal to

S(x) = y ^ ; (3.162)

the generalized sum in Poisson's sense of series (3.161) is

A= lim 7 ^ - = y . · (3.163) o c - * 1 - 0 A τ* z


Σ sin ηθ (~π ^ 0 =£ n) (3.164) n « l

is convergent only for 0 = 0 and 0 = ±π. The corresponding power series

£ xnsinn0 (3.165) n-=l

has the sum, for 0 < x < 1 :

x sin 0 S(x) = 1 - 2x cos Θ + *2

SERIES 145

The generalized sum is thus equal to _ . xs in0 _ sino _ 1 0

A * χ ^ ο 1 - 2 χ α ) 8 β + χ » " 2(1 - cosò) "" T C O t a n T ·

(3.166)

2°. ΓΑβ method of arithmetic means (Cesàro). We take the partial sums sn of the given numerical series (3.156):

s0 = a0, sx = a0 + al9

*2 = ^ο+^Ι + ^ · · · > «*π = Σ ak fc~o

and form their consecutive arithmetic means:

(3.167)

^1 -"* 0» ^2 ~~ ^0 + 1 , An — •y0 + ' y l+ · · · -Mn-1 (3.168)

If the sequence {An} has a limit A as Λ -*· <», we call -4 the generalized sum (in Cesàro's sense) of series (3.156).

This method follows from the general plan if we put

X = n, ω = + h for m = 0, 1, . . . , «— 1,

riOO = » ( 0 for m^ n.

EXAMPLE 41. Let us return to series (3.164). When φ τ* 0:

cos —0 —cosi H-f-y 10 *n =

2s in~0 2 /i 1 Λ sin (n + 1)0 — sin 0 ηΑη = y cotanto * ^ ,

4 sin2 y 0

A = lim Λ(η = — cotan—.

n—► «> 2 2 We have thus obtained a generalized sum in Cesàro's sense equal to the generalized sum in Poisson's sense.

The inter-relationship between the Cesàro and Poisson methods may be explained by quoting the following theorem :


THEOREM 23. (Frobenius). If a series is summahle in accordance with the method of arithmetic means to the finite "'sum" A, it is simultaneously summahle by the method of power series, to the same sum A.

Notice that Poisson's method is applicable to a wider class of series than Cesaro's method, though it does not contradict Cesàro's method when both are applicable.

§ 3· Methods of calculating the sum of a series

We shall discuss in this section some methods of finite summation of series, some estimates for series, finite sums and products, and the problem, of importance for practical computation, of improving the convergence of series, i.e. methods enabling a given series to be replaced by another, having the same sum but more rapidly convergent.

1. Elementary methods of exact summation

It is very rarely possible to sum exactly an infinite series obtained as a result of solving some problem. Some simple methods of strict finite summation are given below.

oo

THEOREM 24. If the terms an of the series £ an can be written

in the form an = 6n—bn+1 and if the bnform a sequence {bn} having a limit a, the sum of the series is

S = g an = b0-a. (3.169)

EXAMPLE 42. Suppose we have the series

n-iy/n(n+l){y/n+y/n+l) here

1 1 1 y/n(n+l){y/n+ N/w+l) y/n χ/fl+ï '

and lim l/y/n = 0, so that S = 1 (b0 = bx = 1).

SERIES 147

THEOREM 25. If the terms of the series £ an are expressible as 71-0

*n = «A+i 4-α2όΛ+2+ . . . ι-αρ*η+ρ. (3.170) where p is some fixed positive integer ^ 2, bnform a sequence having a limit a, and oci are numbers satisfying the condition

αχ+α2+ . . . +«P = 0, (3.171)

the series in question is convergent, and its sum is equal to

S = 0 ^ +(£1+02)62+ . . . + (αχ+α 2 + . . . +ap_1)ip_1 f

+ (02+203 + 3*4 + 0?- l)ap)a. (3.172) EXAMPLE 43. Given the series

« 4/1 + 6 nt^o(Λfl)(Λ + 2)(Λ + 3 ) ,

we have 4n + 6 1 _ 2 3 ^

( Π + 1 ) ( Λ 4 2)(Λ + 3) "" n + l + n + 2 Λ + 3 '

Consequently, ax = 1, a2 = 2, a3 = — 3. Condition (3.171) is satisfied : 1 + 2 - 3 = 0. In addition, bn =

= l / n - 0 a s f l - 00. We now find, by (3.172):

S = l . l + ( l + 2 ) i = j .

THEOREM 26. If the terms of the series £ an con 6e written as n-o n = bv — bn+q,

where q is a positive integer, while the sequence of numbers bQ, bl9 b2,... has a limit ξ, the sum of the series is

00 s = Σ *n = *o + *i+ ··· + V r ^ ·

n-o Tables of series expansions of functions and the relevant sections

of reference works may be used for the finite summation of series. Given a particular series, it should be verified whether or not it is to be found in a reference book, or whether it can be transformed to a familiar series by a change of variable or some other means.

EXAMPLE 44. Suppose we have the series

Y kpk sin foe, | p | < 1. fe«l

We find in ref. 7, formula 1.447 3, which we can write as

£ ph cos kx = — -— 7 ^ v ■ n2—T ( l ^ 1 ^ ^ · ^Ξι 2 1 — 2/7 cos x +/r 2

The uniform convergence of this series and the series of the derivatives of its terms with respect to x follows from comparing them with the majorant series ]P |/?|ftand £ k \p\h respectively. On differen

t i fe«l tiating both sides of the equation with respect to x, we find the sum of the series (see § 2, sec. 1):

v2« , h · i P(\ — P2) s i n * ,x . t .

2. Summation of series with the aid of functions of a complex variable

If the trigonometric series

^ + f tfncos*x==yi(x), (3.173)

oo

£ tfn sin /zx = f2(x) (3.174) n«i

oo

have positive coefficients an9 and the series ]Γ αη//ι is convergent, n«l

(3.173) and (3.174) are the Fourier series of continuous functions. We can sometimes find the functions /λ(χ) and f2(x) — the sums

of series (3.173) and (3.174) — by making use of functions of a complex variable.

Let

2 n-1

be the sum of a series convergent in the circle \z\ < 1. If lim <p(z)= Ul-*i

SERIES 149

= lim φ( /Ό = (pie**), then/i(x)+//2(*) = <P(el% i-e- after separating

the real and imaginary parts of <p(0> we have:

Λ(χ) = Κε{φ(β*)}; (3.175)

/ 2M = I m { ^ ) } . (3.176)

EXAMPLE 45. Let us establish the convergence and sum of the series

, COS* COS2* Λ ^ COSHX , . ,__. 1 + - ί Γ + ^ Γ + · · · = 1 +

η Σ - 1 Π - . (3.177)

^£+^£+. . .»1^ . (3.178)

The convergence of series (3.117) and (3.178) follows readily from a comparison with the series

oo 1

i + Σ - - = ^

which is majorant for the given series. Furthermore, we put oo - n

7 1 - 1 W 1

On expressing z in the exponential form z = re** and letting r tend to 1, we have

e** = £co8 *+*sln * = ecos «[cos (sin JC) +1 sin (sin x)] = (cos x -f / sin x)n

= 1+Σ „, cos x / · \ , · cos'* · / · \ . . yZ COSilX . $ &UMX

ecosxcos (smx)4-ieC0Sxsm(smx)= 1+ £ —■ |-i £ — j — n * l Λ! n - l # '

Hence,

1 + Y ^ Γ ^ = e*°s x c o s (sin *\ (3.179)

0 0 βΪΤΊ MY J ì i ^ l = ecos « sin (sin x). (3.180)

n-«l n!

3. Summation of series with the aid of Laplace transforms

Definition of the Laplace transform. Suppose that, given any x>0, the modulus of the function φ(χ) increases more slowly than some exponential function i.e. there exist numbers M and s0, not depending on x, such that, for any x9

\(p(x)\*cMe*ox. (3.181)

Let p = s+ia be a complex number. The integral

a(p) = f ~ <r P*<p(jt) àx (3.182)

now exists and has derivatives of all orders in the half-plane Re/> >s0 . The integral (3.182) is called the Laplace transform of <p(x). In the accepted brief terminology, <p{x) is the pre-image, and a{p) the image.

Tables have been compiled for the pre-images and images of many functions (see for example ref. 7).

Summation of numerical series (ref. 9). If, given the convergent infinite series

S = £(±l )*e(*) (3.183)

its terms are images of a pre-image φ(ξ), i.e.

a(k) = Γ e-^(e)de9

the following formula holds (preserving the correspondence of the ± signs):

f ( ± !)**(*)= ± Γ 2 τ Ξ Γ Γ · (3·184)

EXAMPLE 46. Knowing that

we have by (3.184):

^ 1 1 C°° sin αξ αξ π Λ π Σ 72 ■ 2 = "" —t—Γ~ = -^-cotanhjra—=-.

SERIES 151

Generating functions. If the terms of the sequence {fk(t)} (k = 1, 2, 3, . . .), defined in some interval a <*</?, are the coefficients or the Taylor expansion of a known function F(x, t) in powers of x fof |;c| < 1, i.e.

F(x, 0 = f **Λ(0 (3·185)

f(x,0 is called the generating function of the sequence {/Ä(0}· For example, one of the following sequences can be taken as {fh(t)}\

(a) the sequence of powers /, r2, i 3 , . . . , where

F{*> t) = τ^Τ7 s Σ Λ*· <3·186> 1 - x* Ä-i and | xt | < 1 ;

(b) the sequence of trigonometric functions

or

where

and

sin t, sin 2/, sin 3f, . . .

cos t, cos 2/, cos 3/, . . .

I- / «Λ x sin f ^ fc . . Λ 1V y l - 2 x c o s i + xa Ä-i

F2(x, t) = -—r 5 = Y xk cos to. 2V ' l - ^ c o s i + x 2

Ätii

(3.187)

(3.188)

where / can vary in any interval and | x | -< 1 ; (c) the sequence of Chebyshev polynomials Tx{x\ T2(x), Tz(x\ . . . ,

for which the following relationship holds:

* * ' > ^ ^ s r ? - j l · ^ (3·189)

where \t\ ^ 1, |x | < 1; (d) the sequence of Legendre polynomials Ρλ(ί), Ρ2(0> Λ(0> · · · ,

where

F(x, 0 = * - 1 = £ x*Pft(f), (3.190)

for |f |^ 1 and IJCI <s 1.

It should also be noticed that the infinite series (3.186)-(3.190) are convergent for x = ± 1 or oscillate between finite limits (i.e. the sum of the first m terms remains less in absolute value than some constant independent of t and m).

The summation of functional series (ref. 9). Suppose that the generating function F(x9 t) is known for an infinite sequence of functions {fk(t)} (k = 1, 2, 3,. . .) (see above). Let the series

S(t, *)= £ a{k)xhfh{t) (3.191) fc=l

be convergent in some interval a < t < ß and for | x | 1. If the coefficients a[k) of series (3.191), considered as functions of the index k9 are images of some φ(ξ)> we have for the sum of series (3.191):

S(t, x) = [~q>(S)F{xe-t9 t)ft9 (3.192) - J : EXAMPLE 47. Knowing that the fraction l/(fc-(-/) is an image of

e~(l for any / s 0 and k > 0, i.e.

1 Γ*

* + '"Ί. and using formulae (3.187), (3.188) and (3-192) for positive integral /, we have: _ . S xk sin kt

= V"M Σ ^ - h ^ Σ CU-D r x r s i n ( r -O i + sin// -( m«l "* |_r«o J In (1 - 2x cos * + *») +cos ft arc tan t

XSmt X ; (3.193) v y l - x cos ί j v y

= x

sin/i 2

xh cos fc? *(*.*) « £ k + l

= *~M Σ l J Σ Qi(-l)Voos(r-/)i-oos[ft -

cos//, / t . ox . T . xsint 1 , . <Λ„Ν — In (1 - 2x cos f+x2) -f sm // arc tan - V . (3.194) Z, 1 —" X COS Γ I

SERIES 153

Formulae (3.193) and (3.194) remain valid for any / and | x | 1, with the exception of points of discontinuity (ref. 11), which may make their appearance for the trigonometric series with x = ± 1.

On making use of generating functions (3.186), (3.189), (3.190), as also (3.192) with positive integral /,' summation formulae similar to 3.193) and (3.194) can be established for the series

^ JÏ_ ^ xkTk(t) ~ xkpk(t) Atk+r & k+i a n a

kèi k+i

4· Integral estimations for finite sums and infinite series

Eulefs formula. If a function a(x) has derivatives up to and including the nth order for m x ^ k (m and k are positive integers), the following formula holds, due to Euler:

~ Γ Γ η ί Ο * ? ^ ( T + l - O ^ t , (3.195) nl J o *-™

where Yn{t) = Bn(t)—Bn, Bn are Bernoulli numbers, Bn(t) are Bernoulli polynomials (see Chapter VI).

First integral estimate. We suppose that a(n) (x) ^ Ott in the interval q^x^p+l(p and q are positive integers). Let Mn denote the greatest, and mn the least, value of Yn{t) in the segmento ^ / ^ 1. The following inequality now holds:

^ *i(p + l, ff)-^l [^-«(p + l ) -^»-! )^) ] , (3.196)

n - l fWhen /i = l we have to assume £ (5„/W) [a0 ' - 1^)-«^1^»!)]—0; in

» - 1 addition, it is assumed here and throughout what follows that /<0)(*) = i? »

ft If e w W s O , the inequality signs of (3.196) have to be reversed.

wheret fp+l n - l Ώ

*η(Ρ+1, q) = *(0Λ+ Σ ^rl^Kp + V-a^Kq)].

Let us quote some values for Mn and mn:

Λ/J = 1, ml = 0; Af2 = 0, m2= j ; Λ/3 = ^ - , m3 = ^ - ;

Λ/4 = 1 , w4 = 0; Λ/5 % 0-0244582..., mb * -0-0244582 . . .

It can be shown that

M2i = Y2l ( y ) > 0 and m2i = 0, if / i s even

M2i = 0 and m2X = Y2l ( y ) < 0, if / i s odd.

If we make the following assumptions: (a) a{ri)(x) ^ 0 for q ^ x < oo; (b) lim 0(m)(*) = 0 for m = 0, 1, 2, . . . , n- 1;

(c) the integral o(x)dx is convergent,

we now get, by (3-196), as p -* «>: M

S «(*) ^ *n( + - , iH^e^fo), (3.197) where

EXAMPLE 48. Knowing that the derivatives of any order of a(x) = = 1/v* are sign-definite in 1 ^ JC ^ 101, we can estimate the sum

100 ]

by making use of (3.196).

tWhen n - l , we have to assume £ ^laC-i)(p+l)-ai"-i)(q)]=Q. " - 1 Βν

SERIES 155

We have, for instance, 18-1 < 5 < 19 for n = 1;

18-55 < S < 18-61 for n = 2; 18-5890 < 5 < 18-5909 for n = 6.

Second integral estimate. If we assume that (-1)™"1 a(2m) (x) ^ Of for # x ^ /? -H1, the following inequality holds:

sim-1(p+ 1, ?) + - ^ j [α(2"ι-1>(^ + 1)-α<^-%)] s

S £ ^ έ ΐ , ^ + Ι , g), (3.198)

where

S* Je "-1 PI °2m-

If we assume: (a) ( - l)m-1a(2m)(jc) ^ 0 t for q ^ x < -h « (b) lim ßö*-1^*) = 0 for k = 1, 2, . . . , m;

(c) integral a{x)dx is convergent,

the following holds:

*m-i(+ ~. i ) - - f e e Ö m " l ) ( i ) ^ Σ fl(fc) ^ *m-l( + ~> i). (2/W)! fctg (3.199)

where

ioo 2m-2 #

EXAMPLE 49. Knowing that, with a(x) = x3'2, the following condition holds: (— l)4 aIV (x) > 0 for all x > 0, we can carry out an estimate of the sum

100

S = f fc3'2,

t If (-l)m"~Ia(2m)W ^ 0, the signs of inequalities (3.198) and (3.199) must be reversed.

2 m - 2 t t When m=1, we have to put £ (BJv\)a<*-l)(q) = Bta(g).

v»l

by making use of inequality (3.198) with m = 2. We have 40501-2260 < S < 40501-2265.

The first and second integral estimates may be used not only for evaluating finite sums and infinite series, but also for evaluating finite and infinite products.

EXAMPLE 50. Let us estimate the size of the product P = pi v

On taking logarithms, we have S = In P = £ In k.

Notice that the following conditions are satisfied for the function a(x) = In x in the interval 1 x ^ p + 1 :

a\x) = ! : > 0, a'\x) = - J L < 0 and *'"(*) = J r > 0.

On using (3.196), say with n = 3, we get

^ + 1, ΐ)_Αα'Τρ+1)-α^)]*1ηΡ*ι 216

S *,(/> + 1, 1) + ^ [*"(/> +1) - a"(l)L <3·200) where

On exponentiating inequahty (3.200), we finally get f£ (2+p)p £ ? (2+p)p

V 2ie(p+1)t ^ p ! ^ v 2 1 6 (p+1)2, where

Λ,.ϋ,+ιΓν+^Ι We next consider transformation of series for improving their

convergence.

5. Rummer's transformation

Suppose we have the series with positive terms:

Rm = f an (3.201)

SERIES 157

and some auxiliary series Bm = Σ bn (3.202)

which is convergent and has a finite sum Bm. Suppose the finite limit exists:

A = lim ^ ^ 0. (3.203)

In these circumstances series (3.201) is convergent (see Test III, § 1) and the following identity holds:

Brr A

!+j„('-^)^ {3-m

which is known as Rummer's transformation for the series (3.201)·

6. Improvement of the convergence of series corresponding to a given convergence test

Transformation (3.204) can be used for improving the convergence of series (3.201). For, by (3.204), the convergence of (3.201) will be improved in the sense that the general term a^^

= ( 1 (bjar))an of the transformed series ]Γ aft* will tend \ A J n«m to zero faster than an, i.e. lim α^/αη = 0. Obviously, the faster

n—►<»

the ratio bjan tends to its finite limit A # 0, the faster the series on the right-hand side of (3.204) will converge.

Transformation (3.204) can be applied several times in succession to the same series (3.201).

Suppose, for example, that the terms of the auxiliary series

B™ = Σ W (* = 0, 1, 2, . . . , />)

are chosen successively so that (1) the sum B™ (k = 1,2, . . . , p) is already known to us;

(2) Ak = lim A£\ where A™ = 8 £ , <#+*> = Λ - ^ W π-»· «e an y Ak j

all the Ah being bounded and non-zero.

Now, on putting A{^)IAk = 1 — é£\ where lim ε^ = 0, we have after the pth transformation of (3.201) : n~* °°

p n(h) oo p Rm= Σ - Γ + Σ « η Π ^ ' · (3·205)

fe-0 Ak n-m fceQ

It occasionally turns out that all the ε™ = 0 for a certain k, and we get the strict sum of series (3.201).

Every transformation of type (3.204) is determined by the choice

of some series £ 6n, which corresponds to some test which is sufficient n * l oo

for the convergence of the series £ an, obtained from Test III. We can 7 1 - 1

say in this sense that, for every convergence test following from Test III, there is a corresponding method of improving the convergence of the series. It may prove convenient in practice to take all the fc(

n0),

b%\ b{%\ . . . equal or different in transformations (3.205), i.e. to choose the stages in the improvement of the convergence in accordance with the convergence obtained by using different tests. We shall consider below, as an example of the application of transformation (3.205), methods of improving the convergence of series corresponding to d'Alembert's and Gauss's tests.

Improvement of the convergence of series corresponding to d'Alembert's test. We put in identity (3.204): bn = <ζη+1—αη = Δαη and take

oo

the series ]T an to be convergent by d'Alembert's test, i.e n - l

lim ?*±1= ρ < i.

Let Q ?± 0; then

Rm = Σ *n = - ^ + Σ ("n-^rX ^ 2 ^

where

A0 = lim ^ . n—»»oo û n

On again applying transformation (3.206) to the series

f <#>, where <#> = a n - ^ ,

SERIES 159

we have

The following notation is used here:

Ax = lim —~- = lim

Thus, /; successive applications of the above transformation to the original series give (using the above notation):

*. - Σ «. - -af-\Pi „Π J-( 1 -^-) ] f l - + Σ Π(ΐ-χ)«η» (3-207)

B(»-*)» 8 - 0 \ Λ β /

where

Λ0 = lim —- and ΛΛ = lim η^ψ

8

(fc = 1, 2, 3, . . . , p).

Euler's transformation. Given the power series

Σ «n*n (3.208) 7 1 - 1

suppose that the following conditions are satisfied;

lim *Ji±l = 11 and lim ^—^ « 1 (3.209)

(Ar = 1, 2, . . . , />)·

fThis guarantees the convergence of series (3.208) with |*|-<1 (see ref. 11, vol. II).

We can now apply to series (3.208) the transformation (3.207), which reduces to the following familiar Euler transformation:

£ Λ αι* pfi Ah ( * \k+1

+ ( r h ) v „Ix (ΑΡχ*)χη- ( 3 · 2 1 0 )

EXAMPLE 51. Let <xn = P(n) be a polynomial of degree m ; it is easily verified that conditions (3.209) are satisfied here for all k s m, while we have in addition Ahacn = 0 for any k >■ m. Now, on applying (3.210), we obtain, for | x | < 1 :

Improvement of the convergence of series corresponding to Gauss's

test. Suppose that the series £ an *s convergent where the conditions

of Gauss's test are satisfied, i.e.

un Λλ + ? ι « λ " 1 + ν ( / ι ) '

where <p(n) and y(w) have orders lower than πλ~1, and in addition, Qi—Pi> 1 (s e e § 1» s e c · 5).

If we put bn = (π+1)αη+1—«ûn, transformation (3.204) can be oo

apphed to the series £ ön. We have

n-m Î i - Λ - Ι

where /?i, /?g» · · · > Λ a r e certain new coefficients not dependent on n. Transformation (3.204) for the series (3.211) can be repeated.

EXAMPLE 52. If transformation (3.211) is applied twice to the series

5 = η?ιφΤΊ)

SERIES 161

we get 2 °° 1

5 = Τ + 6η ? 1 φ + 1)(« + 2)(« + 3ν

If we take into account only three terms of the last series, we get S % 59/60, while the exact sum is S = 1.


S = Σ "n> (3 ·2 1 2) n*=m

where *-i(oc + k)(ßi-k) , ·

we can apply transformation (3.211) several times. We obtain after p transformations of the series (3.212):

a/3 y(y-a-/3)

yiy-«-/5)ktO.U(r+*+i)(y+*+i-«-ft , A (y+J-a)(y+J-^) ^ a,,

11 « ± , _ » _ Λ Z-M 0 + J — 0 ■*■ fl („+,,_,) S-0

7. Abel's transformation

The identity n n - l Σ α Α = <*nßn+ Σ ( a Ä - a Ä + l ) ^ (3.213)

ft-1 ft«l ft

where 2?Ä = £ /?4, is called ^lòe/V transformation. It is the analogue t » l

of the formula for integration by parts of finite sums. On letting k — oo in (3.213), we arrive at infinite sequences { a j and {ßk}. It follows from (3.213), provided {Bk} is bounded and limaft = 0, that

ft-*«»

Σ «kßk = Σ (*fc-«fc+i)*fc, (3.214) fc-i ft-i


This transformation can be utilized for improving the convergence of infinite series (see ref. 1).

EXAMPLE 54. Let us improve the convergence of the trigonometric series

OO OO

£ uk sin kx and j uk cos kx, where lim uk = 0,

with the knowledge that

k i ■ x c o s ( f c + y ) * £ shux= T cot T i -/—

i « l ^ L ~ . X 2 sin —

and

i, sin(*4)· £cosix = -y+-

t e i ^ /% . X 2 s m ~ 2 On using transformation (3.214) after putting ock = uk, Auk =

= wÄ+1-HÄ,weget: , x

Wi COt — _ / i \ oo 2 1 / 1 \ £ wÄsin£x = r + — £ ^ c o s / f c - f - ^ ) * ;

2 2 s i **«i V 2J 2 (3.215)

£ i/fccoste = —Ç- î—- £ /Ji^sinf fc+T )*. (3.216)

Obviously, these transformations of trigonometric series can conveniently be employed in the case when the uk tend to zero slowly and lim àuh\uk = 0. The improvement in the convergence will be

fc->oo

the greater, the faster Auk/uk tends to zero. If transformation (3.214) is applied a second time to series (3.215)

and (3.216), we get oo / u \ x 1 °° £ uk sin kx = f Wi-·—

) cot — £ d2uk sin

(& +

l)x *-i V 2 7 2 4 an» I*"1

SERIES 163

and

Y uk cos kx = - ^ - + (wi- w2) Σ ^ 2 ^ c o s (k + Vx> *-1 2 4 s i n 2 | Ä e l

where zl2wft = wft+2~2wfc+i + wfc *s a difference of the second order. For instance, if uk = l/k, we have zlwÄ = 1/&(&-+-1), ζ ΐ χ = 2/&(Α:+1)(£ + 2), etc.

8. A, N. Krylov's method of improving the convergence of trigonometric series

If f(x) satisfies Dirichlet's conditions, it can be expanded as a. convergent Fourier series:

f(x) = -ττ+ Σ (an c o s nx + bn sin nx\ (3.217) * n - 1

where the Fourier coefficients an and 6n are of order l/n if f{x) is a function of bounded variation. Such a series is slowly convergent. It is possible to speed up the convergence by isolating the slowly convergent part. Let f(x) have bounded derivatives of the kth order everywhere except for a finite number of points x{°\ x{°\ . . . , x{JJ. At these points f(x) has a discontinuity of the first kind, the size of the jump Ai being equal to

h = f(x[0) + 0) -/Ui°> - 0 ) ( 7 = 1 , 2 , . . . m0). (3.218) Suppose that the /th derivative also has discontinuities of the first: kind at the points x(

xJ), x(j\ . . . , x^, with jumps

Ap) = f(J)(x[i) + 0) -/<%Ρ> - 0) (3.219) ( /= 1, 2, . . . , mi; j= 1, 2, . . . , fc).

We can now write/(x) in the form mo 1 mi 1

/(*) = Σ ±t>\0)<O(x-40))t Σ τΗι1)σ^χ-χ^+ · · · m* 1

. . . + Σ - Λ{%(* - *[*>)+?(*), (3.220> 1=1 π

or

where

(3.221)

/ — π— u

Co(w) = Σ — — = < n«l «

2 n — u

2 0

CAU) = / \ f ~ / w π ^ COS

Jo 6 n-1 Λ π 2 (π 4-1/)2

for — 2π < w < 0,

for 0 < w -c 2π,

for κ = 0, 2π, - 2 π .

(3.222)

12 4 π2 (π — ύ)2

12 4

Further integration gives us cr2(w):

for — 2π ^ κ ^ 0,

for 0 ^ w ^ 2π. (31223

^ sin nu 3πν* — 2π2ιι — ι? 12

(0 ^ « ^ 2JT). (3.224)

The function <p(x) is continuous along with its first k derivatives, and its Fourier series

oc °° Φ) = -T + Σ (αη C0S w* + £n Sin «*)

* n«l is rapidly convergent, since the coefficients ocn and βη have the order 1//2Ä+2. If we take into account the expansions (3.222)-(3.224), the series for/(jc) will be

jw = ? > £ *"» ;»»- § IH» i SS5-fc=fì5 -i r i n n f i w iti π n~i ir

_ g l gsinW(,-^))+^ iti π n t i /r

. . . + 4 £ + £ («n cos «x + /în sin nx). (3.225) n= l

SERIES 165

The slowly convergent parts have thus been separated out from Fourier series (3.217).

We often obtain by some method the Fourier series for a function without knowing the function itself; here, if the series is slowly convergent, it is of little use for computing the values of the function, not to mention the values of its derivatives. It is often possible to improve the convergence in this case, by making use of familiar series for σ(χ—x0), etc. This will be shown with the aid of an example (ref. 3).

EXAMPLE 55. We are given the Fourier series

π ^ HCOS/I-y

f(x) = - ! Y 2 sinnx (O^x* n). (3.226) π n - 1 Λ — 1

We separate out the lowest power \jn from the coefficient:

n 1 1 1 Λ22— 1 n nz rfi — Λ3 *

The initial series can now be split into three:

71 7t - COSAI— - COS« —

H \ 2 V 2 ' 2 V 2 · π rtïx n π Λ ^ ir

π - cos n — 2 ?, 2 — Σ „g „a sin**- (3.227)

The first two series are summable in the finite form:

cos n ,71

2 °° 2 Sx(x) = Y sin/zx= ^ n - 1 n

1 οχχ,πί XH-yj-fSin«i ΛΓ-yj sin 71

7 1 - 1

= 4 [ σ ° ( * + τ ) + * ° ( * - τ ) ] ; (3·228>

COS 77 s2(x) = - £ Σ ■

** n - 1 r?

π 2 . — sm nx =

j ^ s i n w ( * + y Wsinn(x-j)

β ^[ σ »( χ + τ ) + σ »(* -τ ) ] · (3·229)

On using (3.222) and (3.224), we get:

Sx(x)=<

x π

χ—π .π

(.-x-f), π

π

S2(x) =

Consequently,

χ3 π 24

(χ-πΤ

6 ; r f 2 4 *

6π + 2 4 ( Λ ; - π )

2

ë x

/(x) = Sfc) + Sax)-±. Σ

π cos η —

π n - l w V - 1) sin nx.

(3.230)

(3.231)

(3.232)

The remaining series is rapidly convergent (its coefficients are of order l/n5) and the expression (3. 232) obtained enables us to evaluate easily the values of the function and of its first derivatives.

In the case when the series given above for σ0, σι% σ2 are insufficient for improving the convergence, the following series may prove useful:

cos x — cos 2x -f Y cos 3x +

= - R e [ I n \l-eix\] = - In 2 sm y (3.233)

SERIES 167

and its analogue: cos 2x cos 3x cos 4x Π 7 ^ + ~ΤΓ" + ~3ΛΓ"

— (1 - cos *) In 2 s m T (π χ\ . ■(T-T)-sinx-fcosx. (3.234)

By using these last series, we can improve the convergence of any series of the form

) cos nx0 ) sin nx -f

-( c( — \ sin nxQ + D( — )cos«x0 jcos nx L

where A(l/n), B(l/n\ C(l//i), D(l/n) are analytic functions of l/n for small values of the argument.

9. A. S. Maliev's method of improving the convergence of trigonometric series

The Fourier series of a function is slowly convergent if it has derivatives of all orders inside the interval (0, 2π), while the function or one of its derivatives has different values at x = 0 and x = 2π. The following method was proposed by A. S. Maliev for obtaining a rapidly convergent trigonometric expansion of a function of this kind.

Let f(x) be given in the interval (0, π), where it has continuous derivatives up to and including the (k— l)th order and the &th derivative satisfies Dirichlet's conditions. This function can be expanded as a series in functions cos 2nx and sin 2nx, or as a series in sin nx only or in cos nx only (depending on whether the continuation of f(x) into the interval (—π, 0) is even or odd). These series are in general slowly convergent. We can arrange for order l/«ft+1 in the Fourier coefficient if we first continue the function into the interval (—π, 0) in such a way that, when continued periodically on to the whole of the real axis, the function has k derivatives, where the kth derivative satisfies Dirichlet's conditions.

For this, we take/(x) in the interval (—π, 0) in the form, say, of a (2k— l)th degree polynomial φ(χ)9 having, along with its derivatives up to the (k— l)th order, values at the points 0 and π, respectively, equal to the values of f(x) and its derivatives at x = 0 and x = n. We can conveniently choose φ(χ) as

φ(χ) = (x+nÂo + Aî- . . . + ( ^ Z % ^ 1 ] +

+ χ*ΪΒ0 + Β1(χ+π)+ . . . + -^^(χ+π?-ι\

The coefficients Λ0, Λΐ5 . . . , ^4Α-1 ; 2?0, 5l5 . . . , 5Λ.1 are now determined successively from the equations:

φ(0) = πΜ ο =/(0) , φ'(0) = ^ - Μ ο + π Μ ! = /'(0),

c>(*-D(0) = C^kfr- 1 ) . . . 2πΛ0 + C\^k(k- 1 ) . . . 3π2^χ + . . . . . . + C^Z\k^Âk^ + C\Âk^ =/(ft-1>(0);

9>(-π) = (-π)*50=/(π), φ'(-π) = ^ ( - π ^ ί ο + ί - π ) ^ =/'(π),

ρ<*-ι>(_π) = C^Kk-1)... 2(-π)£0 + Η-^1Α:(Α:-1)...3(-π)2^1 + . . .

. . .+αζ ί ( -"^^- ι= / ( Α - 1 ) (^ After constructing the polynomial ç>(x), we get the function

x) for —π x ^ 0, for 0 ^ x ^ π.

This function, continued periodically, has k— 1 continuous derivatives and a fcth derivative satisfying Dirichlet's conditions. The Fourier series for ip(x) is as convergent as l/nk+1 and yields/(x) in the interval (0, π).

EXAMPLE 56 (ref. 3).

f(x) = x-j (Ο^χ^π).

SERIES 169

Let k = 3 (for convergence of the order I In4). Now,

+^[-ά_έ (*+ π )-^ (*+ π ) 2} On expanding as a series the function

M f ?>(*) ( - π Ä x Ä 0), VW ΙΛ*) (Osxs*),

, , . 240 „ / l 12\ 1440 „ 1 . 71 n - l , 3 , 5 . . . \" n it J 7f 1 1 - 2 , 4 , « , . . . "

we get

CHAPTER IV

ORTHOGONAL SERIES AND ORTHOGONAL SYSTEMS

Introduction

A sequence of functions {fn(x)} is said to be orthogonal in the interval (a, b) if, for i ^ j9

P/i(*i/K*)<k = 0·

An important role is played in analysis by the representations of functions as orthogonal series:

oo

Σ cnfn(x)> n - l

i.e. series in an orthogonal system of functions. Trigonometric series are the classical example of orthogonal series.

The theory of orthogonal series arose in connection with the solution of problems of mathematical physics by the so-called Fourier method. Linear integral equations with symmetric kernels may also be solved by means of orthogonal series.

The theory of orthogonal systems of functions has a remarkable analogy with the theory of orthogonal systems of vectors (see Chapter II, § 1). This analogy was observed long ago and has been reflected in the terminology. It led in the course of time to the conception of Hilbert space—the infinite-dimensional analogue of w-dimensional Euclidean spaces (see Chapter II), the role of orthogonal systems of vectors being played in Hilbert spaces by orthogonal systems of functions. To provide a valid basis for the theory of orthogonal systems of functions, the basic concepts of analysis had to be gener-

170

ORTHOGONAL SERIES AND ORTHOGONAL SYSTEMS 171

alized; in particular, generalization of the concept of integral led to the Lebesgue integral.

The more elementary part of the theory, and especially the "computational" side of it is described in the present chapter. Unless there is a special proviso, the integral is understood in Reimann's sense throughout this chapter.

«-dimensional vectors can be interpreted as functions defined at n points; this interpretation is discussed in broad outline in § 1, sec. 1 ; it throws into greater relief the analogy with orthogonal series of functions. Biorthogonal systems of functions — the analogues of biorthogonal systems of vectors — are discussed in § 2, sec. 6. The first example of such a system was offered by P. L. Chebyshev in connection with interpolation problems.

Orthogonal systems of functions are connected with problems of approximation of complicated functions by simpler ones, when the measure of the closeness of two functions/^) and <p(x) is provided by their square derivation:

[b[fW-<p(x)]2dx.

The problem of the best approximation (from this point of view) of a function by polynomials led to the creation of the theory of orthogonal polynomials (the simplest system of orthogonal functions). The first example of an orthogonal system of polynomials was the system of Legendre polynomials. The general theory of orthogonal systems of polynomials is due to P. L. Chebyshev. This theory is described in § 3.

The classical systems of orthogonal polynomials are discussed from a unified view-point in § 4, sec. 1-4.

The later sections of § 4 are devoted to concrete systems of polynomials — those of Legendre, Jacobi, Chebyshev, Hermite, Laguerre (as also Chebyshev's analogue of Legendre polynomials for a finite number of points). Good uniform approximations of functions can be obtained in a number of cases with the aid of segments of series in orthogonal polynomials. Such approximations are obtained very often from segments of series in Chebyshev polynomials (see § 4, sec. 7).


§ 1. Orthogonal systems

1. Orthogonal systems of functions defined at n points

Logically, the simplest example of an orthogonal system of functions is provided by a system of functions that are defined at n points. We shall confine ourselves to functions of one variable (everything said can be generalized immediately to functions of several variables).

Let xl9 x2, . . . , xn be a finite set of numbers (or points of the real axis) and let /(χ{) be functions defined at these points. Each such function can be regarded as a vector/with components^ = f(x^) (i = 1, 2, . . . , 72). The length or norm of such a vector is

n

The set of these vectors (functions) forms an w-dimensional Euclidean space, which is denoted by the symbol En(xx, x2, · . . , *n) (see Chapter II).

A basis in En (xl9 x2,.. ·, **) is any system of n linearly independent functions/(1),/(2), . . . , / ( n ) on the set x» x2i . . . , xn.

2. Orthogonal systems in En(xlf x2> . . · , *n)

Functions/Ox:^ and φ{χ^ are orthogonal on a set of points xl9 x2,...,

(/, ψ) = Σ fm = Σ AxiMxd = o> (4·2) i « l i - 1

which corresponds to the orthogonality of the vectors with coordinates fi and ç>4(/= 1,2, ... ,w).

A system of functions/**0 (x^ (k = 1,2,...,/) of En (xl9 x2,..., xj is said to be orthogonal if

( / « . / « ) . ; i i ' ^ 0 f o r * - > · (4.3) [ 0 for k ?* j \

An orthogonal system of n functions is described as complete. There exists in En (xl9 x2, . . . , *„) an infinite set of orthogonal

systems of n functions {/0)} (j' = 1, 2, . . . , n); the functions of each such system form an orthogonal basis in space En (xu x2, . . . , *„). Every function f(xj ofEn (xl9 ..., x^) is linearly expressible in terms of the functions of an orthogonal basis:

Âxù = Σ <HP*\*d- (4.4)

The coefficients Cj in (4.4) are called the Fourier coefficients of the function/in the system of functions/(1),/(2), .. .,/ (n), where

q = ( / , / ü ) )= Σ / (^ ) / ϋ ) ω · (4.5)

The coefficient c^ is the projection of the vector / o n to the base vector fi\

If ||/0>|| = 1 for all the f® (j = 1, 2, . . . ,« ) of an orthogonal basis, the basis is said to be orthonormalized. If the basis/ü) (J = 1,2, . . . , /z) is orthonormalized, we have for any function f(x^ of £Λ (^,

X%9 · . · 9 -Λγ / ·

ii/ii2 = Σ ci> (4.6)

where the c; are the Fourier coefficients of the function/in the orthonormalized system {/0)} (this follows from (4.4) and the linear properties of the scalar product).

If an orthogonal normed (||/0) || = 1) system is not complete, i.e. the number of its elements / < n, instead of equation (4.6) we have the inequality

ll/il2^ Σ 4 (4.7)

Equation (4.6) and inequality (4.7) transform in the limit as /i·*·» to ParsevaVs equation and BesseVs inequality respectively (see § 2, sec. 4).

3. The best mean square approximation

The square approximation of a function/(x) by a function φ(χ) with respect to a system of points xl9 x2, . . . , xn is defined by

- / n

ΙΣ i i / -ç>n- Z ( M ) - ^ i ) ) 2 . (4.8)

We shall discuss generalized polynomials of the /th order, i.e. functions of the form

Σ 4/(ft)(*i)> (4.9)

where {fik)} (k = 1, 2, . . . , / ; / < n) is an orthogonal system of functions of En (xl9 x29 . . . , x^.

THEOREM 1. From among all the "generalized polynomials" of the I'th order (/ ^ n)9 the best mean square approximation of a function f(xj at the points xl9 x29 . . . , xn is given by a polynomial of the form

Σ Ckf(k)M> (4.10)

where ck are Fourier coefficients. This /th order polynomial coincides with the sum of the first /

terms of sum (4.4). This best approximation vanishes when / = n.

4. Orthogonal systems of trigonometric functions

The most important examples of orthogonal systems on a finite set of points are first the orthogonal systems of polynomials introduced by Chebyshev (see § 4, sec. 11), and secondly, orthogonal systems of trigonometric functions.

EXAMPLE 1. The system of functions {cos focj (k = 0, 1, 2, . . . , n) is orthogonal on the system of 2n points xi = in/n (/ = 0, ± 1 , ±2, . . . , ±w—l, — n). Any even function/(x^, defined at these points, is expressible by the sum

2 Ä«I

where ] n - l

ck = — Σ Λ*υ c o s £*t·

EXAMPLE 2. The system of Λ— 1 functions {sin kx{} (k = 1, 2, . . . , 72—1) is orthogonal on the system of 2/2— 1 points x{ = m/n (/ = = ± 1, ±2, . . . , ±/i— 1, —/i). Any odd function φ(χ^9 defined at these points, is expressible as

where

n - l <P(xò = Σ cksinkxi9 (4.12)

A - l

1 n - l Ch = — Σ ^ i ) s i n kxv

EXAMPLE 3. The system of 2n functions cos kx, sin Ix, where k =s 0, 1, 2, . . . , « ; / = 1, 2, . . . , a— 1, is orthogonal on the system of 2/1 points

*« = — ( / = 0, ±1 , ± 2 , . . . , ± B - 1 , -n i .

Any function ipixj, defined at these points, is expressible as

Q.

ψ(*υ = "T + Σ (ak c o s k*% + £* sin kXi). (4.13) ^ fe-1

where 1 n- i

** = T Σ Vfo)cos kxi> \ n - l

bk = -r Σ V(*i) s i n ^ i · /2 i - - n

Formula (4.13) reduces to (4.11) if ψ(χ) is an even function, and to (4.12) if ψ(χ) is odd.

Any periodic function ψ(χ{\ defined at all points xi = fri/n, where / is any integer, is expressible by (4.13), or by (4.11) or (4.12) if it is even or odd.

These formulae, deduced by Euler and Lagrange, transform to (4.42) and (4.43) as n — <». They are at the basis of numerical methods of harmonic analysis


§ 2. General properties of orthogonal and biorthogonal systems

1. Orthogonality. Scalar (inner) product

The orthogonality of two vectors in w-dimensional Euclidean space is well known to consist in the fact that their scalar product vanishes (see Chapter II, § 1, sec. 3). The orthogonality of functions is defined by analogy with the orthogonality of vectors.

The scalar {inner) product of functions f(x) and g(x) on [a, b] is defined by the expression

( / ,* )= [bf(x)g(x)dx. (4.14) ■Ü (We naturally assume here that f(x).g(x) is integrable in [a,b].) If

( / , * ) = Pf(x)g(x)dx = 0, "j> f{x) and g{x) are said to be orthogonal on [a9 b],

EXAMPLE 4. The functions sin x and cos x are orthogonal on the segment [0, π], since

sin2* ί sin x cos xdx = o

= 0. o

The orthogonality of functions with respect to a weight function is a further generalization of the concept of orthogonality.

Let p(x) be a fixed non-negative function in [a, b]. Any two functions f{x) and g(x)9 for which

rbf(x)g(x)p(x)dx = 0, (4.15) ' a

are said to be orthogonal on the segment [a, b] with respect to the weight p(x).

EXAMPLE 5. The functions sin (ware cos x) and cos (ware cos x) are orthogonal on [—1, -fl] with respect to the weight 1/Vl—*2>since

J a

ί +i 1 sin (n arc cos x) cos (n arc cos x) — 7 = dx = -1 V^T-*2

J : = sin nu-cos nu du = 0.


All the integrals considered in this chapter will be assumed to exist in Riemann's sense.

The concept of orthogonality can be generalized in a natural manner for functions /(*), g(x) and the weight p(x), which are Lebesgue but not Riemann integrable.

Letpix) be non-negative in [a, b], non-zero on a set of measure zero and Lebesgue integrable on [a, b]. Now,

p{x) dx =- 0. f. I a rb rb Let/(;t) be such that I /3(JC) dx exists in the Lebesgue sense. Then p(x)p(x)dx Ja Ja

also exists. We say in this case that f(x) is square integrable with respect to the weight p(x) on [a, b] (in the Lebesgue sense), and we write L2

p{x)(a, b) for the set of such functions.

If p(x) a 1, we use the simple notation Lz(a, b). For instance, the function f(x) = x~~112 belongs to L3(0,1), whereas f(x) » x~l/*

does not. The following proposition holds: if fix) and g(x) belong to ΐ£(β)(α, 6)» then

rb f(x)g{x)p{x)dx

1 a Ü exists, and the Cauchy-Bunyakovskii inequality holds:

mb i f(x)g(x)p(x) dx & b rb

P(x)p(x)dx g\x)p{x)dx a j a

(this latter is a generalization of the corresponding inequality for vectors, see Chapter Π, § 1, sec. 2).

The scalar product in L\(z)(a, b) of functions fix) and g(x) is defined by the expression

( / . * ) - I f(x)*(x)p(x)dx.

The functions fix) and gix) of ££(ίΒ)(α, b) are said to be orthogonal with respect to the weight pix) if (/, g) « 0.

Let σ(χ) be a non-decreasing function in [a, b]. Let us consider the functions/(x), for which the Stieltjes integral exists:

rhf\x)da{x). ) a i


f(x) do(x) also exists. ) In this case f(x) da(x) also exists. ) The set of such functions

J: will be denoted by L\{x)(a, b).

There exists for any two functions f(x) and g(x) of L2a{x)(ay b):

rbf(x)g(x)Mx) (4.16) f a

and the corresponding Cauchy-Bunyakovskii inequality holds. Integral (4.16) is called the scalar (inner) product of functions f(x) and g(x) of Llix)(a, b).

The functions f(x) and g(x) are said to be orthogonal with respect to the integral weight a(x) if

- J ! ( / , £ ) = [bf(x)g(x)do(x) = Oy

i.e. their scalar product vanishes. In the case when a(x) has a finite number of growth points: xl9

*2> · · ·> xn> integral (4.16) becomes the finite sum

Ìf(xùz(xùKxi+0)-a(Xi-0)}. (4.17)

If all the jumps of σ(χ) are equal to 1, i.e. afa+0)-afa-0) = 1

for all i, (4.17) becomes

jtflxdgM. (4.18)

The orthogonality off and g is influenced here only by their values at a finite number of points — the growth points of a(x). In this connection, the functions / and g can be regarded as specified only at these points (as in § 1, sec. 1). In this case (4.18) is the ordinary scalar product of the vectors / and g of En(xl9 . . . , *„). The case when a(x) has a finite number of growth points is singular inasmuch as it is only in this case that there exist only n linearly independent mutually orthogonal functions (see § 1, sec. 1).

The above concepts relating to the orthogonality of two functions are united by a common property: the inner product of the functions f(x) and g(x) is in each case equal to zero. The whole of our sub-

sequent discussion (unless there is a special proviso) will be independent of the particular method by which the concept of inner product is introduced. We shall thus use in general the symbol (/, g) to denote the inner product of f(x) and g(x), i.e. this symbol may be defined in any of the ways described. The case when

( / , £ ) = {bf(*)g(x)da(x).

unites all the other definitions of scalar product. The case of a "differential" weight p(x) is obtained from this when σ(χ) has an integrable derivative, and da(x) = p(x)dx9 so that integral (4.16) becomes

rb f(x)g(x)p(x) dx.

J a Orthogonality without a weight is the case when da(x) = dx. The orthogonality of functions defined at a finite number of points corresponds to a finite number of growth points of σ(χ).

In future, we shall therefore use expression (4.16) for (/, g), where necessary.

The scalar product of the function/(x) with/(x), i.e. (/,/), is usually denoted by | |/ | |2and | | / | | =>/(/ , /) is called the norm of the function. This quantity is the norm of f(x) in the corresponding normed space.

The quantity

ll/-*ll = ^ Γ (/-*)■*<*)

is called the root square deviation of functions f(x) and g(x). This is the measure in the accepted sense of the closeness of f(x) and g(x).

2. Orthogonal systems of Bessel functions, Haar functions, etc.

Let any two functions of the sequence

<Pi(x), <P2(x), - - -, <Pn(x\ . . . be orthogonal on (a, b), i.e. (q>i9 φ^) = 0 if / ^ j .

Such a sequence is described as an orthogonal system of functions on (a, b). If the system of functions

<Pi(x)> 9>2(*)> · · · » <Pn(x), . - -

is orthogonal and each of the functions differs from zero, the system of functions

is also orthogonal. Every function of this system satisfies

IIViWII = V(V^T)= 1. (4.19) Such systems are extremely convenient. They are described as

orthonormalized (or orthonormal). The orthogonal system

<Pi(x), 9>2(*)> · · · > 9>n(*)> · · · is said to be complete in the space 1%(χ)(α9 b) if, among all the functions of Ll{x)(a, b), there is none that is non-zero and orthogonal to all the functions of the system

<Pi(x)> %(*)> · · · > <Pn(x\ i.e. the fact that

I fW<Pn(x) da(x) = 0 J a

for all n, implies that /(*) = o

(excluding possibly a set of measure zero). In the case of a finite number of growth points of σ(χ), a complete

system contains as many functions as there are growth points of σ(χ).

EXAMPLE 6. The system of functions

sin jc, sin 2x, . . . , sin nx, . . . (4.20)

is orthogonal on (0, π), since

smmxsinnxdx = 0 for m s* n.

The system is complete. EXAMPLE 7. The system of functions

1, cos x9 cos 2x9 . . . , cos nx, . . . (4.21)

ί


is orthogonal on (0, n), since

cos mx cos nx dx = 0 for m ^ n.

The system is complete. EXAMPLE 8. The system of functions

1, sin x, cos x, sin 2x, cos 2x, . . . , sin nx, cos nx, . . . (4.22)

is orthogonal on (0, 2π), since fin

sin mx cos nx dx = 0 for any wand«, Jo pi«

sin mx sin nx dx = 0 for m ^ n, Cm

cos mx cos «jc rfx = 0 for m ^ n

(m = 0, 1, 2, . . . ; « = 0, 1, 2, . . . ) .

The system is complete. The system of Bessel functions. Let Jn(x) be the Bessel function of

order n, and Al9 ^, . . . , A{, . . . the positive roots of the equation

/»(*) = 0 or of the equation

■£(*) = o. The system of functions

JnttlX), Jni^x), Λι(Α3*)> - - - , Jn(h*\ - · · (4.23)

is now orthogonal in [0, 1] with respect to the weight x, since with

Jn(^ix)Jn(^ix)x dx = 0>

The system (4.23) is complete. The system of Haar functions. In recent times a use has been found

for the orthogonal system of Haar functions. This system consists of piecewise constant functions %\{t), where

i

182

ώ0)(0 =

zMO =

zm =

*i2>( ') =

xm =

um =

zl3,(0 =

MATHEMATICAI

1

1

\ ~1

0

ί ^ - V 2

l 0

f ^ - V 2

1 0

ί 2

- 2

V 0

ί 2

- 2

1 0

( 2

- 2

l 0

for

for

for

for

for

for

, ANALYSIS

t e [o, î];

„[ci),

1

<φ{> ><α·ϊ\-

at other points;

for < € [ i , 4),

for r e ( I , l ] ,

at other points;

for i €[ftj) , for « f H at other points;

for i € [ I , f) ,

for ,€(τ· τ} at other points;

for , i [ i , I ) , for - f il at other points;

(4.24)


for tÇ.

X(A')={ ■2

0

Γ 1 1 [ _ 4 ' 8

(H In general,

χθ)(0) = V2n,

V2" for

ZÎf'WH

for tÇ.

at other points.

~2k-2 2k-.

(4.24)

f €

- V 2 n

0

for tÇ.

\2k-2 2k-\\ 2n+1 ' 2n+1 ) :

(

2n+1

2/c-l 2n+i

at other points [0, 1]

2k Ί 2n+i J »

> (4.25)

The Haar functions are generated from the "polygonal functions" of Schauder's basis. The orthogonality of the Haar system follows from the fact that, given the same n,

so that nXt)tâ\*) = 0 for

iû\ τ8Ρ) = o.

Ars 1,

Given different subscripts m and n, where m>- n, and any j and fc the sub-intervals, where #^'(0 ?* 0 are wholly contained in the intervals of constancy of %},(/) for m >· n, and

Γ. 2ft

&Kt)nKt)dt=x 2«« + i

2 m + i J υ>

where λ is the value of χ$ on the interval of constancy. The system of Haar functions has an important approximation

property: the Fourier series (see Chapter III) of any continuous function with respect to the Haar system is uniformly convergent to it in [0,1].

The concept of orthogonality of functions of one variable on a segment may be generalized in a natural manner to functions of several variables, defined in some domain.

The functions/(P) and g(P) of a point P of «-dimensional space are said to be orthogonal with respect to a domain D of this space if

J. f(P)g(P)dP = 0. (4.26) D

A similar definition can be given of the orthogonality off{P) and g{P) with respect to the weight k(P):

I. f(P)g(P)k(P)dP=0. (4.27) D

EXAMPLE 9. The problem of the vibrations of a plane clamped membrane leads to the differential equation

Αϋ+λϋ=0 (where Δ is Laplace's operator) with the boundary condition U = 0. The eigenfunctions of this equation, corresponding to different values of λ, are orthogonal:

f [ufayWfa ,y)dxdy = 0

EXAMPLE 10. Ltt<pl(x\<p2(x),. ·., <pn(x),.. .be a system of functions orthogonal on the segment [a, b]. The system of functions

Uij(x9y) = ?i(x)9j(y) (4.28) is now orthogonal with respect to area on the square a ^ x ^ b, a^y^b.

For I φί(χ)ψί{γ)φίι(χ)φι{γ) dx dy = J a J a

= <Pi(x)<Pk(x) àx Ja Ja

= <Pi(x)<Pk(x) àx <Pj{y)<Piiy) dy = 0,

if either i ^ k or j ^ /, i.e. if Uitj(x, y) and UKl(x, y) are different functions of system (4.28).

Systems of the form (4.28) are utilized, for example, in the expansion of the symmetric kernel of the integral equation

<p(x) = λ K(x9s)qts)(b+f{x) (4.29)

as a series in the eigenfunctions of the equation.

If {(Piix)} is a normed system of eigenfunctions of (4.29), corresponding to the eigenvalues l/Xi9 while {Ofa)} is a system of eigenfunctions corresponding to the zero eigenvalue, the system of functions (where i and j are arbitrary):

<P\{xyPj(ß\ OfaMs), φάχ)0&), 0{{χ)0^\ (4.30)

is orthogonal on the square {a ^ x ^ b; a^ s ^ b\ since the system {<P%(X\ Oj(x)} is orthogonal on [a, b],

System (4.30) is complete in the space of functions square integrable on (a ^ x ·£ b; a^ s ^ b). Hence the square integrable function K(x, s) can be expanded as a series in this system. The Fourier coefficients of the function K(x, s) with respect to the functions q>i(x)q>j(s) for i 5* j , Ofaypfis) and <pAx)0}(s) vanish for any i and j , while with respect to the functions q>i(x)<Pi(s) they are equal to l/Xi9 by virtue of the relationships

K(x, styiixtyis) dxds = (pax) dx K{x, styis) ds = J a J a Ja Ja

1 rb = j ; <Pi(x)<Pi(x) dx = 0, if i y& j ;

K(x, styi^Ojis)dxds = (pax)dx K(x, s)Oj(s) ds = 0, J a J a Ja J a

K(x, ί ) ^ ( φ ί ( ί ) ά ώ = (pax)dx Κ(χ, .s)<PtCs)ds = J a J a J a J a

i(x)<Pi(x) dx = -£-1 f »

Hence ΛΓ(χ, ) = £ τ 9i(x>Pi(s)·

The concepts of orthogonahty of functions with respect to surfaces and curves can be treated in the same way as the above orthogonality of functions in a domain D of τζ-dimensional space. For instance, the spherical Laplace functions are orthogonal with respect to the surface of a sphere.

3. Linear independence. The process of orthogonalization

A system of n functions is said to be linearly independent if, given any system of n numerical factors, for which

£;'5 o,

the function λ1φ1(χ)+λ2φ2(χ)+· - · + ληφη(χ) is non-zero. If there exists a system of factors λί such that

Ai<Pi(x) + faP%(x)+ · · · +*n<Pnix) Ξ °> (4.31)

where at least one λ{ differs from zero, the system of functions φ^χ), φ2(χ)>..., ψη(χ) is said to be linearly dependent. (In the case of functions of L*(a, b\ equation (4.31) can hold for a linearly independent system on a set of measure zero.)

The concept of linear independence of functions is similar to the concept of linear independence of vectors of 77-dimensional vector space (see Chapter II, § 1, sec. 4).

Let the inner product (q>i9 φ^ be defined for any pair φ{{χ\ <ρ£χ) of the function system <Ρχ(χ), <p2(x), · · ·* <pn (*)·

In analogy with the Gram determinant for a vector system (see Chapter II, § 1, sec. 5), we introduce the determinant

Δη =

(<h> Ψι) (<Ρι> Ψ2) · - · (<h> <Pn) (?2> <PÙ (<?2> Ψ2) · · · (<?2> <Pn)

(<Pn> Ψΐ) (<Ρη> Ψ2) · · · (<Pn> <Pn)

(4.32)

which is called the Gram determinant of the function system. Properties of the Gram determinant. 1°. The Gram determinant

Δη ^ 0 for any system of functions. 2°. Let the system of function, <pl9 φ2,..., φη be defined on the seg

ment [a, b] and let the interval (al9 bj be contained in (a, b). Now, if Δη is the Gram determinant of the system in [a, b] and

(<pi9 <p3) is defined by

(<Ph <Pj) = <Pi(x )<Pj(x) da(x)

while Ζΐ is the Gram determinant of the same system in [av 6J, we have

This property also holds for the Gram determinant of functions of several variables in a domain Q: the Gram determinant can only diminish when the domain diminishes.

THEOREM 2. The necessary and sufficient condition for a finite system of functions to be linearly independent is that its Gram determinant be different from zero.

THEOREM 3. Every finite orthogonal system of functions is linearly independent.

Theorem 3 is an obvious consequence of Theorem 2, since the Gram determinant of an orthogonal system of functions is equal to the product of the squares of the norms of all the functions of the system, each of which is greater than zero.

An infinite sequence of functions <pi(x), 9>2(*)> · · ·> <Pn(x)> · · · is said to be linearly independent if any finite sub-system of it is linearly independent.

Given the system φΐ9 φ2, ..., ψη of linearly independent functions and some function /(*), let us consider the problem of the best root square approximation of f(x) by means of linear combinations of

n the system, i.e. from all the linear combinations £ dj<pi we find the

one that gives a minimum for the expression II n I'

2

/ - Σ dMi II ivml I The coefficients of the best linear combination are found from the system of linear equations

ddk[ f- Σ di<Px t - l

= 0

i.e.

^\{ψ\, 9>2) + <4(ç>2» <Pd +

(Je = 1, 2, . . . , ti),

+ dn(<Pn, Ψι) = (f, Ψύ,] +dn(<Pn> Ψζ) = (/» ψώ>

di(?i> Ψτύ + <k(<p2, ψη)+ ... + dj(pn, φη) = (/, φη).

(4.33)

The determinant of this system is the Gram determinant; in view of the linear independence of the initial system, it differs from zero and (4.33) has a unique solution. Obviously, it is easiest to find the coefficients dl9 d2,..., dn from (4.33) if the initial system is orthogonal. In this case (4.33) becomes the system

dkdP* 9Ù = tt Ψύ (* = 1, 2, . . . , n). (4.34)

By starting out from any (finite or infinite) linearly independent system of functions (or vectors), an orthogonal system can be formed, the elements of which are certain linear combinations of the elements of the initial system. Let us consider the process of forming such a system. It is called a Schmidt orthogonalization process.

Suppose we are given the linearly independent system

<Pi, fo.-ifn. · . . · We put

co (X) - ft(*>

Let

The coefficient c21 can be chosen so that (y>2, Û>I) = 0, by putting

Having chosen such a c21, we put

«^-T - *^ · (V2t % )

Here (y2, V2) ^ 0, since otherwise ψ2(χ) = 0, and this would mean that φχ and φ2 are linearly dependent.

Suppose that functions ωΐ9 ω2, . . . , ωη_1 have already been foxmd. We now put

n - l Vn(x) = ¥>n(*) - Σ cniû>i(x),

i - 1

here the numbers cni are chosen so that y>n(x) is orthogonal to all the

ORTHOGONAL SERBES AND ORTHOGONAL SYSTEMS 189

functions ωχ(χ), . . . , ωη_1(χ). We must take for this: cni = (φη, ω^. We next put con(x) = ψη(χ)/(ψη, y*)1/2> where (ipn, y^ * 0 in view of the linear independence of the functions <pl9 . . . , <pn_r This process can be continued indefinitely.

The system ωχ(χ)9..., ωη(χ), . . . thus formed will be orthonormalized The process of orthogonalization of a vector system is precisely

similar. In the case of a vector system in w-dimensional space, or of a system

of functions defined at n points, such a system contains not more than n elements.

4. Fourier coefficients. Closed systems

Let ^ W , co2(x), . . . , ωη(χ), . . . (4.35)

be an orthonormal system of functions. Given a function/^), suppose that its inner product exists with any function of the system. The numbers ci = (/, ω^ are called the Fourier coefficients off(x) with respect to system (4.35) In the particular case when (4.35) is a trigonometric system, the ci are the familiar Fourier coefficients of the theory of Fourier series.

The Fourier coefficients of functions are the analogues of the projections of a vector on to the vectors of an orthonormal vector system. If all the Fourier coefficients of a function exist, we can formally construct the series

Σ <Wk(*)· <4·36)

This series is called (in analogy with the corresponding series with respect to the trigonometric system) the Fourier series off(x) with respect to the system ωχ(χ), ..., cojjc),....

BesseVs inequality and ParsevaVs equation. We can write BesseVs inequality for any/(x), for which (/,/) and all the Fourier coefficients exist:

Ì > ? ^ (/>/)> (4.37)

from which it follows that the series £ c\ is convergent and

Σ ^ (/, / ) . (4.38)

In the particular case when £ c\ = ( / , / ) , (4.38) is known as ParsevaVs n « l

equation. Let

<*>i(x\ o)2(x)9 . . . , ωη(χ\ . . . (4.39) be an orthonormal system of functions on (a, b)9 and/(x) a function defined on (a, b) for which (/, / ) exists.

When considering the problem of the best root square approximation off(x) by the linear combinations

Σ d&iix), t = l

we arrived (see sec. 3 of this article) at an expression for the coefficients of orthogonal system {co^x)} giving the best approximation (4.34):

A (/> °>k)

which yields in the case of a normed system:

dk = (f> <*>*) = ck. From this there follows:

n THEOREM 4. Of all the linear combinations £ d$o£x\ the finite Fourier

sum t e l

n

.Σ X <*»<(*) has the least root square deviation fromf(x) (i.e. the linear combination for which dx = erf.

The orthonormal system ω^χ), . . . , ωη(χ), . . . is said to be closed if Parseval's equation is satisfied

.Σ <H '-1 J. "ρ(χ)(ίσ(χ) (4.40)

for any function/(JC) of Ll{x)(a, b). In this case,

ί bf\x) da(x) - £ c\ - 0 as n - - , a i = l

which implies, in view of the equation

f\x)do(x)- f c\ = fT/(*) - f c^x)! da(x) la i e l J a L i e l J

that the root square deviation

[T/W-.Σ WW|*(*) (4.41) tends to zero as τζ -* <».

If (4.41) tends to zero as n -* <», we say that the Fourier series oo

i « l

is convergent in the mean to f(x). Consequently, if f(x) belongs to Ll(x)(a, b), its Fourier series with

respect to any closed system is convergent to it in the mean. EXAMPLE 11. The trigonometric system (4.22) is closed in (0, 2π.) EXAMPLE 12. THEOREM 5 (V. A. Steklov). The set of eigenfunctions of the Sturm-

Liouville equation forms a closed system. It was in the context of this theorem that V. A. Steklov introduced

the concept of closure of a system. It follows from ParsevaFs equation that every orthogonal system,

closed in Ll{x)(a,b), is complete, and every complete orthogonal system in L*(X)(a9 b) is closed.

5· Fourier series in a trigonometric systemf

The problem of representing an arbitrary function/(x) by a trigonometric series

a °° "T+ Σ (an c o s nx + K sin nx) (4.42)

£ 71=1

t See Chapter III for more detailed information on trigonometric Fourier series.


arose in the middle of the eighteenth century in connection with the problem of the vibrations of a string. This problem occupied all the greatest mathematicians of this period. The essential step in the solution of the problem was taken by Fourier, who established! that, by virtue ofthe orthogonality ofthe trigonometric system, the coefficient of series (4.42) can be expressed in the form :

1 f2* f(x) dx,

an = — f(x) cos nx dx, 7C '

*-K 0 '2π

f{x) sin nx dx

(4.43)

(hence the name Fourier coefficients in the case of any orthogona-system).

Fourier stated the theorem to the effect that a (graphically) arbitrary function can be represented by a Fourier series.

Historically, the first theorem on the convergence of Fourier series was:

THEOREM 6 (Dirichlet, 1829), Any function that admits of integration in any interval in (0, 2π) and does not have an infinity of extrema in the interval can be expanded as a Fourier series, which is convergent at every point x to the value

/ ( * + 0)+/(x-0)

The crucial point in Dirichlet's proof was the representation of the partial sum sn(x) of the Fourier series and of the remainder in the form of integrals:

2w+l s"(5t) = sf-, sin-

/(a)-(x-oc)

sin x — a dx

t This had been noticed before Fourier by Euler.

and

1 Γ- s m ^ — ( x - a ) f(x) - sn(x) = ^ (f(x) -f(a)) «fa,

J - s i n ( — J together with the following observations:

(1) if 0 < c < -— π, then as « -*<»,

i; sin(2«+l)^ π ^ ünß ^ - J y ( 0 ) '

(2) if 0 < è < c ^ —π and <pG3) is monotonie, then

//?>. sin(2fl + l)/3 Ä Φ(ρ) — . D dp -+ 0 as TI -* oo, r r sinp i: A rather more general theorem may be proved by the same method, THEOREM 7. Iff(x) is a function of bounded variation, its Fourier

series is convergent at every point x to the value— [/(*+0)+/(;c—0)].

If in addition, f(x) is continuous at every point of some interval, the Fourier series is uniformly convergent in the interval.

THEOREM 8 (Lebesgue's Test). Let

and

If

and

l

<p(0 = ¥>x(0 =Λχ+*)+Λχ-*)-2Αχ)

*x(A)= Γΐ?«(0ΙΛ.

Φ χ ( Α ) Λ , Λ ^ τ ^ - - 0 as h - 0 h

dt — 0 ay 77 ·* 0,

/Ae Fourier series off(x) is convergent to f{x) at the point x.

THEOREM 9 (Dini-Lipschitz). Iff(x) is continuous and its modulus of continuity ω(δ) satisfies the condition

ω(<5)1η4--0 as δ - 0, o

the Fourier series off(x) is uniformly convergent. THEOREM 10 (de la Vallée-Poussin). If the function

X(t) = ~ I <px(t) dt rb* is of bounded variation in some interval with left-hand end point t = 0 and %(t) — 0 as t — 0, the Fourier series off{x) is convergent at the point x to the value f(x).

THEOREM 11 (Hardy).. If

/(x + A ) - / ( x ) = o i ï l n T i T V 1 " |

and the coefficients cn of the series have magnitudes of the order 0(n~ò)9 where δ > 0, the Fourier series is convergent at the point x to the value f{x) (see Chapter I, § 3, sec. 8 regarding the symbols o(n) and 0(n)).

6. Biorthogonal systems of functions

A biorthogonal system of functions on (a, b) is a system consisting of two sequences

<Po(*)> <Pi(x)> <Pi(*)> · · · > <Pn(x\ · · · Ì V>o(x)> Vi(x), Ψζ(*)> · · · > Ψη(*)> · · · J

(4.44)

satisfying the following condition on (a, b):

(<Pi> Vj) = àij9

where òi5 is the Kronecker delta (see Chapter II, § 1, sec. 2). Here, the inner product (9 , ψ3) is either φ^ dx or φ^ da(x). In

Ja Ja the latter case we speak of a system of functions, biorthogonal with respect to the weight σ(χ) or p(x), if da(x) = p(x) dx.

Given any function, for which the inner products encountered later exist, we can associate with it the series

/(*) where

Ä = 0 (4.45)

and consider the question of the convergence of the series and the possibility of using segments of it for approximation to the function.

EXAMPLE 13. Chebyshev's biorthogonal system on (—1, 1) with respect to the weight p(x) = 1 is

sin(£-fl)<p <Pk(x) = s i g Q - sin<p

. Jfc- f l i / Λ \ S i n ~ΤΓ" Ψ 1 v μ(α) d 2 d/Ä+i «

where sin ç?

(* = 0, 1, 2, 0.

(4.46)

x = cos <p while d in (4.46) runs over all odd divisors of k+ 1, including </ = 1 and d=k+l if this is odd; the function sign x was introduced in Chapter I, § 2, sec. 1 ; μ{ά) is the Möbius function, i.e.

Mi) = i; μ(α) = 0, if a is divisible by a square different from unity; μ(α) = (— l)r, if a is not divisible by a square different from unity, and r is the number of prime divisors of a differing from 1. The functions <pft(fc) of this system are piecewise constant, while

yk(x) are polynomials in x having k simple zeros in (—1, +1). The first few functions of this system are given below :

1 9 Ό = 1 ,

<Pi(x) = \ j

<p2(x) = 1

- 1

+ 1

- 1

for for

for

for

for

— 1 < x < 0, 0 < x < 1,

, 1

1 1 2 2

1 y <X < 1.

Vo = j

y>i(x) = x,

ΨΪ(Χ) = 2 (-4).

9z(x) = i

(-· -£ V2'

7Γ')

Λ ( χ ) = 2χ(2χ2-1) .

—1 in

+ 1 in

- 1 in

+ 1 in

EXAMPLE 14. Markov's biorthogonal system in (—1, +1) with respect to the weight pipe) = lj\J\—x2 is

9>o= 1, V>o = π l v ( - l ) h kep

<pk = sign cos k% ¥>* = -y Σ — J ~ c o s d '

(4.47)

where x = cos ç>, <i in (4.47) runs over all add divisors of the number k that do not contain square factors, and A is the number of prime factors of the form 4/n+l contained in d (m ^ 0) (see Chapter I, § 2, sec. 1, regarding the function sign x).

Here, as in the Chebyshev system, <ph(x) are piecewise constant, while rpk(x) are polynomials in x. The first few functions of this system are given below:

Ψο = ι ,

**»-{+!

φ2{χ) = <

/ + 1

- 1

+ 1

in in

in

in

in

( -1 ,0 . ) (0, 1)

( - ; - *

("vr v? (*■ 0·

ViW = 2 '

V2W = T ( 2 x 2 - l ) ,

»(*)={

- 1

+ 1

- 1

+ 1

in

in

m

m (

-■·-£ - # , o),

o,f), £..).

Vai-* ) - 2 * ( * · - | ) .

Series in Chebyshev and Markov functions are similar to series in sines only or in cosines only.

On passing from polynomials and the interval (—1, +1) to trigonometric sums and the interval (—π, +π), we can combine both systems into one, which generates series corresponding to trigonometric series of a general type.

A biorthogonal system is called a Chebyshev-Markov system with respect to the weight a(x\ if <pk(x) is a polynomial of degree k, and

y)k(x) = sign Qk(x), where Qk(x) is a polynomial of degree k.

THEOREM 12. The biorthogonal Chebyshev-Markov system for a given function σ(χ) is unique; the polynomial Qk(x) appearing in the expression for ipk(x) is the polynomial of degree k with first coefficient 1 for which

I Qh(x) I da(x) = min P | Pk(x) \ da(x). Γ \Qk(x)\da(x) = min Γ J - i «x) J - i

The system of polynomials q>h(x) is determined after this by means of a process similar to the Schmidt orthogonalization process (the metric of space L\{x) is the natural one when considering series in the Chebyshev-Markov system).

§ 3. Orthogonal systems of polynomials Various systems of orthogonal polynomials were introduced by

Legendre, Jacobi, Chebyshev and other mathematicians The first of these historically was the system of Legendre polynomials. Chebyshev's studies laid the foundations of the general theory of systems of orthogonal polynomials.

Let the system Po(x)> PiW, - - · · i « ( 4 · · · > (4.48)

where Pn(x) is a polynomial of degree n, be orthogonal on {a, b) (in particular, the interval may be infinite) in the sense that

(Pm, Pn) = P Pm(x)Pn(x) da(x), (4.49)

where σ(χ) has an infinite number of growth points (in particular da (x) =p(x)dxy where p(x) is integrable in (a, b)). Apart from a constant factor in the polynomials, the system (4.48) can be obtained by means of orthogonalization (see § 2, sec. 3) of the system of functions: 1, xl9 x2,... xn,... Let us agree to write Pn(x) in future for polynomials of orthogonal system (4.48) defined by the condition that their first coefficient is equal to 1 :

P0(x) = 1, Px(x) = x + a10, . . . j _ } (4.50)

. . . , P n M = * n + 0 n , n - l * n ~ 1 + . . . +*n,o> J and Pn(x) for polynomials of the orthonormal system, i.e.

bp*(x)da(x)=l. (4.51) a

The Fourier coefficients of f(x) in an expansion in polynomials {Pn(x)} have the form

cn= Pf(x)Pn(x)da(x).

In particular, if da{x) = G(x) dx, we have Cb φ(χ)Ρη (x) dx, <p{x) = f(x)G(x). (4.52)

J:

cn =

Notice, that, to evaluate integral (4.52), it is not necessary to find the values of Pn(x) throughout the interval. On writing

q>i-D(x) = (%(£)</£ and for i>s φ^^χ) = f* <p(~i+l)(£)άξ, Jo Ja

we obtain, by integration by parts,

cn = φ{-^)Κ&)-φ(-2)Ψ)Ρή&)+ . . . h t - n y - » - ^ ) ^ « . Therefore, to find the coefficients c0, cl9 . . . , cm9 we only need to find,

by successive integration οΐφ(χ), the values of 9?(""x)(e), <p( 2)(b\ . . . , <p(~m)(è), the values of the polynomials Pk(x) (k = 1, 2, . . .,w), and their m derivatives (for k = 0, 1, . . . , m) at the point b.

1. Zeros of orthogonal polynomials

THEOREM 13. The n-th degree polynomial Pn(x) of the orthogonal system (4.48) has n real distinct simple roots, all of which lie in (a, b).

THEOREM 14. The zeros of the polynomials Pn{x) and Pn-X{x) of system (4.48) alternate: between any two zeros ofPn(x) there is a zero ofPn-ii*)*

It follows from Theorem 14that ΡΛ(Λ:) and Pn_1(x) have no common zeros.

THEOREM 15. If λ(χ) is the number of changes of sign in the series P0(x), Ρχ(χ), . . . , Pn(x), the number of roots of the polynomial Pn(x) in the interval (α, β) is equal to the difference λ(<ζ)—λ(β) (the property of so-called Sturm systems).

2. Recurrence relations for orthogonal polynomials

Let Ρη^.1{χ\ Λι+ι(*)> Λι(*) be a ny three successive polynomials of system (4.48), satisfying condition (4.50). In this case the following relationship holds:

Λι+2(*) = ( ^ « η + Α + ι Μ - 4 + Λ ( 4

where the parameters an+2 and An+1 are given by:

α π + 2 =

*π+ι —

J> (x)da{x)

Γ P*+1(x)ck(x)

Cbp*(x)da(x) a

Obviously, A,

(4.53)

(4.54)

n+l"

The parameters αΛ+2, λη+1 are expressible in terms of the coefficients of polynomials -Pn+2> Λι+ι> Λι·

If

then

_ ft-i

j«=0

an+2 — an+l, n~~an+2, n+l> *n+l — ûn+l, η-1""~ατι+20π+1, n""ûn+2, η·

It follows from formulae (4.54) that, if (a, b) is a finite interval, we have for all n:

where a < a. n+2 ' è, ο^Λ, + ι ■ (4.55)

where c2 = max(|ß|, \b\). )

For examples of such recurrence relations see § 4, sec. 5-10.

3· Power moments. The expression of orthogonal polynomials in terms of power moments

The numbers

" ■ J ! xn da{x) (4.56)

are called the power moments of the weight σ(χ). In the case of the system

1 , X, X , · · · > X 9 · · ·

the pair-wise inner products are expressible in terms of the power moments:

(*w, xn) = I xmxndG(x) = μη+ηι.

Gram's determinant of the system of powers is »

4,=

μ0 μλ . . . μη

μλ μ2 . . . μη+ι

Ρη /^π+1 · · · ^2π

^Ο. (4.57)


The process of orthogonalization of a system of powers yields the expression for the polynomials Pn(x) of (4.51):

Pn(x) = 1 V ^ r A - i

and for the polynomials Pn(x) of (4.

Pn(x) = 4 i - l

μ0 μλ . . . μη_ι μι μι . . . μπ

/^η /"η+Ι · · · ^ Λ - Ι

of (4.50):

1 i"0 · · · Ä i - l 1 U l . . . fti *

1 /*n · · · /*2n-l *

1 Λ:

xr

•

(4.58)

(4.59)

The parameters a and A of the recurrence relation (4.53) can also be expressed in terms of the power moments:

1 n + 2 J TI+1

μο /h

μη μ-η+2

μ-n+i μ2η+1 ^271+3

"4, μο μι

μη-1 /^η+Ι μη μη+ζ

μη ^271-1 ^271+1

(4.60)

where η == 0, 1, 2,

„ _ /*ι ι _ 4 ι+ι4 ι - ι α - 1 — ~ > Λη+1 —

while zl.^ = 1 zi2 (Λ = 0, 1, 2, . . . ) ,

4. The connection between orthogonal polynomials and continued fractions

The fraction

Ρ η (0-Λι(*) t-x

is a symmetric polynomial of degree n — 1 in t and x. Hence the function

Rn_l{x) = Γpn(f)-PAx) da(t) (4-61) J a t — X

is a polynomial of degree n—1 in x. The polynomials An+1(.x), i?n(x) and jRn-1(jc) satisfy the same recurr

ence relation (4.53) as polynomials Ρη+2(χ)* in+iW» ^nW· Ήώ fact indicates (see Chapter V) that Ρη^(χ) and Pn(x) are respectively the numerator and denominator of the wth order convergent of a Chebyshev type continued fraction:

λ° - + - ^ - + . . . + K . (4.62) X — (%i X — Ä2 X — a n + l

The polynomials Rn_x(x) are described as polynomials of the second kind with respect to the weight σ(χ). They are the denominators of the continued fraction

λι -+-Α-+ . . . + — **—+ .. . (4.63) x — α2 x—α3 χ — αη+1

and are orthogonal with respect to some other weight. If the interval (a, b) is finite, the sequence Λη-1(χ)//>

η(χ) of convergents of the continued fraction (4.62) is convergent for all x lying outside (a, b) to the value of the integral

f» da(t) Ja*-< (4.64)

When the interval {a, b) is infinite, the continued fraction (4.62) is not always convergent. The question of its convergence is related to the determination of the corresponding problem of moments.

Regardless of the convergence, the continued fraction (4.62) is related to the integral (4.64) by the following property: the expansion of the convergent An^OO/P^x) of (4 62) as a series in negative powers of x coincides as far as terms of the order l/x2n (inclusive) with the series

^ + f + f + - - . + ^ + ^ T + · · · . (4-65)

(4.66)

which can be obtained if the function l/(x— t) in the integral (4.63) is expanded as a series in powers of l/x and formally integrated term by term (the convergence of series (4.65) is of no consequence).

A continued fraction of the Chebyshev type, possessing the above property, is described as associated (assoziierte, Perron, ref. 11) with the series (4.65).

In the case σ(χ) = const, the continued fraction (4.62) can be transformed, for x < 0, into a continued fraction of the Stieltjes type

1 1 1 1 axx+a2 -}-<73*+tf4 -f . .·.

such that the wth convergent of (4.62) coincides with the 2/zth convergent of (4.66). In view of this, the expansion as a series in negative powers of x of the 2nth convergent of the fraction (4.66) coincides with series (4.65) up to and including the terms l/x2n. A fraction of the Stieltjes type, possessing this property with respect to the series (4.65), is described as corresponding to it (correspondierende, Perron, ref. 11). We know from the theory of continued fractions that every continued fraction of the Stieltjes type (with av^0) has a corresponding series, and every continued fraction of the Chebyshev type has an associated series.

The converse problem is of great importance in analysis: transformation of a previously assigned series in negative powers of x into a corresponding or associated continued fraction, yielding rational approximations for the function given by the series.

Not every previously assigned series has a corresponding or an associated continued fraction. A series possessing this property is described as seminormaL

THEOREM 16. The series

x x* + - ■ + x n + l is seminormal if and only if all the determinants

<Pn

Co

Cl c2

cn-\ Cn

c n - l cn · · · c2n-2 are non-zero.

and Wn =

W i - l

cn-l cn c2n~3

In this case, the coefficients of continued fraction (4.66) corresponding to this series are expressed in terms of <pn and \pn as follows

al — Ψΐ> a2v — ~ ~ 9 a2u+l ~

where <p0 = 1, ψλ = 1. The fraction (4.66) corresponding to the series can be transformed

by a contraction (see Chapter V, § 1, sec. 3) into an associated fraction.

5. The conversion of orthogonal expansions into a sequence of approximating fractions

A problem can be posed similar to that of converting the series

CQ C-y C^ Cft

into a continued fraction

_L _L _L a1x+a2+a3x+ ...,

namely, conversion of the series

R(x) = f <***(*). (4-67) fe-0

where {cok(x)} is a system of functions orthogonal with respect to some weight in (a, b), into a sequence of rational fractions of the form

-ffl /».(*) _ £ aiCûi{x)

i -0

(4.68)

Here, in analogy with the conversion of power series into continued fractions the numbers a{ and bt are chosen so that the first n+1 terms of the expansion of the fraction (4.68) in an orthogonal system {ω{(χ)} coincide with the same terms in the series (4.67). The fraction (4.68) will also be termed in this case the n-th convergent.

The problem of finding such a convergent is not solvable in the general case. However, the problem has a solution for certain systems namely those satisfying the relationship

ω^ω^χ) = Ahcok+1(x) + Bkœk(x) + C^^x). (4.69)

Numerous orthogonal systems satisfy (4.69), including in particular all orthogonal systems of polynomials, the system of trigonometric functions {cos nx}, of Bessel functions {Jn(x)}9 and so on.

The formal process of converting the series (4.67) into the fraction (4.68) can be accomplished for these systems as follows. The fraction (4.68) is sought as the nth convergent of a continued fraction of the form

OQ ω^+^ ωλ{χ)+<Χι ωλ(χ)+<χζ ^ ^ ^ 1 + β1ω1(χ) + γ1 + γ2 +β2ω1(χ) + γζ + ...

The requirement that the series (4.67) and the fraction (4.68) correspond can now be written as

Λι(*)-?π(*) Σ CkCOk(x) = 4 ^ n + l ( * ) M + V W * ) + · · · (4.71) Jt-0

Comparison of the coefficients of like functions ω0(χ\ . . . , ωη(χ) in (4.71) leads to the following expressions for the coefficients <xi9 ßi9 yK (/ ss 1, . . . , n) of continued fraction (4.70):

« ι = 1 , & = 7ι = 0, γ^Β,, γη= " l ^ ? " 1 , "π-2

(4.72)

using these, we can find PJx) and qn(x) from the recurrence relations

Λ, (*) = *nPn-i(x) + lo>i(.x)+&JPn_2(x) + γηΡη_3(x), Ì ?nW = <*n0n-l (*) + I»i(*) + W î n - î W + Tuln-zi*), J


where we have put

?_2 = °> ?-i = °> ?o = 1, tfi(*) = 1, P_a = 0, P . i = Q, P0 = C0, P^x) = C0 + C^(x)

(here, y4n, 2?n, Cn are the coefficients of relationship (4.69)). To obtain rational approximations of sufficiently rapid convergence

by this method, an effective approach is often to employ expansions in an orthogonal system of Chebyshev polynomials of the first kind (see § 4, sec. 7).

6. Orthogonal polynomials and quadrature formulae of the Gaussian type

The formula

£ ΜΜχ) * X 4n)/(*(»n)) (4.74) i - 1

for approximate evaluation of the definite integral is called a quadrature formula of the Gaussian type if the base-points x [ n ) ( / = l , . . . , n) and coefficients A^ (/ = 1, 2, . . . , 77) are chosen so that (4.74) is accurate when f(x) is any polynomial of degree not exceeding 2w—1.

Y(n) Y(n) v(n) •*1 > "^2 ' * ' ' » n

Pn-l(*) = y ^(n) 1 Pn(x) Ä " ' x + xW

i.e. An) = *n_ U*in)) β (4J5)

THEOREM 17. If (4.74) is a quadrature formula of the GalLrsian type,accurate for polynomials ofdegree not exceeding 2n-1, its base-points

are the n roots of the polynomial Pn(x) of the system of polynomialsorthogonal in (a, b) with respect to the weight u(x), while the coefficients A~n) are the coefficients in the expansion of the n-th convergentRn_1(x)/Pn(x) of the continued fraction (4.62) into a sum of simplefractions

Alternatively, (4.75) can be written as

r* pn{t)d*{t) 4»> = e(f-*in))/\;(o

In the particular case da(x) = dx and [a, b] = [0, 1], (4.74) is the ordinary Gaussian quadrature formula, the base-points of which are the roots of the Legendre polynomial of degree n.

7, The closure of an orthogonal system of polynomials

The necessary and sufficient condition for closure of a system of polynomials {Pn(x)}, orthogonal in (a, b) with respect to the weight σ(χ) in the space £^χ)(α, b\ is that the problem of moments for the sequence of moments of the weight a{x) be determinate or that σ(χ) be its extremal solution (see ref. 7).

The problem of moments for a finite segment [a, b] is always determinate, so that orthogonal systems of polynomials with respect to any weight on a finite segment are closed. In particular, the systems of Legendre, Chebyshev and Jacobi polynomials are closed (see §4).

The problems of moments are determinate for the Laguerre weight (see § 4, sec. 9) on (0, -f «>), and for the Hermite weight (see § 4, sec. 4) on (-co, +«>), i.e. these systems of polynomials are closed.

Hence the Fourier series in orthogonal polynomials of a function f(x) of L^x) are convergent in the mean to this function in the case of all the classical weights.

A discussion of the convergence of series in orthogonal polynomials {Pn(x)} at every point and of the uniform convergence requires asymptotic estimates of \Pn(x)\ as n -*■ <».

8. Christoffel's formula· The convergence of Fourier series in orthogonal polynomials

Let {Pn(x)} be an orthonormal system of polynomials in (a, b) with respect to the weight σ(χ), and let f(x) be any function of the space L^a, b).

Let us write sn for the nth partial sum of the Fourier series of f(x) with respect to the system {Pn(x)}:

'«(*) = Σ W ) . (4.76) fe-=o

If we make use of the expressions for the Fourier coefficients, sn(x) can be written as

Φ) = Γ/(0 i f h(t)Pk(x)) do{t). (4.77) The expression

Kn(t, x) = f Λ(θΑ(*) (4.78)

is called the kernel of the integral (4.77). The following formula holds for the kernel:

Kn(t, x) = 7 ^ P ^ i ^ P ^ x ) ~ ^ i ^ ) P n W 9 (4.79)

where λη+1 is given by (4.54). Formula (4.79), obtained by Christoffel for the case Û = — 1, t = + 1 , do(x) = <£x, and generalized by Darboux to the case of any weight, is known as the Christoffel-Darboux formula.

Formula (4.77) and the relationship

Γ0 Kn(t, x) da{x) = 1 I o Ü

lead to a formula for the remainder of the Fourier series of f(x) in the system {Pn(x)} :

(4.80) where

/( ')- /(*) 9>*(0=* / - *

The next two theorems, on the convergence of the Fourier series of/(JC), follow from (4.80).

THEOREM 18. If all the polynomials Pn(x) are bounded at the point x and the function φχ{ί) belongs to L^t) (a, b), the Fourier series

Σ ckpk(x) fc«o

is convergent at x and

Λχ) = Σ W ) .

DEFINITION. The sequence of functions ψχ{χ\ ψι(χ\ ...9φη(χ), . . . is said to be uniformly bounded if there exists a constant M such that | φη(χ) | < M for all the functions of the system.

THEOREM 19. If the polynomials Pn(x) are uniformly bounded on fa b)> given any function f(x), integrable on (a, b) with respect to the weight σ(χ), its Fourier series in the system {Pn(x)} is convergent to the value f(x) at every point x for which

I 9x(t)da(t)

exists. The following theorems establish the convergence of the Fourier

series of f(x) in an orthogonal system of polynomials {Pn(x)}9 having regard to the structural properties of f(x) instead of the properties of the system.

THEOREM 20. Iff(x) satisfies the Lipschitz condition with index 1

Ι / ( * 2 ) - / ( * ΐ ) Ι < ^ Ι * 2 - * ΐ | β ,

then the Fourier series off(x) in a system of orthogonal polynomials {Pn(x)} is convergent almost everywhere on (a, b).

DEFINITION. The function Ln{x) = | Kn(t, x) \ da(t) is known

as the Lebesque function of the orthonormal system {Pn(x)}. THEOREM 21. Let f{x) be continuous and let its best approximation

by polynomials of degree n

En(f)= inf max|/(x)-en(;c)| Q»(*) (a, ft)


satisfy the relationship

Ln(x0). En(f) - 0 as n - «>.

Γ/zew *Ae Fourier series of f(x) in the orthogonal system {Pn(x)} is convergent, at x = ;c0, to the value f(x0).

§ 4. Classical systems of orthogonal polynomials

1. Pearson's differential equation

The equation

ρ' _ α0+α ιχ ( _ oc(x)\ Q A> + Ä * + Ä * l ß(x)

is known as Pearson's equation. Solutions of the equation are called Pearson functions. The equation was introduced by Pearson for representing empirical laws of distribution. When ax = — 1, a0 = 0,

A) = 1> ßi — & = 0> the solution is the classical function e 2

— the density of a normal distribution law. The weights of all the most important systems of orthogonal poly

nomials are Pearson functions. JacobVs weight is

ρ(χ) = (1-*) λ (!+*)". (4.82)

where λ > — 1, μ > — 1; ρ(χ) is defined on [—1, 1] and all its moments exist there. Pearson's equation for Jacobi's weight is

j £ = (μ-λ)-(μ + λ)χ ^ ρ l—x2

Chebyshev's weight is a particular case of that of Jacobi, corres-

ponding to λ = -—, μ = — ;

(4.81)

(4.83)

(4.84)

Pearson's differential equation for Chebyshev's weight is

0 ' _ x (4.85)

Legendrés weight Q{X) = 1 is a particular case of that of Jacobi, corresponding to λ = 0, μ = 0. Pearson's equation for it is

— = 7-^-2· (4.86) ρ 1 — x2

Any Pearson function, having all moments, and for which the denominator ß(x) in (4.81) has real distinct roots, is reducible to any of the above functions by means of a linear change of the independent variable.

When the denominator β(χ) has multiple or complex roots, the Pearson functions may not be weights of an orthogonal system of polynomials, since not all moments exist on the interval on which they are defined.

The Chebyshev-Laguerre weight: Q(X) = J A T ^ , where λ > - 1 , μ > 0, (4.87)

is defined on (0, + «), Pearson's equation for the Chebyshev-Laguerre weight is

£ - * Z £ 2 . (4.88) Q X

Any Pearson function, having all moments on (0, + «>), for which the denominator β(χ) in (4.81) is a polynomial of the first degree (/J2 = 0), is reducible to this function by means of a linear change of the independent variable.

The Chebyshev-Hermite weight: ρ(χ) = e"*2 (4.89)

is defined on (— «>, -j-oo). Pearson's equation becomes

Q

Any Pearson function, having all moments on (—<», +<»), f°r

which β(χ) = const, in (4.81) (/3X = β2 = 0), is reducible to a Chebyshev-Hermite function.


Thus all the Pearson functions that can serve as weights of an orthogonal system of polynomials are reducible to one of the above basic weights.

2. The differentia] equations for corresponding classes of orthogonal polynomials

The polynomials of orthogonal systems whose weights are Pearson functions satisfy linear differential equations of the second order, to which various physical problems often reduce — a fact that ensures their importance in applied mathematics.

If the weight ρ(χ) of an orthogonal system of polynomials satisfies

Q ßo+ßit + ßz*2 '

then the nth degree polynomial of this system satisfies the differential equation

β(χ)/' + [φ)+βϊχ)&'-Υιν = 0, (4.90) where γη = Η ^ - Κ Π + Ι ) / ? ^ and α(*), β(χ) are as in (4.81).

EXAMPLE 15. For Chebyshev's weight,

oc(x) = x9 αχ = 1, 0 ( * ) = 1 - χ » , Ä = - l .

The equation for Chebyshev polynomials is of the form (l-x2)y"-xy' + n2y = 0.

3. The expression, by means of the weight, of a polynomial of the nth degree belonging to an orthogonal system of polynomials

Let us consider an orthogonal polynomial system with Pearson's weight ρ(χ) from (4.81). Let Pn(x) be the nth degree polynomial of the system, orthogonal with respect to the weight function ρ(χ). We can write Pn(x) in the form


The formula was obtained by Rodrigues for Legendre polynomials (in 1814), while similar formulae were later obtained for other polynomials. Rodrigues' formula was first published in the general form (4.91) in ref. 5.

If An = 1, the coefficient of the highest term in Pn(x) is

2 n = ft (ai + fc&)> (4.92)

while for x71'1 it is frn = ^Wa"un· ( · 9 )

The coefficient of the highest term in the normalized polynomial is

2° = aï =

■1 Q(X)ßn(x)dx

4. The generating function of an orthogonal system of polynomials with Pearson's weight

Let us take an orthogonal polynomial system {Pn(x)}9 where Pn(x) is defined by (4.91) with constant An = 1.

The generating function of the system is the function ψ(ζ, w) of two complex variables z and w such that

Vfr H>) = f ^ J l ^ v v » . (4.95) 71*0 n i

THEOREM 22. G/ve/2 a/ry orthogonal system of polynomials {Pn(x)} with weight function ρ(χ\ satisfying condition (4.81), there exists a generating function (4.95), which is given by

^w^àj^k' ( 4 · 9 6 )

where | w is the root of the quadratic equation | - z -w /3 ( | ) = 0 (4.97)

that is close to z for small w.


EXAMPLE 16. Let us find the generating function for the Legendre polynomials.

Equation (4.97) becomes in this case

wf2-j-!-(z + w) = 0, where

!„ = - ^ ( - l ± V l r4wz+4vt^)

is chosen in such a way that èw is close to z for small H>, i.e.

!„ = ^ ( - 1 + 7 1 + 4 ^ + 4^ ) .

On applying formula (4.96) and putting ρ(χ) = 1, we get

ψ(ζ, w) = * s (4.98) Vl+4»vz+4w2

The derivatives of polynomials orthogonal with respect to Pearson's weight are also orthogonal polynomials with respect to the weight

. . if φ) + β'(χ) , 1 ei(x) = e x p | J β{χ) dx^

5. Legendre polynomials

The first system of orthogonal polynomials was historically the system of polynomials with the weight function Q{X) = 1 on [—1, + 1], introduced by Legendre in 1785.

We introduce the following notation: let Ln(x) be the Legendre polynomial, in which we have not fixed the factor, to an accuracy of which the system of orthogonal polynomials is defined, let Ln(x) be the nth degree polynomial with highest coefficient equal to 1, and let Ln(x) be the normalized Legendre polynomial.

Rodrigues' formula is

£η(χ) = Λ ^ ~ 1 ) " ] . (4.99)


Formulae (4.92)-(4.94) lead to the expressions forLn(x) andLn(x:) r « ( * ) = (^£ ( * 2 - i ) n ' (4·ιοο)

î „ W = ^ ( ^ ï ï ^ ( ^ - l ) « . (4.101)

The explicit expression for a Legendre polynomial is

!*(*) = Λ ftI ( - 1)" Κζ_£γ. C^n"2ft· (4·102) It follows from (4.102) that Ln(x) is an even function in x when n is even, and odd when n is odd.

A recurrence formula. For the Legendre polynomials, we have in (4.54):

* - o i - (* + 1)2 7 1 + 2 ~* υ ' ^ + 1 ~ (2Λ + 1)(2II 4-3) »

i.e.

W * ) - ^ + ΐ ω - ( 2 / , ) " ^ 2+ 3 ) Ζ η ( χ ) . (4.103)

Obviously, Z.0(*) = 1> Ai(*) = x> w e further obtain from (4.103):

£2(^ = 1 ( 3 ^ - 1 ) ,

A W = -r(5x3 — 3x),

35 L4(x) = ^-(35**-3Qx» + 3),

£5(ΛΤ) = - L (31 5X5 - 350*3 + 75*) and so on.

The continued fraction (4.62) for a Legendre polynomial becomes

1 ± 2_ (n + 1)2

1_J__il-il_ (2«+ 1)(2« +3) m) X X X X ' ' ' X

The denominator of the nth convergent of (4.104) is Ln (x).


The continued fraction (4.104) is convergent to 1η(χ+1)/(χ— 1) at every point x lying outside [—1, -hi].

The generating function. Let H(x9 w) be the generating function of the Legendre polynomials (4.95); (4.98) now gives

and

where

or

H(x9 w) =

1

1 >/l f4w* + 4H>2

_ _ Y Ln(x) ^n y/l-i-4wx+Avfi n-o n\

(4.105)

H(x,w) = l~2xw-f2(3jc2-l)H^-4(5^3-3^V + + 2 ( 3 5 * 4 - 3 0 * 2 + 3)Η>4-Γ- . . . .

The generating function of the Legendre polynomials is often written in the form

so that

H(x, t) = , , where / = - 2w9 yfl-lXt + t*

= Σ «*)**. y/l-2xt + t2 fe-o

where lh(x) is given by (4.99) with Ak = 1/(2Ä:)Ü. The polynomials Ln(x) and LJx) are expressed as follows in terms

of ln(x):

Ln(x) = (2n-l)!! 'nW;

Ln(x) = J^p-ln(x). (4.106)

The polynomials ln(x) satisfy the recurrence formula

(«+2)/„+2(x) = (2η + 3)χ1η¥1(χ)-(η + 1)Ιη(χ). (4.107)

The differential equation for Legendre polynomials is

(1 - x*)y" - 2xy'+n(n + \)y = 0.

The integral form of Legendre polynomials. The polynomial

1 dn

/ n W ] = ! ( 2» ) ΪΓ"ώ?^~ 1 ^

can be written in the form

k (x) = - (x + Ι ' Λ / Ϊ 3 ^ COS φ)η d<p (4.108) π Jo

The integral in (4.108) is known as a Laplace integral (it has a real value for real x, in spite of the integrand being complex). The polynomials ln(x) are uniformly bounded on the segment of orthogonality [-1, + il-

it follows from (4.108) that

I'«(*)l 3 1 ;(4.109) for

|*| S 1.

A stricter inequality holds for points lying inside the interval ( - 1 , +1):

l ' ^ ' ^ v i b · (4110)

Turarìs inequality is

for n 2= 1, - 1 ^ X^L 1. The following are expansions of certain functions in Legendre

polynomials:

A 2*(2n-2)...(2»-2fc + 2) + * V U (2B + l)(2» + 3 ) . . . (2» + 2fc+1) '2ftW'

3 2« +3 k(x) +

A 2n(2n-2)...(2n-2k + 2) + ^ we + V (2n + 3){2n + 5), . .(2n+2k + 3) '2k+lW'

7ra=7,?o( 4"+ 1 ){^^rr2 k W' |x|<1' V î 3 ? 2ft-o' 22k+1À:!(fc + l)! "*+1V" '

V1 Χ " Τ { Τ h i ( 4 f e + 1 ) 2*"«*!(* + l)! /2ft(X)J'

π ~ /(2fc-l)!!\2r , . . . , Λ1 . . , arc sin x = — £ l ^ \ [l2k+1(x) - Incidi 1*1*1.

The convergence of Fourier series in Legendre polynomials. THEOREM 23. Iff(x) has a continuous second derivative in [— 1, +1],

it can be expanded as a uniformly convergent series in Legendre polynomials in [—1, +1].

By (4.106) and (4.109), we have the following inequality for normalized Legendre polynomials:

| 2 » ( * ) | * ^ 2 ϋ + Ι for |* | s i . (4.111)

Formula (4.111) and the corresponding inequality for the kernel Kn(t, x) of Ln{x) give an inequality for the Lebesgue function (see § 3, sec. 8):

L„(*)-(n + l)2. (4.Π2) A consequence of Theorem 21 (see § 3, sec. 8) and (4.112) is:

THEOREM 24. Every continuous function f(x), the best approximation of which satisfies the condition

lim «*£„(/) = 0 (4.113)

can be expanded as a uniformly convergent series in [—1, +1] in Legendre polynomials.

The following is a consequence of Theorem 23 and formulae (4.106) and (4.110):

THEOREM 25. A function f(x), the square of which is integrable in [— 1, 1], can be expanded as a Fourier series in Legendre polynomials convergent to fix) at every point x for which the integral exists:

REMARK. Condition (4.114) is satisfied, in particular, if the finite derivative/'(x) exists at the point x.

THEOREM 26. Letf{x) be a function, the square of which is integrable in [—1, 1], and suppose the left- and right-hand limits f(x—0) and /(x—0) exist at the point x In this case, if the integrals

j* //(0-/(*-Q)p, and ^-f^VJdt are finite, the Fourier series in Legendre polynomials is convergent

at the point x to— [/(*-0)+/(;c+0)]

THEOREM 27. Iff(x) satisfies the Dini-Lipschitz condition in [—1, + 1]:

lim ω(<3) In (δ) = 0

{where ω(δ) = sup {|/(*ι)—(Λχύ 1} is ^e oscillation of

f(x))> it can be expanded at every point of the interval as a Fourier series in Legendre polynomials, the convergence being uniform in every segment [-1 + A, 1 -A] (A > 0).

6. Jacobi polynomials

The Jacobi polynomials are polynomials orthogonal in [—1, +1] with respect to the weight

ρ(χ) = (1-χ)λ(1+*)<*, (4.115)

where λ > — 1 , μ > — 1. The weight function (4.115) is Pearson's weight (see § 4, sec. 1). The above definition specifies the Jacobi polynomials apart from a constant factor. If this factor is not fixed,


we shall denote the polynomials by J(n,fl)(x); we write ^'μ>(χ) if the coefficient of xn is unity, and j£> μ\χ) if the polynomial is normalized.

The Legendre polynomials (see sec. 5) are a particular case of

Jacobi polynomials, with λ = μ = 0. The case λ = μ = —and 1 2

λ = μ = ——-are specially treated later (see sec. 7, 8). The polynomials corresponding to these values of λ and μ are called Chebyshev polynomials (of the second and first kind respectively).

In general, the case λ = μ has certain special features. The Jacobi polynomials with λ = μ are said to be ultraspherical.

Rodrigues* formula. Formula (4.91) gives for Jacobi polynomials:

Jfri*(x) = An{\ +*)-* | l [ ( l -*)*+» (1 +*)*■«], (4.116)

where we have for /^' μ)(χ) : _ ( /XA + y + ii + 1)

and for )%>μΧχ):

A - f _ n » / 2 - ^ ^ « n + i ) ^ + f i t + l)/XA + /i + 2n+l)

(4.118) These formulae are suitable for n > 0. They are also true for n = 0, provided λ+μ-fl τ* 0; if λ+μ+1 = 0, they lose their meaning at n = 0, though it is clear that

7M*)=i. 4/2 explicit expression. If 4n = (—l)n/2nw! in Rodrigues' formula

(4.116), we shall write $*μ\χ) for the corresponding Jacobi polynomial. It is often convenient to discuss this Jacobi polynomial. For it, the following formula holds:

Αλ'μ)(*) =

(4.119)

A recurrence relation. The general recurrence relation (4.54) for three consecutive polynomials of a system of orthogonal polynomials becomes, in the case of the system of Jacobi polynomials;

■Jftft*) = ( ^ - « η + Λ ^ Μ - ^ + ι Τ ^ ^ λ (4.120) where

μ2-λ2 α η + 2 = (λ + μ + 2η + 2)(λ+μ + 2η + 4) ' 2 (λ + η + 1)(μ+*+1)(λ+μ + η + 1) Λ(η , η ^ 1 " {λ+μ + 2η + 1)(λ+μ±2η+2γ(λ+μ + 2η + 3)*Κη^1)- !

(4.121) Since

7<λ><*> = 1 and 7ίλ'«> = χ- / ~ ^ Λ ,

all the Jacobi polynomials can be obtained successively from (4.120). These expressions are very unwieldy, however; e.g.

2 w T λ + /ζ + 4 τ ( λ + / ι + 3)(λ + μ + 4)

and so on. The explicit formula (4.119) is more practical for their actual computation. Complete tables of Jacobi polynomials are available (see ref. 3). The Jacobi polynomials satisfy recurrence relations, not only with respect to the parameter w, but also with respect to the parameters λ and Λ, λ, μ and n, these being consequences of Gauss's formulae for the hypergeometric functions; the Jacobi polynomials are a particular case of the latter.

The generating function. Let I(x, w) be the generating function of the Jacobi polynomials (see sec. 4). Formula (4.96) now gives

I(x, w) = , (1 +2w + Ji +4wx +4νν2)-λ χ y l + 4xw + 4w2

χ ( ΐ -2νν + νι+4Η'χ + 4>ν2)"μ (4.122) and

oo Ί(λμ)(χ\ Kx, w) = Σ , " * > (4.123)

where ~ dn Jfrr* = (l-x)-k(l+x)-*^[(l-x)<*+n>(l+x)*+*]. (4.124)

The generating function for the Jacobi polynomials is often written in the form

/(*, 0 = , 2 λ 4 μ (l-t+y/l-2xt + t*)-kx yj\-2Xt+t2

χ(ΐ + ί + νΐ -2χί-Κ 2 )~ μ , (4.125) where t = — 2w; the coefficients of the expansion in powers of t are now the polynomials j£> μ\χ) (see (4.119)). In the case of ultra-spherical polynomials (λ = μ), the generating function is simplified if we introduce, instead of polynomials j£· μ\χ)9 the polynomials

r(x+j)r(n + 2x) l

γ(η\χ) = — -r prJk Hx)> w h e r e « = λ + y ,

in fact,

Γ(2α)Γ(>ζ+α + ^

(1-2χί+ **)-« = Σ )4β)(*)'η· (4.126)

By using the ultraspherical polynomials y£, N. Ya. Sonin obtained an analogue of Taylor's formula:

Λ*+α) = 2 T « f (n+ v) % ^ ^ ^ / ( * ) , (4.127)

where /fe(a) is Bessel's function, D = rf/rfx. A differential equation for Jacobi polynomials. Differential equation

(4.90) for polynomials orthogonal with respect to Pearson's weight function, becomes, in the case of Jacobi polynomials:

(1-χ?)γ" + [(μ-λ)-(μ + λ + 2)χ)γ' + η(λ + μ+η + l)y = 0. (4.128) Inequalities for Jacobi polynomials and the convergence of the

Fourier series. Given the condition a = max {λ, μ} ^ - y (4.129)

the following theorems hold.

THEOREM 28. The greatest value of the modulus of j£> μ\χ) is attained in the segment [— 1, 1] at one of the points x = ± 1.

THEOREM 29. Given condition (4.129), the normalized Jacobi poly-nomials satisfy the relationship

I 4 λ ' μ)(*) I < Mn 2

for all \x\^ 1. Here M is a constant, depending on λ and μ. It follows from Theorem 29 that the Lebesgue function of the

Jacobi polynomials satisfies

Ln{x) < MJ*+\

which leads, in conjunction with Theorem 21 (see § 3), to

THEOREM 30. Let a ^ — —- and let p be a positive integer not

less than 2cr-f2. Every function /(*), defined in [—1, 1] and having a continuous derivative of order p, can be expanded as a uniformly convergent Fourier series in the polynomials j£' μ\χ).

7. Chebyshev polynomials of the first kind

Chebyshev polynomials of the first kind are a particular case of

Jacobi polynomials (see § 4, sec. 6), corresponding to λ = μ = .

The polynomials J^ 2 ' 2 ' were first discussed by P.L. Chebyshev in 1857, when solving the problem of the best approximation of continuous functions by polynomials. We shall denote them by Γη(χ); these polynomials were obtained by Chebyshev in the form

Tn(x) = cos (n arc cos x). (4.130)

Formula (4.130) defines Tn(x) only on the segment [—1, 1], But definition (4.130) can be extended to all values of x by means of the familiar trigonometric formula.

cos ηφ = cosn <p—C% cos11"2 φ sin2 φ -f- Q cosn-4 φ sin4 φ — . . .

(this completion of definition (4.130) will be assumed without making any special proviso in future).

The polynomials Tn{x) possess many remarkable so-called extremal properties in addition to their general properties as polynomials of an orthogonal system; these extremal properties will be described after dealing with the general properties.

Rodrigues' formula is r-w = ^ ^ £ ( 7 r b ) " · <4·13»

We shall write Tn(x) for the polynomial with highest coefficient equal to unity, and Tn(x) for the normalized polynomial.

An explicit expression. Formula (4.130) yields an explicit expression for the Chebyshev polynomials, where

1 Tn(x) = 2^n c o s (n a r c c o s *)»

[2 Tn(x) = /—cos (n arc cos JC).

Expression (4.130) can also be written in the form

Tn(x) =j[(x+ Φ^ϊ)η+ (x- V^T)"]

(4.132)

or

m (4.133)

Tn(x) = Σ ( - D" —u C*_k2»-«*-i^-«».

As follows from (4.130), the zeros of Tn(x) are the numbers (2k-l)n xin) = cos: . " In

The recurrence formulae

(k = 1, 2 ri).

Tn(x) = 2xTn_x{x)-Tn_2(x\

Tn(x) = xTn.y(x)-jTn.z(x)

(4.134)

(4.135)

(4.136)

follow from (4.120) with λ = μ = — - . Since T0(x) = 1, T^x) = x,

it follows from the equation (4.135) that

and so on. In the case

(4.62) becomes

since here,

λ« = #»ο

7i(*) = 2x«-l , Τ3(χ) = 4*3-3χ, Γ4(χ) = 8χ4- 8^ + 1, Γ5(Λ:) = Ιόχϊ-ΙΟχϊ + δχ, Τύ (χ) = 32*β - 48χ4 + 18^+1

of Chebyshev polynomials, the continued fraction

1 1 1

- _ ± _ ± _ ± _ (4.137)

= jr = 1 . , X< = —, ώ = 0. L V l - * 2 4 ' '

The denominator of the nth convergent of the continued fraction (4.137) is Tn(x). The continued fraction is convergent, for all x outside the interval (—1,1), to the function n/y/x2—l.

The generating function. The coefficient of tn in the expansion of

^ '> - T^F <4-1 3 8> in powers of t is the polynomial Tn{x)\

The differential equation for Chebyshev polynomials is (1 - x2)y" - xyf + n2y = 0. (4.139)

Expansion of a function as a Fourier series in Chebyshev polynomials and the comparison of this with the expansion as a Maclaurin series. Wide use is made of Fourier series in Chebyshev polynomials Tn(x) for uniform approximation of functions. Notice that the expansion of a function/(x) in [—1, 1] as a Fourier series in polynomials Tn(x) reduces to the expansion of /(cos x) in [—π, π] as a Fourier series in cosines.

For instance,

eacos9=: /o(f l ) + 2 Σ ln(d) cos ηφ, n=l

where 7n(a) is Bessel's function; hence the substitution x = cos 9? gives

* * = /0(e) + 2 £'»(«)Γ»(*). n - l

An expansion may similarly be obtained for f(x) = xn from the familiar trigonometric formulae:

COs2n + Q n

c o s » - ^ = -55 =5- Y <&-i cos (2/2-2*- l)cp.

The substitution x = cos 9? gives

1 (n"x \ ^^δηί Σ 2C2nr2n_2fe(*) + Qn li v2n

1 ^ 1

z fc»0

(4.140)

Similarly, expansions in Chebyshev polynomials Tn(x) follow from the expansions in cosines (with | JC J < 1):

Ivi — —±— V ^'in ^ 1 1 "" π^π&Μ-Ι '

4 A sign* = — £ (-1) m+i π n - l

Γ»η-ΐ(*) 2 n - l '

COSÛX = J0(a) + 2 £ (-1)ηΛη(α)Γ2η(χ), n-x

sin ax = 2 £ ( - ^ + ν , , - ^ Γ ^ ί χ ) , n - l

arcsinx = - £ ( - l)»+i ■7»-1 W

W n - l (2n-1)2

(4.141)

(4.142)

(4.143)

(4.144)

(4.145)

e°* = T0(a) + 2 £ In(fl)Tn(x), (4.146) TI» 1

l n ( l - 2 ^ + 92) = - 2 g 7 ? n W (l?l < 1). (4.147)

In general, if Ck is the Fourier coefficient of/(cos x) in the system {cos ηφ}, we have

2 Ρπ 2 Γ1 dx Ck = - / ( c o s ^ c o s ^ ^ = - / ( * ) Γ ^ * ) — = = . (4.148).

TTJO JrJ-i V 1 - * Roughly speaking, when such functions are represented by the sum

of a like number of terms, the limits of the accuracy are 271""1 times better for an expansion in Chebyshev polynomials than for a Taylor series.

For example, when | JC| ^ 1 follows from (4.146) that, in the case of f(x) = e™ with large n,

[ an an+2 "I an

7ύ2« + (η+ l)!2»+* + · · -J % ^ 2 ^ · '

while at the same time

/<n>(0) _ an

If we substitute the Maclaurin series for /(*)

/(*) = / ( 0 ) + / ' ( 0 ) x + Œ - x 2 + . . m +(^jfl f . . . ,

under the integral sign in (4.148), in view of the orthogonality of the polynomials Tk(x) and the expressions (4.138), (4.140), we get an interesting connection between the coefficient Cn in the expansion of f(x) in Chebyshev polynomials and the corresponding coefficient of Maclaurin's formula for f(x)9 viz

_ 1 //<">(0) /<"+*>(0) /<"+"(0) \ C n - 2 \ ^ n\ + (* + 2)!l! > + 4)!2! + '")' ^ ^

It follows from (4.149) that, if the principal part of/(n)(0)/n! + . . . ) reduces to/(n)(0)/w! for large n, we have

1 fin)(0) C n % y ~ ^ r · ( 4 · 1 5 0 )

It follows from (4.150) that, given | JC | < —, the expansion of a

function as a Maclaurin series is in general better, whereas the expan

sion in Chebyshev polynomials is better for — < \x\ < 1, since

max\CnTn(x)\ = C n ^ n < max j < l * l £ i

The convergence of Fourier series in Chebyshev polynomials. In view of the fact that

\Tn(x)\*l9

it follows from Theorem 19 § 3 that: THEOREM 31. Given any function f(x\ integrable in (—1, 1) with

respect to the weight l/y/l—x2, its Fourier series in the system {Tn(x)} is convergent to the value f(x) at every point x for which the integral

p fix)-fit) dt

exists. The Lebesgue function of the system of Chebyshev polynomials satisfies the inequality

Ln(x) ^ 2-f-lnw.

We therefore have (see Theorem 23 § 3) : THEOREM 32. Every function f(x) for which

lim En(f) In n = 0, n—»-oc

can be expanded as a uniformly convergent series in Chebyshev polynomials.

The extremal properties of Chebyshev polynomials. THEOREM 33 (Chebyshev). Of all the polynomials with highest

coefficient equal to unityy the polynomial Tn(x) has the least deviation from zero, i.e.

max | f n (x ) |< max |βΛ(χ)|

for all polynomials Qn(x) of degree n, having highest coefficient equal to unity.

COROLLARY. Since

max | f n ( * ) l = = = ï »

we have for any polynomial Qn(x) with highest coefficient equal to unity:

max | ρΛ(χ) | > ; — j .

The following interpolation property of the zeros of a Chebyshev polynomial is a consequence of the theorem. Suppose we form, with respect to the points xl9 x2, . . . , xn an interpolation polynomial ôn-iW °f degree n—1 for a function/(x), n times differentiable in [—1, 1]. The remainder term of the accurate interpolation is now given by

Unix) =/ (*)-ôn- l (*) = · ^ ρ ( * - * ΐ ) ( * - * 2 ) . . . (*-*»),

where Xi < xi+1 and ξ ç (xl9 xn).

If the interpolation base-points xl9 x2, . . . , xn are zeros of the Chebyshev polynomial Tn{x\ then

max | (χ-Χχ) (x — x2) · · · (* — **) 1 «€[ -1 ,1 ]

has a minimum value. Thus, if the coefficient of/(n)(|) in (4.148) changes only a little in

relation to the variation of x in (— 1, 1), the interpolation base-points which are roots of the polynomial Tn{x\ yield the least value of the remainder term.

THEOREM 34 (Chebyshev). Of all the polynomials Qn(x), subject to the condition βη(£) = M, where \ξ\ < 1, the polynomial ΜΤη(χ)/Τη(ξ) deviates the least from zero in the segment [—1, 1].

THEOREM 35 (A. A. Markov and V. A. Markov). If the polynomial Qn(x) satisfies, in [—1, 1], the inequality

\QnM\^M,

the derivative of order k of this polyomial satisfies in [—1,1] the inequality

the sign of equality being obtained in (4.151) only for the polynomial Tn(x) at the points x = ± 1 .

8. Chebyshev polynomials of the second kind

The polynomials of the second kind with respect to the weight 1/Vl—*2 (see § 3, sec. 4), i.e. the numerators of the convergents of the continued fraction (4.137), form an orthogonal system in [—1, 1] with respect to the weight ρ(χ) = Vi—*2 and are therefore the

Jacobi polynomials corresponding to λ = μ = —-. They are known

as Chebyshev polynomials of the second kind. We shall denote them by Un(x).

The Chebyshev polynomials Un(x) and Τη^(χ) are connected by the relationship

Un(x)= Cn^Tn+1(x). (4.152)

An explicit expression for the polynomials. We have from (4.152):

Un{x) = Cn *»[ ("+!>"*«»*] . (4.153)

Formula (4.153) defines Un(x) only in the interval (—1, 1), but this definition may be extended to all x by means of the familiar trigonometric identity

sin(w -f \)φ = sinn+1 φ— C%+1 sin71*"1 φ cos2 <p+ + Q+i sinn~3 φ c°s4 φ—

In C/n(jc), let the coefficient of xn be unity, while \\Un\\ = 1.

Now, - w _ | s m K 2 ± ^ £ o M ; (4154)

V π yjl — x2

Recurrence formula. As the numerators of continued fraction (4.137), the polynomials Un(x) satisfy the same recurrence relation as Tn+1(x):

Ûn+2(X) = *#Λ+ΐ(*)—^π(*λ

where £/<>(*) = U U±(x) = x; hence

U2(x) = x*-j,

Uz(x) = x3 — *,

- 3 1 i/iOc) = A:4 —-r-Xo-f-TT anc* so on.

4 4 * 16 4 continued fraction for polynomials Un(x). These polynomials

form an orthogonal system with respect to the weight ρ(χ) = Vi— *2

and are in turn the denominators of the nth convergent of the continued fraction

ÜL 1 1 A JL — x — * — χ — . . . , (4.156)

where

A0 = ~ = P V î 1 1 ^ ^ whüe k=j ( / = 1 , 2, . . . ) .

The numerators of the nth convergents of the continued fraction (4.156) are Unmml(x). For all x lying outside the segment [—1, +1], the continued fraction is convergent to the function

/(χ) = ( χ - ν ^ ^ Ί ) π .

Fourier series in Chebyshev polynomials Un(x). The following inequalities hold:

\ΰη(χ)\^^(η + 1) ( - Ι Ξ Χ Ξ Ι ) ,

| ^ n W I ^ . / - - 7 = = ( - 1 < * < D , (4.157)

The next theorems are consequences of (4.157) and the general theorems (see § 3, sec. 5).

THEOREM 36. Every function, having a continuous derivative of the third order, can be expanded as a uniformly convergent series in polynomials Un(x).

THEOREM 37. Every function f(x) of Lfc—*(—1,1) can be expanded as a Fourier series in the orthogonal polynomials Un(x) at every point x, and for every f(x) the following integral exists:

L ^ i f 0 ) ' ^ * · (4·ΐ58)

THEOREM 38. If a function f(x), defined on [—1, 1], satisfies the Dini-Lipschitz condition

l i m o s i n i = 0, (4.159)

it can be expanded in the interval (— 1, 1) as a Fourier series in polynomials Un(x), the convergence being uniform in any interval (— 1 + +Λ, 1-A).

Given suitable convergence conditions, (4.152) enables us to differentiate the expansion

/ (* )= ΣαηΤη(χ) (4.160) n-o

term by term, to obtain the expansion

n«=l


Expansions (4.139—4.142) given above in polynomials Tn lead to the following expansions in polynomials Un:

eax

sign x

sin ax

cos ax

and so on.

An extremal property of polynomials Un(x).

THEOREM 39 (Chebyshev). Of all the polynomials Qn(x) of degree n with highest coefficient equal to unity, the least value of the integral

Γ \Qn(x)\dx

is given by Qn{x) = Un(x).

9. Laguerre polynomials

The polynomials orthogonal in the interval (0, + °°) with respect to the weight function ρ(χ) = xae~x (a > — 1) are usually called Laguerre or Chebyshev-Laguerre polynomials.

These polynomials were first encountered in the case a = 0 in the analytical mechanics of Lagrange, then in Abel's posthumous papers. Chebyshev discussed the polynomials in 1859, and for them obtained a recurrence formula and an expansion as a continued fraction; it was only in 1878 that they were considered by Laguerre. The case of any a > — 1 was first discussed by Sokhotskii.

Some authors only speak of Laguerre polynomials in the case a = 0, and refer to generalized Laguerre polynomials when α?ί0. We shall denote the latter by Z,£(x).

= T Σ nIn(a)Un^(x\

4 ~ 2nU2n_x(x) = ^ y

= T l(-l)n2nJ2n(aW2n-i(x), a n - 1

4 l ("l)n+1(2«-lV2n.lW^2(n-l)W a n - i


Rodrigues' formula is

Ufa) = Anx-«é*^(x«ine-*) (4.161)

The polynomial L„(x) with highest coefficient equal to unity is obtained from (4.161) with An = (— l)n, while the normalized L^(x) is obtained with

A = (-1)»

A recurrence formula: K+2(x) = (x -«-2n-3)I« + 1 (x ) - ( i i + l)(n+a + l)ZsM. (4.162) In the case a = 0 this reduces to

W * ) = (*-2«-3)Zn+1(*)-(* + l)2Z„(xX from which we find, using L0(x) = 1, -L x) = x— 1,

Z2(;c) = x2-4;c-f-2, L3(;t) = x3 —9*24-18χ-6, and so on.

A continued fraction for polynomials L„(x). In the case a = 0, the continued fraction (4.162), of which the denominator of the nth convergent is Ln(x), takes the form

1 l2 22 32

x— 1 — x— 3— x — 5 — x —7 — . (4.163)

The continued fraction (4.163) is convergent for all x not lying in (0, -foo) to the function

/w = A. Jo * - '

The continued fraction for any a, Γ(χ + 1) a+1 2(a + 2) 3(a + 3)

x _ ( a i . l ) _ x _ ( a + 3 ) - x - ( « + 5 ) - x - ( a + 7) (4.164) is convergent for any x not lying in (0, + «>), to the function

/(*) = A. Jo *" '

A differential equation for L„(x) :

xy" + (a + 1 - x)y' + ny = 0. (4.165) Sokhotskii obtained a formula for the power xn in terms of Laguerre

polynomials: n Γο fx\

χη = Γ(η+α + 1) Y η / η7*

The generating function is

xt (-i)vi;(*) τΰ-0

( 1 - 0 — » β χ - { = T K~ ' , - (4-166>

Formula (4.166) was first obtained by Sokhotskii. The generating function can also be written in the form

(ly/tx) *-° « ! Γ(α + /ι + 1 ) where 7β(ζ) is BessePs function. Formula (4.167) was obtained by N.Ya. Sonin. Sonin also obtained the relationships

1 dLl+1(x) _ - dLttx) (n + 1) rfx - ^ w <fc » n(n+*)L<n%(x) __ d [T*{x)\

x71-1 1 [ *n J (4.168)

Closure of the system [L°n(x)}. The problem of moments for the sequence

μη= Γ e~xxa+ndx

is determinate, from which it follows, by Riesz's Theorem, (see § 3, sec. 7) that the system {L'n(x)} is closed in the space L2_x^ (0, + °°).

The Laguerre polynomials can be obtained from the Jacobi polynomials y^·μ) by means of a passage to the limit:

/£(*)= lim # . 1 0 ( 1 - ? * ) (4.169)

(K. A. Posset's formula).


10. Hermite polynomials

The polynomials, orthogonal on (— «>, +°°) with respect to the

weight ρ(χ) = e 2 are known as Hermite polynomials. This name is sometimes given to the polynomials orthogonal with respect to the weight e~*2. Such polynomials are first encountered in Laplace's works, then in those of Chebyshev (1859). Chebyshev obtained a Rodrigues' formula and an expansion as a continued fraction. They were considered by Hermite five years after Chebyshev. Rodrigues' formula is

Hn(x) = Ane> — U 2 J , (4.170)

when An = (— l)n we get Hn(x) — the polynomial with highest coefficient equal to unity; when An= (— l)n/\/n\ V2^weget the normalized Hn(x).

The recurrence formula is

HM(x) = χΗη(χ)-ηΗηΛχ); (4.171)

since H0(x) = l9Hx(x) = x, we get

Ηάχ) = χ?-1, H3(x) = x3 — 3χ9

H4(x) = x 4 -6* 2 + 3, Ηδ(χ) = x5—10*3-]-15x, and so on.

The generating function is

/*""*= Σ ^ τ ^ . (4.172) n-o Λ!

The continued fraction (4.62), of which the denominator of the nth convergent is the polynomial Hn(x\ is

^ L ! L . (4.173) X — X- X— X— . . .

Fraction (4.173) is convergent for all non-real x to the function

ί du -oo X—t

A differential equation for the Hermite polynomials: y"-xy'±ny = 0. (4.174)

The polynomials Hn{x) can also be obtained from the Jacobi polynomials /η

λ' k\x) as λ -+ «> :

T-J ï -1 /Mvi r ) ] · (4·175>

Closure of the system of Hermite polynomials. The problem of moments for the sequence

μη = xne 2 dx

is determinate. Hence it follows by Riesz's theorem (see § 3, sec. 7) that the system (Hn(x)} is closed in the space L2

t (— <», + <»). e έ

11. Chebyshev polynomials, orthogonal on a finite system of points

In a paper of 1855, "Continuous fractions" (O nepreryvnykh drobyakh), and in certain other works, Chebyshev considered the expansion of the sum

t - i ζ — Χχ

as a continued fraction and investigated the properties of the denominators of its convergents.

If σ(χ) is a step function with growth points xl9 x2,..., xN, the jumps at which are equal to mx, m2, . . . , mN respectively, then

£*5>_r*w. (4.176)

The denominators of the convergents of the continued fraction (of type (4.62)), corresponding to integral (4.176), are now polynomials orthogonal with respect to the weight function σ(χ).

The orthogonality of Ps(x) andPÄ(x) with respect to σ(χ) implies in this case that

£ 0\xi)P9(xoPk(xd = 0 s*k. i « l

The continued fraction (4.62) is now finite, the system of orthogonal polynomials is also finite and contains precisely N polynomials P0(x), Pi(x)> ...» -Pjv-i (*)· Chebyshev applied the results of his investigations to interpolation by the method of least squares which consists of the following. The values of the function f(x) are specified at the points xl9 x2, . . . , xN. Among all the polynomials of a given degree n < N9 we have to find the polynomial Qn(x) such that the sum

ΣΐΛχυ-QntxùT (4.177) i«=l

is a minimum (see § 1, sec. 3). Let Θ\χ^ = 1 in (4.176); then

f[/(*i)-Ôn(*i)l2= Γ (/(*)-Ön(*))2<fc(*). i - I J -oc

And, as follows from § 2, sec. 4, (4.177) takes its least value when Qn(x) is the nth segment of the Fourier series of f(x) in a system of polynomials orthogonal with respect to the weight function a(x).

Using a system of equidistant base points on the segment [0, 1]:

Xi=jf (*= U 2, ...,N).

Chebyshev formed an orthogonal system of polynomials {0(*^2 = 1}. These are also known as Chebyshev polynomials. We shall denote them by Pk j^x) (the polynomial of degree k in the system {x{ = = i/N}):

^fc,iv(*)= l+alx + a%x(x-l) + a3x(x--ì)(x-2)+ . . . . . . +akx(x- 1) . . . (x—fc + 1),


where „ _ {-iyckc>k+1 " n(n-l)...(n-s+l)

In particular,

PO.N(X) = 1,

P2,N(x) = 1-6^ + 6 ^ - ^ ,

P (ï\ - 1 n * i 10 ^ * " 1 ) gn * ( * - ! ) ( * - 2 )

_ 1 4 0 * ( * - ! ) ( * - 2 ) + 7 0 * ( * - ! ) ( * - 2 ) (*-3) N(N-l)(N-2) x JV(JV- l)(iV-2)(iV-3) '

P5,„(*)= 1 - 3 0 ^ + 2 1 0 - ^ ^ -

5 6 0 x(x-l)(x-2) | 6 3 0 * ( χ - 1 ) ( χ - 2 ) ( * - 3 ) N(N-l)(N-2) ' N(N-l)(N-2)(N-3)

-252 *(*~ 1} ( * ~ 2 ) ( *~ 3 ) (*~ 4 )

' N(N- 1) (tf - 2) (N- 3) (iV- 4) '

and so on. The interpolation polynomial of degree m with respect to the system base points {x4 = i/N} for/(x), denned by the method of least squares, is the segment of the Fourier series of f(x) of order m with respect to the system {P^^ix)}:

Qm(.x)= ZCkPk,N(x),

where

= {1f(x)Pk,Ni


The expression for Ck does not depend on the degree of interpolation m. On increasing the degree of the interpolation polynomial new terms are simply added:

Ôm+i(*) = Qm(x) + Cm+1Pm+lfN(x).

Chebyshev showed that the case when σ(χ) is continuous can be obtained by means of a passage to the limit from the case when σ(χ) has a finite number of growth points.

In particular, the Legendre polynomials can be obtained from the polynomials PktN(x) as N — «>. If Lk(x) is a Legendre polynomial, normalized by the condition that Lk(l) = 1, we have

(4.178)

CHAPTER V

CONTINUED FRACTIONS

Introduction

1· Notation for continued fractions. Basic definitions

An expression of the form

A _L α ΐ η

o2 + . '.On

is called a continued fraction. In view of the unwieldinessofthe above method of writing, several

authors have proposed different methods, for instance:

, (Pringsheim)

(Müller)

(Rogers) (5.1)

We shall use the last notation. Pringsheim also proposed writing a continued fraction in the form [b0; ajbv]~. If al9 and Ζ>α in fraction (5.1) follow a different law to the remaining an and bn, Pringsheim used the notation [b0; αφχ, ajbv]~.

The fraction ajbn is called the n-th partial quotient of the continued fraction (5.1); an and bn are the terms of the n-th partial quotient; #ι> #2» az> · · · a r e called the partial numerators of the continued

A . α 1 **

. +r2-+ . . . κ

. +*η+ . . .

241


fraction; bXi b2, bz , . . . are its partial denominators; b0 is called the zero partial quotient.

All the terms of the partial quotients will be assumed finite; all the partial denominators are usually assumed to be non-zero.

A continued fraction having a finite set of partial quotients is described as terminating.

A continued fraction having an infinite set of partial quotients is described as non-terminating.

2. A brief historical note Algorithms similar to continued fractions were employed even by the mathe

maticians of antiquity (Euclid's algorithm, the Archimedean approximation to yjl). Of the mediaeval mathematicians, Omar Khayyam (c. 1040-1123) came close to continued fractions when trying to generalize Euclid's algorithm to the case of incommensurable magnitudes, but continued fractions as such first appeared in the "Algebra" of the Italian mathematician R.Bombelli, published in 1572. A number of outstanding mathematicians of the seventeenth century, including J. Wallis and Chr. Huygens, occupied themselves with continued fractions, but the founder of the theory of continued fractions as an independent branch of mathematics was Euler. Almost all the great mathematicians of the eighteenth century and the first half of the nineteenth century made some contribution to the development of the theory. Interest has recently returned afresh to continued fractions, owing to their great theoretical and practical value. In particular, continued fractions are employed in various approximate computations. For instance, with their aid we can compute approximately the values of many functions, the power series expansion of which are slowly convergent or even divergent.

§ 1. Continued fractions and their fundamental properties

1. The evaluation of convergents. Convergents

The terminating continued fraction

0 + * ! + * * + . . . +*n " O n is called the nth convergent of the continued fraction (5.1). We put here

ßo 1 ' ß - i 0 *

CONTINUED FRACTIONS 243

Basic recurrence relations:

n - hn Λ,α n (« = 1, 2, 3, . . . ) . > (5.2)

Equations (5.2) enable the convergents to be evaluated successively. It is useful here to employ the following scheme:

°°^b1+b2+...+bn+ . . . ' 1 b0 P1 P2 Pn

0 1 Ôl Ô2 ' " O n " " EXAMPLE 1.

V2 = l + (V2-l)=l+T^7f =

, 1 1 1 + 2 + 2 + 2 +

1 1 3 7 17 0 1 2 5 12

1 1-5 1-4 1-417

1 1 2 + 2 + . . . 41 29

1-4138

99 70

1-41429.

The difference between neighbouring convergents is

ΤΓ-ΤΓ^ = (- Vn+1 ^n"'^ (n = 1, 2, 3, . . .)· (5.3) \ln S^n-1 Vn- l^n The difference between convergents whose indices differ by 2 is:

£ ± ì - £ - ì = ( - i r i ^ A i (n = 1, 2, 3, . . . ) . (5.4) VSn+1 V^n-1 k£n-li£n+l

Continued fractions whose partial quotients have positive terms. It follows from equations (5.4) that, if all the terms of the partial quotients are positive, the convergents of even order form a monotonically increasing sequence, bounded from above by the number bo+ajbi. Such a sequence has a limit. Consequently, lim P2nIQ2n

71-X» exists in this case. Similarly, if all the terms of the partial quotients are positive, the convergents of odd order form a monotonically decreasing sequence, bounded from below by the number bQ. Such a

sequence also has a limit. Thus lim <Ρ2η-ι/ί?2π-ι a^so exis t s · Thus the value (if it exists) of a continued fraction, of which all the terms of the partial quotients are positive, is always greater than any convergent of even order and less than any convergent of odd order. The number lim PJQn is taken to be the value of a continued frac-

tion (5.1).

2. Transformations of continued fractions

The fundamental identity transformation is , tfi a2 an

Plal PlP2a2 Pn-lPnan ^ ^ PA + P2b2 + . . . + PnK + . . . '

where pl9 p2,... are any numbers, finite and non-zero. Ordinary continued fractions. By using transformation (5.5), we

can always reduce a continued fraction (5.1) to the form

1 1 1 , r ~

where „ _ L . „ α2αΑ · » · a2h-2b2k-l „ _ WZ - - - a2k-\b2k α 0 — D0i a 2 Ä _ ! = — — , a2ft - — — .

*1*3- · · « 2 Ä - 1 *2*4 · · · Û2fc

We call (5.6) an ordinary continued fraction, while the numbers ocl9 a2 , . . . are the partial denominators of the ordinary continued fraction.

An ordinary continued fraction with positive integral partial denominators is said to be regular. Only regular continued fractions are usually considered in the theory of numbers.

Continued fractions, of which all the partial denominators are equal to 1. Such fractions are obtained from the fraction (5.1) by a transformation (5.5) in which we put pn = l/bn(n = 1, 2, . . . ). When b0 = 0, such a fraction has the form

1 + 1 + . . . + 1 +...' ^°

where

c'i = -fr, cn= τ—V- (« = 2, 3, . . . ) .

The terms of the partial quotients of fractions (5.6) and (5.7) are connected by the relationships

ci = — , cn = - — — (n = 2, 3, . . . ) . <*1 α η - 1 α η

Daniel Bernoulli's continued fraction. The continued fraction, the convergents of which are equal to K0, Ku K2,..., has the form

Kx — KQ Kx — K2 (Κχ — K0) (K2 — 3 )

QKn-2 — ^n-z) C^n-l ~" ^η) . . . 4- Kn — Kn^2 -f.

EXAMPLE 2. The continued fraction for which Kn = 1 / ( H + 1 ) 2

(/Î = 0, 1, . . . ) , has the form _ _ 3 _ Ü 1 7 W (2/1 - 3 ) (2κ+1)

4 . 1 - 4 . 2 - 4 . 3 - 4 . 4 - . . . - 4M - . . . * 1 1 3 15 1 4 27 2 4 0 " *

3. Contraction and extension of continued fractions

If we take as KQ, Kx, K2,... some subsequence of convergents of fraction (5.1), we say that the fraction (5.8) has been obtained by means of a contraction of fraction (5.1). Whereas, if we take as KQ9 Kl9

K2,... a sequence which includes in particular all the convergents of (5.1), we say that the fraction (5.8) has been obtained by means of an extension of fraction (5.1).

After putt ing Kn = P2JQW we have

*ι -h *2 + · · · + * η + . . . bxb2 + a2 -α2αφι αΑαφφ§

- (b2bz + az)bA + V 4 - (bibs + α5)ό6 + 64tfe - .

^271-2α2η~ 1^271-4^2η . . . — (62η-2^271-1 + α271-ΐ)^271 + ^2η-2α2η "~

(5.9)


EXAMPLE 3.

/ T i , 1 1 2 1 1 1

1 7 41 239 7 5 29 169

On putting Kn = P2n+1/Q2n+V we have

£>o + bx + b2 -f . . . + i n + . . . Û! -

- (6 A I" al)h + V s - ( V>4 + aÒh + Μδ ~ α1η-\αΐγβίη-Φΐη+1

. . . — (Ö2n-i02n + β 2 Λ η + 1 + ^2n-An+l " · · ■

Here, the sign = indicates that the convergent of zero order of the continued fraction on the right-hand side of the last equation is the fraction 1/0, and not 0/1, since this continued fraction has for its convergents the convergents of odd order only of continued fraction (5.1).

EXAMPLE 4.

1 + Λ / 2 = 2 + 1 + 1

+ 2 + . * 5

,.~ 2

1 5 0 2

1 - 6 ·

29 12

1 - 6

169 70

1 - 6 - .

985 408

If we replace the sign = by the usual equality, we get the expansion

0_ 5_ 30 175 1020 1 2 11 64 373

The connection between ordinary and singular values of a continued fraction. If b0 = 0, the continued fraction (5.1) can be considered in two ways, depending on whether we take its zero order convergent

as equal to 0/1 or 1/0. In the first case the values of the continued fraction are described as ordinary, and in the second as singular. On writing K and K respectively for the ordinary and singular value of the same continued fraction, we have

bx + b2 + . . . 0 ax αφ2

bx + b2 + . . . 1 a± axb2 + a2

1 b± * A + a2 0 b± bxb2

The continued fraction for which K is the ordinary value is

bxa2

K — SL ~ a> * 1 _

Ο α± αφ2 + α2 1 bx b-J)2

It follows that the connection between K and K is given by

K = ^ι

4. The transformation of a continued fraction resulting from a theorem of Stolz The following is well known in mathematical analysis:

STOLZ'S THEOREM. Let

Lim Pn = co, Lim Qn = «, ρ η + 1 > ρ η n-><» n->oo

/or α// Λ. Then limf= ^ r ^ '

z/ iAe limit on the right-hand side of the last equation exists. By using this theorem and the basic recurrence relations (5.2),

we can transform the fraction (5.1) into a continued fraction, whose nth order convergent is (Λι-"Λι-ι)/(οη-Οπ-ι)> where Pn and


Pn_1 are respectively the numerators of the «th and (n—l)th order convergents of the continued fraction (5.1), while Qn and ßn_x are the denominators of these convergents. This continued fraction has the form

h ,ai an _ h , ai a2 + b2-l

az±blZ\_a an + bn-\ a2 + b2-\ 2 βη-ι + * η - ι - 1

——r r 4- · · · + °n — * τ —r r ■

EXAMPLE 5.

/3--1+I I I I

1 A £ 1 11 1 1 3 4 11

~ 0 + 1 + 1 + 3 + 1 + 3 + ..'. 2. 1 1 1 ?! 1 0 2 2 14

.4 more general transformation of continued fraction following from Stolz's theorem. We apply Stolz's theorem to the sequence

7nPnl (« = 0 , 1 , 2 , . . . ) , VnQn

where PJQn (» = 0 , 1 , 2 , . . . ) are the convergents of (5.1), and γ0 ylt y2>... are any non-zero numbers. Now, if the inequality ynQn > ^n-iôn-i 's satisfied for all n, we have, by Stolz's Theorem:

lim £ L = lim ynPn-rn-ιΛ,-ι t n-»-~ört n-»-~ T n ö n - y n - l ß n - 1

if the limit on the right-hand side exists. By using this equation and the basic recurrence relations (5.2),

we can transform (5.1) into a continued fraction, whose «th order

convergent is ( ^ η - ^ - Λ - ι ί / ^ η ο η - ^ η - ι ο η - ι ) · T h i s continued fraction has the form

an

= 0oH yA-y0 + y · ^ - !

V2r3 3+rir3 3~rir2 ri^^+ro^-rori

| Yzbz-Yi [ y<> TWfr+rr/sfrs-rira + --y2 y2 yiy2^2+yoy2*2-yoyi

r n - 2 y n - i ^ - i + y n - 3 y n - i * n - i - yn-3yn-2 « n - l

+ y A i-Yn-Yn-1

EXAMPLE 6.

V2 = > 4 +

=^+

1 2 +

2 5 11 + 3 -f 2 + 12 ·

1 5 1 3

15 11

235 165

yrt-3 r n -Yn-1 Yn-ìYn-

{Yn = n + l ; 1

. . . + 2 + .. 95 319

■f 26 + 44 +

7535 5335

.lYn<*n+Yn-■1«η-

η :

=

-l + yn-

= 0, 1,

- 2 y A - y n - 2 y n . -3yn- l^n- l -yn-

. 2, . . . )

-3w + l)(«2 + "-2(/22-3)

- 1

-3yn-2 4 - . . .

- ΐ )

This gives us an expansion of y/l as a non-periodic continued fraction. In number theory there is a theorem which asserts that every irrational square root can be expanded as a periodic continued fraction and that, conversely, the value of every convergent periodic continued fraction is some irrational square root. But it is not always pointed out that an irrational square root can also be expanded as a non-periodic continued fraction, there being an infinity of such expansions. The expansion, of \Jl obtained by us as a periodic con-


tinued fraction is more slowly convergent than the initial expansion of y/2.

Corollaries of Stolz's theorem and transformations of continued fractions following from them. Let the sequence {PJQn} satisfy the conditions of Stolz's theorem. The sequence {PnlQn} n o w a l s o

satisfies the conditions of the theorem. Therefore

lim 5 L = lim Ρ»+ Ρ»-ι > Qn n-+°°Qn + Qn-l

We can obtain, on the basis of this equation, the following transformation of a continued fraction:

bx + ...+bn + ... ο τ * ! + 1 -

{affilia bn-an + l

b2-a2+\ b2-a2+\ 2 *η - ι -βη- ι + 1

3 +1 - T Γ-ΓΪ +----rbn+\ £ 2 - ß 2 + l ' " n * Λ - ι - * Λ - ι + 1 + .

EXAMPLE 7.

/ 3 - - 1 4 - 1 ! l l

V 3 " 1 + T+T+T+T+. . . - " = i 1 1 1 1 1 1

2 - 3 + 3 + 4 + 3 + 1 + . . . The following equation may also readily be obtained from Stolz's

theorem: lim £ - = lim ynPn±Vn-kPn-k $

where {yn} is a sequence, the terms of which are all non-zero. We can obtain on the basis of this equation as many different identity transformations of a continued fraction as desired.

A further transformation of continued fractions. Let us consider the identity

0°+Τι+Τ2 + ...-Τ+Ίζ+Ίζ+... ( 5 · 1 0 )

CONTINUED ENACTIONS 251

Let us suppose that the convergents of even order of the fractions appearing in this identity are equal if their indices are the same. Now, on performing a contraction of the continued fractions and using the basic identity transformation to make corresponding terms of partial quotients of the contracted fractions equal, we obtain the following two systems of equations, connecting the terms of the partial quotients of the initial continued fractions:

c2czdA = 0^364,

and c 2n-2 c 2n- l"2n = α2η-2α2π-1^2π-4^2η Π = 2, 3, . . . .

£ ^ 2 + ^ 2 + ^2 = £ΐ^2 + 02> ì (^271-2^271-1 + ^ 2 7 1 - 1 ) ^ + ^71+2^271= > (5.12)

= (*27i-2Ô27l-l + a27i- l)^27i+è27l-202n (« = 2 , 3 , . . . ) . )

Systems (5.11) and (5.12), taken together, contain In equations with An unknowns cn and dn (n = 1, 2, . . . ) . It is thus impossible, in the general case of equations (5.11) and (5.12), to express cn and dn(n = 1, 2, . . . ) in terms of al9 cu^ . . . , bl9 b29... or, conversely, an, bn(n= 1, 2, . . . ) in terms of cl9 c29..., dl9 d29 But if b0 = = l> cn = - α π , d2n = b2n, </2η_χ = Αη62Λ^1 (η = 1, 2 , . . . ; λη Φ 1), the identity (5.10) takes the form

1 - L Ü El ÎUZL __ *i + *2+ . · · +bn + ... ~~

__ _1. al^2 a2 a3^4 αφ2

~~ 1 - α ^ 4 - ^ 2 + 2^2- 1 -^^^ + 2αφι + 2α£2-~Τ~- . . . g 2 n - 1^2n g2n^27l-2

• · · -" ^27i-2^2n-1^27i + 2 û 2 n _ 1 è 2 n + 2^2nÔ2n-2 ~~ 1 ~~ · · ·

(5.13) EXAMPLE 8.

/2 = i+I I = 1 1 1 1 1 1 1 v ^2+2 + ... 1 -8-1 -8 - 1 - 8 - 1 -...

1 1 8 7 48 41 280 239 1 1 6 5 34 29 198 Ï69


If the equations

°0 — 1» "1 — 1 1 — · · · — 1 — . . . , \J*M) °2 bA °2n

are satisfied, then equation (5.10) takes the form

1 ,fi f2 fn_ = è1 +è2 + . . . +in + . . .

1 ax 0 3 tf2 tf2n+1 A 2/1

1 — ΐχ + 62 + *3 + · · · + *2n + *2n+l + (5.15)

5. The properties of regular continued fractions

A continued fraction in which all the partial numerators are equal to unity, and all the partial denominators are positive integers, is described as regular^ or arithmetical. It follows from equation (5.3) that all the convergents of a regular continued fraction are nonreducible.

Euclid's algorithm and the expansion of rational numbers as regular continued fractions. Let u and v be two given positive integers, where u > v. On dividing u by v, we have

u Wi

where q0 is the quotient and ux the remainder. On dividing v by ul9 we have similarly

V Wo — = ?i+—. ux ux

Proceeding in this manner, we get

U2 «2

and so on. This process of successive divisions is known as Euclid's algorithm. Since u, ul9 u^,... is a monotonically decreasing sequence of positive integers, the process is finite, i.e. there exists an index n such that un_Jun = qn (consequently, un ^ 0, wn+1 = 0). Hence un is the greatest common divisor of the numbers u and v. It is often denoted by (w, v).


EXAMPLE 9. Find the greatest common divisor of the numbers 816 and 323.

The computation is usually set out thus: 8161323

"646 2 3231170

~"l70 1 1701153

£53|Γ7 1 0 9

(816, 323) = 17. Thus the expansion of 816/323 as a regular continued fraction has the form

816 1 1 1 3 2 3 + 1 + 1 + 9

2 1 1 1* 1 1 2 19 *

Thus Euclid's algorithm enables us not only to find the greatest common divisor of two positive integers, but also to expand their ratio as a regular continued fraction.

The solution in integers of an indeterminate equation of the first degree with the aid of Euclid's algorithm. The equation ax+by = c, where ay b and c are known, while x and y are unknown, is known as an indeterminate equation of the first degree. Such an equation has an infinite set of solutions. But it is often required to find only the integers that satisfy the equation, i e. in conventional language, it is required to solve the equation in integers. We consider here only equations for which a, 6, c are integers. The equation ax+by = c has an integral solution only in the case when c is divisible by (Ö, b). We can thus always assume that a and b are mutually prime. In this case the general solution in integers of the equation has the form

* = (-l)n-icôn_1 + , ô n ,

y^i-lTcPn^-tP» where t is an arbitrary integer, P n - 1 and β η - 1 are the numerator

and denominator of the penultimate convergent in the expansion of a/b = PJQn as a regular continued fraction.

EXAMPLE 10. 43x+37>> = 21.

illiZ !! _ i -L -L £7J_6_ 1 37 " + 6 -t-6

ALL6 1 1 11 0 6 1 6 37

x = -21.6 + 37* = 37/—126, y = 21.7-43* = 147-43ί,

i.e. x = 3 7 * - 1 5 , ^=18-43* ,

where again, * is any integer. The convergents of a regular continued fraction as best approxima

tions. Suppose we have expanded any real number A as a regular continued fraction. Let us write PJQn for the nth order convergent of this continued fraction. The following inequality now holds:

1 ßnßn+l'

These convergents are best approximations to the number A in the sense that no rational fraction with denominator not exceeding Qn can differ from A by less than the fraction PJQn.

The expansion of an irrational number as a non-terminating regular continued fraction. Every real irrational number can be written uniquely as a non-terminating regular continued fraction. Conversely, every non-terminating regular continued fraction (such a fraction is necessarily convergent, by Seidel's convergence test, see § 2, sec. 2) is the expansion of one and only one real irrational number.

Periodic regular continued fractions. The regular continued fraction

<7oH — is said to be purely periodic if the sequence q0, ql9 ?i + ?2 + · · ·

q2, . . . of its partial denominators consists of a repetition, with the same period, of the n numbers q09 ql9..., #n_r

If the repetition starts with some qk(k ^ 1) instead of with q0, the regular continued fraction is described as mixed periodic.

A On


Similar concepts of periodic continued fractions may also be introduced for continued fractions of a general type, although regular periodic continued fractions are primarily discussed in number theory.

The expansion of irrational square roots as periodic continued fractions. Every periodic continued fraction (not necessarily regular) represents an irrational square root. An important role is played in number theory by a theorem of Lagrange, which states that every irrational square root can be expanded as a periodic regular continued fraction.

For example, VI±i= 1.1 I I

2 1 + 1 + 1 + 1 + . . .

/ Is-3+I 1 I I 1 I I I I v ^ 1 + 1 + 1 + 1 + 6 + 1 + 1 + 1 + 6 + . . /

Notice that the expansion of V13 as a continued fraction of general type,

VÎ3 = 3 + (VÏ3-3) = 3+ ί _ = 3 + — i l i V W 3 + VÏ3 3+3 + 3 + 3 + . . .

is obtained more simply and is more rapidly convergent. Thus the expansion of irrational square roots as regular continued fractions is more of theoretical rather than practical interest.

6. Equivalent and corresponding continued fractions

When transforming the power series AQ+A1x+A7x*+... into a continued fraction, two cases can be encountered: (1) the convergents of the continued fraction coincide with the partial sums of the initial power series; (2) the convergents do not coincide with the partial sums of the initial power series. In the first case the continued fraction is described as equivalent to the initial series, and in the second case as corresponding to the initial series. The expansion of the wth convergent of the corresponding continued fraction as a power series coincides with the initial power series up to and including the term in xn. It is clear that an equivalent continued fraction is only another form of writing a power series and does not yield new appro-


ximate expressions for its sum. -As regards convergence, a power or numerical series and the continued fraction corresponding to it can behave differently. They may be both convergent, or both divergent, or one be convergent and the other divergent. The domains of convergence of a power series and the continued fraction corresponding to it may thus be different. For instance, there are power series with radius of convergence equal to zero, which can be transformed into corresponding continued fractions that are convergent in a fairly wide domain.

Equivalent continued fractions can sometimes be readily transformed into corresponding ones by means of the transformation (5.13). Since it is far easier to construct the equivalent rather than the corresponding fraction, the application of transforming equivalent fractions into corresponding ones is of practical interest.

The construction of equivalent fractions. For this purpose it is easiest to employ Eulero identity:

Σ°° „ v n „ , clx C2X ci^zx cn-2cnx CnX = Cn -f- —7—

n-0 1 - C1 + C2X"- C2 + CZX- . . . - C n _ ! + C n X - . . . (5.16)

This identity can also be given the form

Σ CQ CIX CQC^X C^C^X CnXn = - p ί — —

n«= 0 1 — ^0 ~f~ ^1X — ^Ι Ί" ^2X "~" ^2 ~f~ ^2X — * · *

cn-2cnx

• ~~cn-l + cnx ~~ (5.17)

EXAMPLE 11.

&ictenx = x--y + - — + . . . =

9x2 (2H-1)2;C2

1 +3-x2 + 5-3x2 + . . . +2n+ l - (2w- l )x 2 + . . In particular, when x — 1 we have,

£ L - 1 Ü. Ü Ü. 2L (2n-lf l ~ l + 2 + 2 + 2 + 2 + . . . + 2 + . . . '

11 i ϋ 76 789 1 1 3 15 105 985

The last expansion was first obtained by Brouncker (1620-1684). This relationship is regarded as historically the first expansion of a transcendental number as a continued fraction.

On applying transformation (5.13) to the expansion of arc tan x as an equivalent continued fraction, we get the continued fraction

1 - . . .* 15x-5xz + 3x5

15

which is no longer equivalent. In particular, when x = 1 we have

ZL=:1_jL JL J L — — (4/2-l)2

4 "" 2 4 - 1 - 152 - 1 - 4 0 8 - . . . - 1 l_ 22 13 1426 789 1 24 15 1680 945

(4/1+1)2

- 8(8/22 + 8/i + 3 ) - ...*

7. The formation of corresponding fractions. Viskovatov's method

The terms of the partial quotients of the corresponding continued fraction can be expressed in terms of the coefficients of the terms of the initial power series, although high order determinations appear in the relationships thus obtained. This makes such relationships of little practical use in the majority of cases. It is therefore better in practice to use a method of successively obtaining the terms of the partial quotients of the corresponding continued fraction from the terms of the power series. Such a method was proposed in principle at the beginning of the nineteenth century by the Russian scholar V. Viskovatov. Viskovatov's method amounts to the identity,

_ ^10 + ^ 1 1 * + *12*2 -f-ftis*3 + - ■ - _ ^10 ^20* <*30* ^00+ α 01· Χ + α 02· Χ : 2 ^" α 03· χ 3 + · · · α 0 0 + α10 + α20 + · · · '

where a m n = am—1, 0 a m - 2 , n + l ~ ~ a m - 2 , oam—1, n+1·

arc tan A: = x—

x T

5 J C 3 - 3 ^ 15 + 9.x2

lSx^.4x2 + 3x5

15 + 9X2

Computation of the coefficients ocmn may be conveniently set out in accordance with the scheme:

a 00 a 01 a 02 · · · a10 a l l a12 · · · a20 a21 a22 · · ·

*30 *31 *32

EXAMPLE 12. Let us expand the expression (1—x)/(l — 5x+6x2) (*< 1/3) as a continued fraction. We have

Consequently,

1 - * 1-5χ + 6χ2

0 1 1 ]

1 1

- 4 - 2

-12 .

1 Ax Γ - i

l Ax I - 1

1 l - 4 x

- 5 6 - 1

6

2x _ 3 4 _

X + 7 -

2 + x 2 - 7 x 2-

12x - 2 ~ 3;c T

2 - 2 * -10x + 12x2

If, when computing the coefficients amn, it turns out that ochQ = 0, then the (&-f 2)th row of the scheme is obtained by means of a shift of the (Ä:-h l)th row one place to the left; the (fc+3)th row is obtained by combining the (k+2)th and the fcth rows in accordance with the general rule, the (fc-f 4)th row by combining the (&-f 3)th and (A:-j-2)th, and so on. The expansion in this case has the form

αιο f(x) = "1 0 "20·* )X <*3o* * 0 0 + α10 + α20 -f

afe-i, o* afe, i*2 afe+i, ix afe+2, l* . - + «Ä-2, 0 + a Ä - l , 0 + afc,l + afe+l,l +

EXAMPLE 13. Let us expand the expression (1 — 3x3)/(l—x2—Ax4) (x2 < (VÏ7—1)/8) as a continued fraction, we have

1-3*3

1 - Χ 2 - 4 Λ 4

=

0 1

5x + 1

3+9X+15*2

3 + 9x+12x2

1 x2

1 - 1 - " 1 x2

1 - 1 1 1 1 1-x2

3x 5x 25* = 1 - ^ 3 + ^ 5 +

3x + 1

1+3* ' l-t-3*-je

12* 5

15 + 9X + 27*2

15 + 9χ + 12χ2 + 36χ3-

1 1 0

- 1 - 3 - 5 25

300 -300.15.

0 - 1 0 0

- 1 3 3 - 4 4 3

15 - 1 5

_

Γ

300* 15* 25 - ~ Γ ~

5χ 3 +

3 + 34* 3+4x-3;c2-f5;e3

3* I

15-45*3

^6Öx* 15-15^-60x*

0 - 4 - 3 - 4

8. Appell's method

In 1913 Appell proposed the following method enabling any positive number to be expanded as a continued fraction.

Let N be any positive number. Let a be the square root of N, computed with a deficiency of not more than 1. Thus N = a2+R, where 0 < R < 2a+1. We shall assume a = Oif JV< 1. We put i* = (2a+l)/Nlf where # χ > 1 . Let ax be the square root of Nl9 computed with deficiency of not more than 1. Then N± = a\+Rl9 where 0 < i?x< < 2 Ö 1 + 1. We put R± = (2ax+ l)/N2

anc* continued this process inde-

finitely. Generally speaking, N is now expanded as a non-terminating continued fraction.

If iVp = N, N is expanded a pure periodic continued fraction. If Np = Nm (1 ^ m < p), Nis expanded as a mixed periodic continued fraction.

EXAMPLE 14.

5 = 22 + l

EXAMPLE 15.

2 =

6 :

-*4 -4· -'4·

5

3

»

Ni

5 2

= N,

2

= 12 +

5

5

3 2 ;

=

3 2

22·

=

^5

5 5 + 22 + 22 +

l 2 ^ 1 + 6 '

= N.

3 3 3 5 3 3 2 = I2 4— -— Z- -— _- —-r l 2 + 1 2 + 2 2 + l 2 + / l 2 + l 2 - f . . . *

\ second period EXAMPLE 16.

1 + VÎ3 _ 1 2 | 3 1 1 2 ~ "Μ 2 +1 2 + 1 2 + . . /

Appell also pointed out the more general relationship

N = an + - <x\ + n% "2 T · · ·

Here al9 a^,... are positive integers, while a may also be equal to zero. Here we can take the «th root of the number with a deficiency or with an excess of not more than 1, or we can even alternate in various ways the approximation of the root with a deficiency and an excess in the expansion obtained.

§ 2. Fundamental tests for the convergence of continued fractions

1. The convergence of continued fractions

It was mentioned above that a continued fraction for which lim ri-*··»

PJQn exists and is finite is described as convergent. The value of the continued fraction is in this case taken equal to this limit. But it does not follow from the convergence of a continued fraction that lim

ri-*·«» PJQn is equal to the magnitude which has been expanded as the continued fraction.

Essentially and non-essentially divergent continued fractions. If lim PJQn = + oo or lim PJQn = — <=», the continued fraction is

Ti —*eo 7l_>.oo

said to be non-essentially divergent, whereas if lim PJQn does not exist, the continued fraction is said to be essentially divergent. The concepts of essential and non-essential divergence were introduced by Perron.

Unconditionally and "conditionally convergent continued fractions. Unconditionally and conditionally convergent continued fractions.

It is well known that the convergence of a series or infinite product is not affected by the removal of a finite set of terms. In continued fractions, however, the removal of a finite set of partial quotients (excluding the zero partial quotient) can turn a convergent fraction into a non-essentially divergent fraction. Pringsheim therefore introduced the following concepts: a continued fraction [ajbv]~ is said to be unconditionally convergent if, for all m ^ 1, the fraction [ajbv]°^ is convergent. Whereas if the latter fraction is divergent for at least one value of m, the fraction [ajbv]* is said to be conditionally convergent. It follows from this that it is not in general possible to quote convergence tests for continued fractions in the limiting form as used for series. The convergence conditions, connecting, say an and bn9 must be satisfied for all positive integral n. It

•must be emphasized that the removal of a finite set of partial quotients can turn a convergent continued fraction only into a convergent or a non-essentially divergent fraction.

A necessary test for the convergence of a continued fraction [l/av]~.


KOCH'S THEOREM. The convergence of the series ]£ ΙαΛΙ *s

7 1 - 1

sufficient for the limits lim P2n = P, lim P2n+1 = P\ lim Q2n = g, n -x» n->eo n-^oc

lim ß2n+i = ß' *° be finite, and the relationship P'Q—PQ' = 1 io be 71 —>· oo eo satisfied. It follows from this that the divergence of the series £ | αΛ |

n - l is necessary for the convergence of the fraction [l/a„]~.

Uniform convergence of a continued fraction. If the terms of the partial quotients of a continued fraction are functions of a finite or infinite set of variables, the continued fraction is said to be uniformly convergent on the set E of variation of these variables when its convergents PJQn tend uniformly to a limit in £, i.e. when, given any ε > 0, we can find a number N such that, for n^N9Qn is non-zero on the whole set E and the following inequality holds:

n l i m 7Γ on λ->~ βλ

ε. (5.18)

It follows from this definition that the series (see [3]):

PN y (Ρ·Κ Ρλ-ΐ\ ■_ PN . y / | \ λ - ι g l g 2 - · - **λ ÔN λ « Ν + ΐ \ ο λ δ λ - 1 / ôiV λ-iV+l δλ-ΐΟλ

(5.19)

is now uniformly convergent in E to lim PJQn, since PJQn is its

partial sum, while lim PJQn is its sum. Conversely, condition (5.18)

follows from the uniform convergence of this series, i.e. the uniform convergence of the continued fraction. It also follows from this definition that the values of the continued fraction and of the series (5.19) coincide for any x £ E9 i.e. that the continued fraction and the series (5.19) are identically equal on the set E.

A condition for the convergence of a continued fraction to the function which can be expanded as this continued fraction. The uniform convergence of the continued fraction

TJT+ J"r^ ( c „ * 0 ; H = 1 , 2, . . . ) (5.20)

on the set E is a sufficient condition for this fraction to be convergent on the set E to the function K(x)9 which can be expanded as the continued fraction.

It follows from this theorem that, if the fraction (5.20) is uniformly convergent for | x | < ρ, it is convergent for | x | < ρ to a regular single-valued analytic function, which can be expanded as the fraction and which can be expanded as the power series corresponding to the fraction (5.20), this series being convergent to the same function for | x | < ρ. The condition is therefore obtained in order that the continued fraction and the power series which correspond to each other are convergent to the same function. Notice that the domain | x | < ρ can be replaced by aily domain Tcontainingzero as an interior point.

If zero is a boundary point of the set E, then ρ = 0, i.e. the power series corresponding to the fraction (5.20) is divergent everywhere except at zero. But the fraction (5.20), uniformly convergent on the set £, is nevertheless convergent on this set to the function that we have expanded as this continued fraction. This function can be expanded in the present case in a neighbourhood of zero as a divergent power series, i.e. it is not analytic. The convergents of the fraction (5.20) are thus approximate expressions for a non-analytic function, i.e. we have to some degree solved the problem of the approximate evaluation of non-analytic functions.

The condition for identical equality of two uniformly convergent continued fractions. If the values of two continued fractions, uniformly convergent inside a domain T containing zero as an interior point, are coincident in a domain S which is contained wholly in Γ, then these fractions are identically equal inside the domain T.

Uniform convergence of the fraction [cirx:/l]~+1. The uniform convergence of the fraction [cpx/l]~+1 inside a domain T containing zero is sufficient for the uniform convergence inside T of the fraction K(x) = [cjl; cvx/l]~, after the possible exclusion of a finite number of points x' of non-essential divergence. The fraction [cjl; cvx/l]£ is convergent here inside Γ to a single-valued analytic function K(x), regular inside T, except at points x\ which are the poles of this function.

2. A necessary and sufficient test for the convergence of a continued fraction in which the partial quotients have positive terms (Seidel's test)

oo

As has been shown, the divergence of the series £ | αη | is necessary n«i

for the convergence of the fraction [l/an]~. Seidel, and independently Stern, showed that if all the terms of the partial quotients of the fraction are positive, the divergence of the series is necessary and sufficient for convergence.

EXAMPLE 17. The continued fraction

is convergent by virtue of Siedel's test, since the series 2+2+ . . . is divergent.

Notice that the convergence of a continued fraction in which the partial quotients have positive terms depends on the divergence of a certain series, i.e. the behaviour of the entire set of terms of the partial quotients, not on the behaviour of each of them. A convergent continued fraction in which the partial quotients have positive terms is therefore unconditionally convergent.

On passing from a continued fraction of the form (5.6) to a continued fraction of the general form (5.1), the following statement of Seidel's test is obtained; this statement was proposed by Stern.

The divergence of at least one of the series

£ T ; " Q ; r ^ Σ aT ' · 'a" *»+* <5·21> n i l "2"4 · · · a2n n - l "i"3 · · · G2n-M

is necessary and sufficient for the convergence of the continued fraction (5.1), of which all the terms of the partial quotients are positive.

3. Tests sufficent for the convergence of continued fractions in which the partial quotients have positive terms

Establishing the divergence of one of the series (5.21) is usually fairly difficult in practice. Thus it is often more convenient to make use of various tests which are sufficient for the convergence of conti-


nued fractions with partial quotients having exclusively positive terms. Some of these tests are given below.

oo

1°. Divergence of the series £ (bn_1br)/an is sufficient for the

convergence of the fraction (5.1) when the partial quotients have positive terms (Stolz).

2°. Divergence of the series £ \/(*n-i*7i)/ön *s sufficient for 71-2

the convergence of the fraction (5.1) when the partial quotients have positive terms (Saalschütz and Pringsheim).

3°. Divergence of the series oo 1

Σ n * 2 i , #n

*π-Αι

is sufficient for the convergence of the fraction (5.1) when the partial quotients have positive terms (Arndt).

Notice that, if

bl + b2 + · · · +*n t-then

K=h + .^

-K= -bo + - ί » ! + - 6 2 + . . . +-bn+,

Hence all the tests for convergence of continued fractions in which the partial quotients have exclusively positive terms may readily be extended to continued fractions in which all the partial numerators are positive and all the partial denominators negative.

4. First set of tests sufficient for convergence

Let the numbers rlt r2,... satisfy the following conditions (Scott and Wall),

0 »ill+Cil s kiI . 2) r 2 | l + c 2 + c3| s k , | , . ( 5 2 2 ) 3) rn\l+cn + cn+1\ «s rnrn_2\cn\ + \cn+1\ (« S 3), · 4) rn m 0 ( « s 3),


where c2, c 3 , . . . are the partial numerators of the fraction

1 c2 cz cn K = -r -r -T- - f · (5.23) 1 +■ 1 + 1 + . . . + 1 + . . . v '

If the series ]T r^ . . . rn is convergent, then the fraction (5.23)

is convergent, while

\K\* l + E ^ . - . r « . (5.24)

If the numbers rl9 r 2 , . . . satisfy conditions (5.22) and at least one of the cn is zero, the fraction (5.23) is convergent.

If cl9 c2,... are functions of several variables, the above convergence test takes the following form. If the numbers rl9 r2,... satisfy

o©

conditions (5.22), and the series £ rxr2... rn is convergent on a set E,

the fraction (5.23) is uniformly convergent on E, inequality (5.24) being satisfied.

In particular, the set E can be chosen on the basis of the following theorem due to Scott and Wall :

The set of conditions

Pl> 1, \cn\ ^ ^ L (n = 2, 3, . . . ) (5.25) Pn-lPn

where px, p2 are the terms of some numerical sequence, is sufficient for uniform convergence of the fraction (5.23) and for the following inequality to be satisfied:

1 * 1 — Λ Λ - 1

i L 1 + Σ ( Λ - 1 ) 0 > 2 - 1 ) . . . 0 > η - 1 )

(5.26)

Inequality (5.26) becomes equality if cn = (1 —p1)lpn_ipn (n = 2, 3 , . . . ) .

If ρχ = 1, Scott and Wall's theorem takes the following form.

The following conditions:

( 1 ) A = 1 ;

(2) Pn-lPn

' (n = 2, 3, . . . )

(3) the series Σ (p2-l)(pz-l)... (pn+1-l) is convergent;

are sufficient for the uniform convergence of the fraction (5.23) and for fulfillment of the inequality

Ι * | 3 ΐ + Σ ( Α - 1 ) ( Α - 1 ) . · . ( Λ » + ι - 1 ) .

This becomes equality if cn = (1 —PyòlPn-ιΡη (Λ = 2, 3, . . .). In particular, of pn = (2/i-t-l)/(rt+fc), the following theorem is

obtained, by virtue of (5.25) and (5.26). The condition

KISS η2-(Α:-1)2

4 M 2 - ! (« = 2, 3, . . . ) (5.27)

£y sufficient for the uniform convergence of the fraction (5.20), while we have: when k ^ 2,

1*1 3 2-k 1 - 1 S 2 - f c 2 - k

n**i 1 = / C 2 + ÄT w-f-1-/:

n + k

(5.28)

An investigation of the series in the denominator on the right-hand side of the last inequality shows that the series is convergent for k > 1 and divergent for k ^ 1. In addition, it follows from inequality (5.27) that — 1 < k < 3. Hence, when — 1 < £ ^ 1 (5.28) takes the form

1*1 2-A: (5.29)

and only retains the form (5.28) for 1 < k < 3. VorpitskiVs test. When fc=l/2, we have /?n=2. In this case

(5.27) takes the form

|cn | ^ j (Λ = 2, 3, . . . ) , (5.30)

while IK I ~ 2 in accordance with (5.29). But we can replace (5.29)by the stricter inequality:

IK -~I s!3 - 3 •

VAN VLECK'S THEOREM. The condition

(5.31)

(n = 2, 3, ...) (5.32)

is sufficient for the fraction K = [1/1, cnz/l];c to be convergent inthe circle Iz I <: 1/4g to a regular analytic non-rationaljunction, which isalso equal to the series corresponding to this fraction, while IK- 4/31 ~ 2/3.

The proof of this theorem follows from Vorpitskii's test.

5. Tests for the convergence of continued fractions periodicin the limit

The fraction [a&-Ib,.l~, for which av ~ 0, lim aJ! = a, lim b" = b,v~oo v-.oo

is described as periodic in the limit. Such fractions are of great practicalimportance, since they are connected with the expansions as continuedfractions of a large number of commonly encountered functions.

The convergence tests for continued fractions periodic in the limitfollow from the theorem:

The condition lim sup {I c" I} ~ g is sufficient for the fraction

[ ~. c"zJool' 1

2

(5.33)

to qe convergent in the circle 1z I <: 114g (excluding possible poles) to aregular analytic non-rational junction, the poles of the latter beingthe points ofnon-essential divergence of the fraction. In a neighbourhoodof zero this function is equal to the series corresponding to the continuedfraction (5.33).

Having chosen as the g in this theorem any positive number, assmall as desired, we pass to the following theorem.

The condition lim c = 0 is sufficient for the fraction (5.33) to be

uniformly convergent in any finite domain of the complex plane, excluding a finite set of points of non-essential divergence, to an analytic function, which is regular in a neighbourhood of zero, and is regular in the remainder of the domain except for the above-mentoined points of non-essential divergence of the fraction, these being the poles of the function. The point z = ^ is essentially a singular point of the function.

To generalize this theorem, we assume that lim cu = c τ* 0. We

thus arrive at the following theorem. The condition lim cv = c ?± Q is sufficient for the uniform converg

ence of the fraction (5.33), excluding a finite set of points of non-essential divergence in any domain T c T, where T is the complex plane with a cut along the real axis from the point (—l/4c, 0) to an infinitely remote point, such that the cut does not pass through zero.

If the cut starts from zero, the following Stieltjes test holds. Suppose the following conditions are satisfied:

(1) ocl9 a2,. .. are real and non-negative; (2) «i, a 3 , . . . , a2n+1, ...are not all zero;

oo

(3) the series ]T ock is divergent.

Then the fraction [z/afc]~ is uniformly convergent in any finite domain lying inside the complex plane, cut along the negative part of the real axis.

§ 3. The expansion of certain functions as continued fractions

1. Lagrange's method

Lagrange proposed the following method of solving differential equations with the aid of continued fractions. Suppose we have a differential equation in y and x, and assume that y % ξ0, when | x | % 0. We now put>> = ! 0 /0+>Ί ) a n <i substitute this relationship into the original equation. We obtain a differential equation in yx and x. Suppose yx % | x when \x\ % 0. We put yx = ξι/(1+^2) anc* repeat the process. All in all, we arrive at the solution of the original

equation in the form of the continued fraction [ | n / l ]~; ξη is more conveniently sought in the form anxv*, where vn ^ 0.

Many differential equations can be solved by this means, but it generally proves difficult to find the dependence of | n on the index n, i.e. to find the general partial quotient of the continued fraction.

2. Fundamental differential equation

Let us apply Lagrange's method to the equation

(a +*'xW +'iß+ß'x)y + yf = ax, y(0) = 0. (5.34)

Having put y = öx/(oc+ß+y^)9 the equation is reduced to the form*

(B+v'xW+lB+ß-V+ßycfo+rt = Ka + flKa'+iffO + y«]*.

On repeating this process, we get the continued fraction

δχ [(α + β)(α' + β') + γδ]χ (αα'- αβ'+α'β + γδ)χ y~~oc+ß + 2oc+ß + 3α +β + . . .

[(nx+ßKnx'+ß^ + ytyx (n2ococ'-nocß' + m'ß + vo)x ...+ 2nx+ß 4- (2n + l)oc+ß + . . . *

It is readily seen from this that the differential equation

(ρ+Λ'χ^χγ'-Κβ + β'χ^γ + γ? = δχ\ y(0) = 0 (5.35)

has the solution

= òxh [(koc+ß)(koc' + ß') + yö]xk

y koc + ß + 2koc+ß +

(Pocx'-kocß' + koc'ß-ryo^ + 3&α 4 ß 4- . . .

[(η^ϊβ)(ηΜ + β')+γδ)χ* . . . + 2nktx+ß +

(κ2Α:2αα' - /zfca/?' -f nkoc'ß 4- y<$)*fe

4- (2« + l)fca+£ + . . / ( 5 > 3 6 )

Almost all the differential equations, the solutions of which were

expanded as continued fractions by Lagrange, Euler and others, are particular cases of equation (5.35). It is therefore natural to refer to (5.35) as the fundamental differential equation.

3. The expansion of a power function as a continued fraction

Let us put y = (1 + x)p9 where v is any real number. Then (1 + x)y' =

= vy = vy, y(0) = 1. On putting y = 1+ vx/(l+z)9 we reduce the differential equation to the form

(1 + X)XZ' + [1 - (1 - v)x]z + z2 = (1 - v)x, z(0) = 0.

This is a particular case of equation (5.35), in which

jfc = a = a' = j9 = y = l , ß' = - ( l - v ) , <5 = 1-v.

We thus have, from expansion (5.36):

1 + 2 + 3 + (2 — ν)χ (n — v)x (n 4- v)x

2 f . . . + 2 + 2/j + l + . . . ' (5.37)

This expansion was obtained by Lagrange. In view of the above-mentioned convergence tests, it is convergent on the complex plane cut along the real axis from x = - o o t o x = —1.

The power series into which the function y = (1+x)p can be expanded is convergent in an open circle of radius 1 with centre at the origin. The expansion of the function y = (1+x)p as a continued fraction is therefore convergent in a much wider domain than the expansion of the same function as a power series.

EXAMPLE 18. On putting x = 1, v = 1/3, we have

V 2 = l + l , | 4 5 3/2-1 3/1 + 1 3 + 2 + 9 + 2 + . . . + 2 + 3 ( 2 / J + 1 ) +

l 4 10 106 l 3 8 84


On contracting the expansion (5.37) in accordance with formula (5.9), we get

2vx (1 - v2)x2

( l f x ) " _ 1 + 2 + ( 1 - Î 0 X - 3(2 + x) - . (4 — iP)x2 (n2 — v*)x2

- 5(2 + x) - . . . - ( 2 « + 1 ) ( 2 + J C ) - . . . *

This expansion is convergent in the same domain as the expansion (5.37).

Irrational square roots may be expanded as continued fractions by means of the following elementary method.

Let V* % ß. Then >Jx = a+(*Jx—a) = a+(x—a2)/(a+y/x). Consequently,

y[x = tf-f-:

2a + 2a + . . .

This expansion is convergent on the complex plane, except the negative part of the real axis.

4. The expansion of a logarithmic function as a continued fraction

Expansion (5.37) can be put in the form

( l + x ) v - l _x (l — v)x (l+v)x (2-v)x ; " τ+~"y~+—τ~+~2~

(2 -f v)x (n — v)x (n -j- v)x + 5 + . . . 4- 2 + 2A? + 1 + ...."

On setting v = 0, we obtain, since lim [(1-f x)"— l]/v = ln(l+jc):

, ,Λ . x x x 2x 2x nx nx ln(l+x) = I f 2 + 3 + 2 + 5 + . . . + 2 +2/7 + 1 + . . . * (5.38)

This expansion and the method of obtaining it from expansion (5.37) were found by Lagrange. Expansion (5.38) is convergent on the complex plane cut along the real axis from # = — « > t o * = — 1 .


EXAMPLE 19. On putting x = 1, we have: 1 1 1 n n

In 2 = 1 + 2 + 3 + . . . + 2 + 2*1 + 1+ . . .

IL 1 L 1 1 3 10

5. The expansion of an exponential function as a continued fraction

On replacing x by xjv, expansion (5.37) takes the form

1 — v l+i> 2-v ■x x x x \ Λ x v v v

1 + — = 1+-Γ v J 1 + 2 + 3 + 2 2 + v n—v n + v x x

+ 5 + . . . + 2 + 2 Λ + 1 + . . .

On letting v tend to oo, we obtain in the limit: „ Λ x x x x x x x /,. *Λχ

1—2+3 — 2 + 5 — . . .—2+2/2 + 1—.. .

This expansion is convergent on the entire complex plane. The expansion and the method of obtaining it from expansion (5.37) were discovered by Lagrange. On contracting the continued fraction (5.39) in accordance with formula (5.9), we arrive at the expansion obtained by Euler:

2x x2 x2 x2

+ 2 ^ + T + T Ô + . . . +2+(2«+iy+...* This expansion is also convergent on the entire complex plane.

6. Expansion of the function 7=arc tan x as a continued fraction

The differential equation for this function has the form

" - Ï T F · ' ( 0 ) = 0·


On setting y = x/(l+z), this equation reduces to the form

(1 + x2)xz' + (1 - x2)z + z2 = jc*.

This is a particular case of equation (5.35), in which

α = α' = / S = y = ó = l , β' = - 1, Λ = 2.

We therefore have, from expansion (5-36):

x x x 2 Ax2 9x2 n2x2 ,e ,ΛΧ arc tan x = - = - — - — - — — . . (5.40) 1+2 1 + 3 + 5 -f 7 + . . . +2ii +1-f . . . v J

This expansion was discovered by Lambert (1728-1777). It is convergent on the whole complex plane except for two cuts along the imaginary axis from — oo / to —/, and from / to + °° *. This expansion is thus convergent in a far wider domain than the power series into which the function y = arc tan x may be expanded (this latter series is convergent inside the unit circle with centre at the origin).

The terms of the partial quotients of fraction (5.40) satisfy conditions (5.14). We can thus obtain a further expansion of arc tan x on the basis of equation (5.15):

x3 9*2 4x2 (2n + l)2x2 (2n)2x2 arc tan x = x — — -r- -=- —-A ;— -: =-3 + 5 + 7 + + 4/14-1 + 4 w f 3 -{-. . .

7. Expansion of the function y — continued fraction

'X

Λ/0 + r ) as a

The differential equation for this function has the form

On putting x

y = 1 + r '

the equation reduces to the form

(l+xk)xz' + (l-xk)z + z2 = **.

This is the particular case of equation (5.35) when a = a' = jS = y = = 5 = 1 , /?'=— 1, k is any number. We therefore obtain, from expansion (5.36):

t dt x xk k2xh (£ + 1)2** 0l+tk 1 +fc+l-f-2A;+l + 3fc+l + . . .

ifllftc» (nk + l)2xk

. . . + 2 Λ Α : + 1 + (2Λ + 1)Α: + 1 + . . / l '

This expansion was obtained by Lagrange. It is convergent on the plane of the complex variable xk, cut along the real axis from — <» to - 1 . When k = 1, expansion (5.41) becomes the expansion for ln(l -f x); when k = 2, it becomes the expansion for arc tan x.

On substituting xp+1 for x in the integral A/(l -f th) and setting

k(p-{ 1) = q, we obtain from (5.41) the expansion

i '* P& x*>+1 {p+\fx* qW 0l+tq p-tl + q+l+p + 2q + l+p + . . .

«V** (nq + l+p)2x* (5.42) ...+2nq + l+p+(2n + l)q + l+p + . . . '

For example,

'* /2fL = 1 i . 11 f! 16/r8 (4n+3)2

M4 ~ 3 + 7 + 11 + 15 + . . . + 8/1 + 3 + 8/1 + 7 + . . . "

1 3 30 378 7140

On using conditions (5.14) and equation (5.15), we can write (5.24) in the form

i

i tPdt _XP+1 χΡ+ι+« {q + X+pfx* 0 l+ί« p+\ q + l+p+ 2q+l+p

q2x* (nq +1 +pfx* n2q2x« + 3q+l+p + ... + 2nq + l+p + (2/i+l)g + l +p +


For example fi t2dt 1

1 3

1 49 16 ~7 + 11 + 15 + .

4 93 1459 21 378 6006

(4« + 3)2 16«2

. . + 8n + 3 +8/J + 7 +

8. Expansion of tan x and tanh x as continued fractions

The differential equation for y = tan x is of the form

/ = i + y\ y(0) = o. On putting y = x/(l+r), this equation reduces to

xz' + z + z2 = — x2. This is the particular case of (5.35) when a' = j3 /=.0,a = )5 = y = 1, ô = — 1, k = 2. We thus obtain from expansion (5.36):

t a n x = _ _ _ _ _ _ ^ _ _ _ _ ^ (5.43)

This expansion was discovered by Lambert. It is convergent on the whole plane of the complex variable x9 except for the points at which tan x becomes infinite.

On replacing x by x/i in expansion (5.43) and multiplying the resulting equation by /, we obtain

"V* "V" Ύ Ύ^

tanh x = -7- — -r- r . (5.44) 1 + 3 + 5 + . . . + 2 / 1 + 1 + . . . v '

9. Expansion of Prima's function as a continued fraction

The integral /•oo

fo-i g-i Λ (a > 0, * > 0) Jx

is known as Prima's function. Let us introduce the function

y = je1"0** i0-1*?-' du

which satisfies the differential equation xy' — (l — a+x)y= —x.

On putting x = l/t, y = l/(l+w), this equation takes the form

t2u' + [1 - (1 - a)t]u = (1 - a)t, w(0) = 0. This is the particular case of equation (5.35) when a = 0, k = a' = = £ = y = l , ß ' = - ( l - a ) , δ = 1 - α . We thus obtain from expansion (5.36):

_ (1 -d)t l_ (2-d)t nt_ (n + \-a)t u - i + i + i + . . . + 1 + 1 + . . . ·

On returning to the original notation, we get xae-x ι _ α i 2 - a I X H- 1 f X + 1 + . . .

rt n -L 1 — /7

(5.45) . . . + * + 1 + . . . On contracting this expansion on the basis of formula (5.9), we get

« Ί tj *"*"* l ~ a 2 ( 2 - a ) ta~1e'tdt = — — Λ:+1 —Ö — x-f3 — <z—x-f5 —a — . . .

" ( "~ a ) , (5.46) . . . — x + 2/i + l—a — . . .

We call I (el/t) dt the integral exponential function and denote it P («70* by Ei(x) (see (6.393)). We obtain from expansion (5.45) with a = 0 and x < 0:

EiW = e»fI | I » ». V (5.47) ^χ — l — χ - . . . — l — χ— . . . y The function

Ei< : i < " " ) = J ! ^ is called the integral logarithm and is denoted by ii(x) (see (6.394)). We obtain from expansion (5.47):

t . , v x 1 1 n n n (*) = , T i — T" i — lnx— 1 — lnx— . . . — 1 — lnx— . . .

10. Expansion of the incomplete gamma fonction as a continued fraction

To obtain an expansion of the incomplete gamma function, i.e. of the integral

* p-ie-1 dì (a > 0), J: I o

we introduce the function

y = χα^ 'x t«-^e-{ dt (x > 0), o

which satisfies the differential equation

xy' + (a-x)y-l = 0, J<0) = 1 .

On putting y = 1/(Ö+J>I) in this equation, we get

xy'i+(*+x)yi+A = -αχ· This is the particular case of equation (5.3 5) when/: = a = /?' = y= l , a' = 0, β = a, δ = — a. We thus have from expansion (5.36)

I ai - a - l + û + 2 + α - 3 + û +

2x nx {a + n)x + 4 + S - . . . +2n + a—2« + l - fa+ . . . '

11. Thiele's formula

The Taylor series

f(x+h) =f(x)+-Lf-{x)+J*.f»(x)+ . . . + ~ f*\x)+ . . .

corresponds in the theory of continued fractions to Thiele's formula h h h

f(x + h)=f(x) + rf(x)+2rrif(x)+ 3rr2/(x) + .,

+nrrn_1f(x)+ . . . '


Here rnf(x) is called the inverse derivative of the n-th order of the fune-tion /(*). The inverse derivatives are defined by the relationships

and in general, for n ^ 2,

rnf(x) = ^ i - a / M + ^ n - l / C * ) ·

For example,

re* = e~x, r2ex = -e* , . . . , r2n^ = ( - l)71^, r2n+i^ = ( - l ) n ( « f l ) ^ .

Hence (2/z + l)rr2ne* = ( - 1)» (2/* + 1)«-«

(2 i + 2)rr2n+1e* = 2 ( - i r + V .

If we use these relationships and replace x by 0 and h by x in Thiele's formula, we again arrive at expansion (5.39).

Thus Thiele's formula is another source of obtaining the expansions of various functions as corresponding continued fractions.

12· Fractional approximations for sin x and sinh x

The general form of the expansion as a continued fraction of sin x is not known. Only a finite set of partial quotients of the expansion can be found by Viskovatov's method. For example,

£ x* 7£> Πχ2 551χ2

1 + 6 - 10 + 98 - 198 + . . . " £ 6x 60*-7x3 5880*-620s3

1 6 + x2 60 + 3χ2 5880 + 360x2 + ll;c4

x* 3x* ΠΛ? 25^ * 6 + 10 - 42 + 66 - . . . "

£ ó x - s 3 60*-7x3 2520*-360χ3 + 11χ5

1 6 60 + 3JC2 2520 + 60χ2

sin*

and

sin*


It follows from the approximation

S m ^ . 6 Ô T 3 ^ ( 5 · 4 8 )

that sin;z:/4 % 0-7071, if we put π/4 % 0-7854. Hence, in view of the periodicity of the function y = sin x and the relationships

sin ( -~π—x J = cos x and cos x = Vi —sin2*, the approximation

(5.48) enables four-figure tables of this function to be evaluated with the aid of a calculating machine. In the same way, by using the higher-order convergents, we can evaluate tables of y = sin x with any number of correct decimal places with the aid of a calculating machine.

Since sinhx = — i sin zx, the expansions obtained for sin* lead us to the following expansions for sinhx:

x x 2 Ίχ2 \\x2 551r sinh x = — -T- τττ 1 - 6 + 10 - ' 98 + 198 - . . .

£ 6x ωχ-Ίχ3 5880χ + 620;τ> 1 6-x2 60 -3x2 5880-360x2-f-llx4

and

x3 3x2 llx2 25x2

x_ 6X + X3 60.x+ 7X3 2520X + 360JC2 + 11X5

1 6 60-2x2 2520 -60χ2

13. Fractional approximations for cos x and cosh x

The general form of the expansion as a continued fraction of cos x is unknown. Only a finite set of the partial quotients of the expansion can be obtained by Viskovatov's method. For example,

sinh


1 x2 Sx2 3x2 313x2 COS X = — 1 + 2 - 6 + 50 - 126 +

_1_ 2 12 - 5x2 600-244χ2

1 2+x2 12 + JC2 600 + 56x2 + 3;c4

and x2 x2 3x2 \3x2

COS X = 1 — — — -rrr- -7^7-2 + 6 - 10 + 126 - . 2_ 2 - x 2 12-5Λ:2 120-56Λ:2 + 3Λ:4

1 2 12+x2 120 + 4*2

Since coshjc = cos ix, the above lead to the expansions:

1 x* 5x* 3x2 313*2

cosh* = 1 . 2 + 6 - 50 + 126 -

1 2 1 2 + 5Λ? 600 + 244*2

1 2 - x 2 1 2 - x 2 600 - 56X2 + 3JC4

and , , x2 x2 3x2 I3x2

COSh X = l + — -r ΤΤΓ -Γ7Γ7-2 - 6 + 10 - 126 + . . . J_ 2 + x2 1 2 + 5X2 120 + 56Χ2 + 3Λ:4

I 2 1 2 - x2 1 2 0 - 4x2

14. Fractional approximation for the error function

The general form of the expansion as a continued fraction of the

error function erf x = (l/y/n) e~t2dt is unknown. Only a finite

set of partial quotients of the expansion can be obtained by Visko-vatov's method. For instance

I '* „ . x x2 x2 39X2 739X2 e~lt dt = — — —

0 1 + 3 - 10 -f- 7 - 234 +

x 3x 3 0 * - x 3 210χ + 110χ3

1 3 + x? 30 + 9x2 210 + 1 8 0 ^ + 39x4


15. Conversion of Stirling's series into a continued fraction

Let us take the Stirling series (see (6.509) and (6.516)):

where

ΙηΓ(χ) = ( x - y ) ΐ η χ ~ χ + γ 1 η 2 π + /(χ),

J(x\ - B2 , J 4 . | B*n i ΛΧ) ~ 1.2* + 3.4X3 + * ' · + ( 2 H - l)2/ix2^1 +

and i?2, 4> · · · a r e Bernoulli numbers (see Chapter VI, § 2, sec, 1, Ie):

1 D _ 1 „ _ 1 * _ 1 » _ 5

By using Viskovatov's method, we can convert the Stirling series into a continued fraction, the general partial quotient of which is unknown:

In Γ(χ) = ( * - y J l n x - x + y 1η2π +

J _ 2__ j>3_ Π70 22999 + 12x+ 5χ + 42χ + 53x + 429* + . .."

16. Fractional approximation for the gamma function

If we evaluate the coefficients in the expansion of the gamma function:

Γ(1+*)= £cnx«, where

] n oo i

^ = 0 « 0-57722 (Eulefs constant, see (6.56)), we obtain approximately:

Γ(1+χ) ^ 1-0·57722χ + 0·98906*2 -0'90748*3 + 0·98173χ4- . . .


On applying Viskovatov's method to this approximation, we obtain in particular:

^ 1+0·15141*+0·24392** _ 1 ( 1 +X) ~ 1+0·72263χ-0·32456*2 _ Γ ( Λ ) ' ( 5 ' 4 9 )

This approximation is sufficiently accurate in many cases, as is clear from the following table:

X

-0-5 -0-4 -0-3 -0-2 -0-1 0

Γ(1+χ)

1-7725 1-4892 1-29806 1-16423 1-06863 1

n*>

1-7767 1-4902 1-29823 1-16425 106863 1

X

0-1 0-2 0-3 0-4 0-5 0-6 0-7 1

A1+*)

0-95135 0-91817 0-89747 0-88726 0-8862 0-8935 0-9086 1

T(x)

0-95135 0-91816 0-89743 0-88711 0-8858 0-8927 0-9071 0-994

17. Fractional approximation for the logarithm of the gamma function

Approximation of this function on the basis of Stirling's series requires a knowledge of In x. However, it is possible to obtain a simpler approximation by starting out from the expansion

InΓ(1+χ)= -Cx+Σ L_LLiLJnj (5.50) n-2 n

where sn = £ l/mn, C is Euler's constant. On evaluating the m—l

coefficients, we obtain approximately:

1ηΓ(1 +χ) ^ -0·57722Λ:+0·82247Λ:2-0·40068Χ3 + 0·27058Λ:4~

On applying Viskovatov's method to this approximation, we obtain in particular:

-0·57722χ + 0·59769χ2

ll+X) ** 1 -f 0-38941*- 0-13928X2 "

284

Hence

MATHEMATICAL ANALYSIS

-0-25068X+0-25957** 1+0-38941X-0-13928X2

The accuracy of this approximation is clear from the table.

(5.51)

X

-0-5 -0-4 -0-3 -0-2 -O-l 0

lgr(l+*)

0-2486 O-l 730 , 0-11329 0-066039 0-0288268 0

T(x)

0-2469 01725 0-11321 0-066029 0-0288266 0

X

O-l 0-2 0-3 0-4 0-5

lgr(l+x)

7-97834 l-96292 l-95302 1-94806 1-9475

T(x)

7-97834 1-96293 l-95305 1-94818 1-9479

18. Fractional approximation for the derivative of the logarithm of the gamma function

In accordance with (5.50), the function

Ψ(*) = -£ ΐη /χ ΐ+*)

has the expansion

ψ ( * ) = - 0 + *2χ-*3*2+ . . . +(-l)n + 1*n+i*n+ · . · ,


Ψ(χ)* -0 ·57722+1·64493Λ:-1·20206Λ? + 1·08232Χ3- . . . .

On applying Viskovatov's method to the last approximation, we obtain in particular:

, - 0-57722 + 1-25687* _ y W ~ 1+0·67228χ-0·16670** ~ 1W'

The accuracy of this approximation is clear from the table.

(5.52)


X

-0-5 -0-4 -0-3 -0-2 -0-1

0

Ψ(χ)

-1-964 -1-541 -1-225 -0-9650 -0-7549 -0-57722

T(x)

-1-938 -1-538 -1-218 -0-9647 -0-7549 -0,57722

X

0-1 0-2 0-3 0-4 0-5

Ψ(χ)

-0-4238 -0-2889 -0-1692 -00614

0-0365

T(x)

-0-4237 -0-2890 -0-1687 -00600

00396

19. Obreshkov's formala

A knowledge of the general form of the partial quotients of the continued fraction into which a given function can be expanded is not sufficient for a determination of the general form of the convergent of this expansion. Nevertheless, the general form of the convergent can be found in certain cases. The most general approach to this problem uses Obreshkov's formula, wltfch is one of the generalizations of Taylor's formula. Obreshkov's formula is A r? rv-v> m n> (x—XQy

»-0 um+ft

(x-Xo^W(jc ) = ™ Om

1 (k+m)\

p-0 ^wi m+k v\ -/(>0(*οΗ

(x - t)m (x0 - f)Ä/(m+fe+1)(0 du (5.53)

Determination of the rational fraction approximations of a power function with the aid of Obreshkov9 s formula. We put x0 = l,/(x) = xn

in (5.53), n being any real number. Now,

xn %

m Cv Cv

(*-Dr

When m = fc, this equation becomes Ä CVCV

K-D" v - 0

Qft (A:- D»

In particular, when k = 1, 2 — n+nx

x. n + (2-n)x

For instance, when x = 2 and n = 1/3, we have V2 ^ 14/11 % 1*273 (the accurate value is 1*2599.. .).

Determination of the rational approximations of the exponential function with the aid of Obreshkov's formula. We put x0 = 0, f(x) = e in (5.53). Now,

m Cv YV

Σ ^τη Λ

In particular, when m = k, 2k(2k-l). . . (fc+l)+c£(2Â:-l) . . . (fc+l)*+C*(2fc-2). . . (fc+1)** f . . . x*

2k(2k-l).. .(fc+l)-Cj(2fc-l)... (k+l)x+cl(2k-2).. .(fc+l)*«- . . . +(-1)***

(5.54) Using the relationship

tanh x = oy t -, e^ + l

together with (5.54), we get C\{lk-\) (2fc-2)... (/c+l)*-C|(2Jfe—3) (2fc-4)... (fc+l)4*3 + . . .

tanh* ' ik(2ik-l). . .(ife+l)jc+Cj(2,c-2)(2,c-3).. .('c+I)2c2-hC J(2fe-4)... £+1)8*4+. . .

On replacing x by *x in the last equation and dividing the right-hand left-hand sides by /, we get the general expression for the convergents of the expansion of tan x as a continued fraction.

Determination of the rational fraction approximations of the logarithmic function with the aid of Obreshkov's formula. We put x0=1, f{x) = In x in (5.53). Now,

v « i <-m+fe v v=l cm+fe vx

In particular, when m = k,

For instance, when k = 1,

When k = 2, —if-4

ln*~^y-(8*-*2-D

§ 4. Matrix methods

1· Extraction of the square root by means of second-order matrices

The theory of continued fractions is built up with the aid of the fundamental recurrence relations (5.2). It seems natural to consider generalizations of continued fractions, based on other linear recurrence relations, connecting the numerators and denominators of neighbouring convergents. Let us consider in more detail the relations

Λι = αηΛι-ΐ+βιβη-1> \ , Λ _ Λ / r „ η -ν Ρ Mtn f ( « = 1, 2, . . . ) . (5.55)

These equations can be written with the aid of matrices in the form

(αΐ)=(;ιΧ::) <—·>- « We put <xn = a, βη = w, γη = 1 <5n = a (/i = 1, 2, . . . ) . Relations (5.55) and (5.56) now become respectively

Qn = Λι-l + tfßn-l J and

GiK :)(£:) <"=-···>· « When a = 0, the process (5.57) is divergent. We shall therefore

assume that a ^ 0. If lim Pn/ßn exists, it can be equal either to y/u

or to — \/u. Consequently the process (5.57) is divergent when a < 0. Let us use the notation

We now have four cases :

1) a > x2, x2

2) 0 *2»

* 2 n - : 2 n - 2

Ô2TI-2 Qln

3) xx < a < 0,

On

-* 2 n + l

Ô271+1

< ^2n

Ô n - l '

0271-1

^ 2 t t - 2

On — xi'y

2 n + l

4) a < xl5

ß2n

Ôn-l ~ßn 71-1

Ô271-2 lim ^ = χν

r example:

1) a > χ2,

I (H) 5 47 455 4409 1 10 97 940 5 4-7 4-6907 4-69043 V22 < 4-69043;

2) 0 < a < ΛΓ2,

/5 27\ 5 V 5) 1

5 5 2 _ 2 6 265 2702 _ 1351 10 ~ 5 51 520 ~ 260

5 5-2 5-19607 5-19616 5-19607 < V27 < 5-19616;

3) Χχ < a -= 0,

1 2\ zi J- Zi IL 1 -l) 1 - 2 5 -12

-41 12 29

- 1 -1-5 -1-4 -1-416 -1-4137 - 1-416 < - V 2 < -1-4138;

4) a x\, (-2 3\ - 2 7

I) 1 - 4 - 2 -1-75 -1-73214 <

- 2 6 15

-1-734

- V 3 .

97 - 5 6

-1-73214

2. Solution of quadratic equations with the aid of second-order matrices

Let us put αΛ = a9 βη = — q> yn = 1, δη = a+/? in (5.55). Relationships (5.55) and (5.56) now become

and

Pi

'» = Λι-1+(« + *)β»-1 J ( }

(5.60) .la+WVßn-i/

If lim P n /ß n exists, it can only be a root of the quadratic equation

x2+px+q = 0." Consequently, when p2—4q < 0, the process (5.59) is divergent, since the roots of the quadratic equation are complex in this case, while a sequence with real terms cannot have a complex limit.

Let us write

* i = -p->Jp2-4q

x* = -P + \/P2-4

271+1

£211 + 1

Λι-ι Ô n - l

Ô27I-1

lim -^- = *2; i - x » Vin

3) *! < a ΧχΎχ^

2 n - l

£ 2 n - l

271+1

22π+1 * 1

■* 2n ^ 2 n - 2

Ö 2 n (?27i-2

Λ \ ■* 71 — 1 "* 71 4) ß < jcl9 τ^—L < τ ~ < *Λ ;

l· Hm rf- = xv

ß«-l " ß» " *'

For instance, the equation

x2 + 2 x - l = 0 has the roots

xlfi= -l±y/2 or Xi^ -2-414 and x2 % 0*414.

1) a > x2,

fl= 1, 1 1 \ J _ 2 _ 1 1 2P__ _1-1 3 ) 1 4 ~ 2 7 24 ~ 12 '

1 0-5 0-43 0-417

2) ^ 4 ^ - < a < x2,

' ^1 2) 1 2 5 12 0 0-5 0-40 0-417

3) Xy < a < *l + *2

ö = - 2 , - 2 1\ - 2 12 29

1 0 / 1 - 2 5 - 1 2 ' -2 -2-5 -2-40 -2-416

4) a < xj,

/~ 3 Λ Zl i°_-_L ZU 58 _ 29 a ~ ~ 3 ' ^ 1 - l j 1 - 4 ~ ~ - 2 7 - 2 4 ~ - 1 2 " - 3 -2-5 -2-428 -2-416

3. The connection between matrix methods and the theory of continued fractions

Let us find the conditions under which the equations (5.55) can turn into equations (5.2). For this, we replace n by n— 1 in equations (5.55), find Qnmml and P n - 1 from the relationships obtained, and substitute them in equations (5.55). On introducing, for brevity, the notation a n - 1 i n _ i -Äi - i r n - i = An_v we get

Pn = L+Ê^yn^ês^pM (5.61)

On = f ^^ + Oß^-^^ßn.,. (5.62) \ 7 η - 1 / Yn-1

and

Equations (5.61) and (5.62) show that Pn and Qn can be taken as the numerator and denominator of the /zth order convergent of a continued fraction provided that

7n *n-l4A=A*n-l+"n, J*~ = " ^ · («2*2). ^n-l -r^n ~ a vn-l-r^n> o — 7η~1 Pn-1 Pn-1 7n- l

These conditions are readily written in the form

a n — K α π - 1 — ^η-Ι α1 "" Ai Αι-1 " " A

£n _. ßn-l __ _ A. rn 7π-ι " " ri '

Hence the matrix

/ α η ßn '

\Ά R rr α 1 - ^ A \ £ Pn a n D r n / VA A /

(5.63)

(5.64)

is equivalent to a continued fraction, the partial numerators an and partial denominators bn of which are given by the equations

o o an = -rL-Wn-iyn-l-<*n-lK-l)> K = f2- Òn_x+<Xn_x (it ë 2).

Pn-1 Pn-1


We can find from the relationships

(:t)0H:K;) and

U «Ai/ U n i I ^ / the zero and first partial quotients of the fraction. Hence the matrix

(5.64), in conjunction with the matrix ι ° £° ι, leads to the continued

fraction Ä

x | «ι«.ι·Α-«^».+^ Α(/?ιΤι~*Α)

Pi

Λ (Αι-ΐΤΆ-ΐ-Οη-Α-ΐ) Äi-l ο

Ρη-1

(5.65)

EXAMPLE 20. The matrix ί I leads to y/ΐ and is equivalent to

the continued fraction /=■ , 1 1 1

V 2 - 1 + T + T + . . . + T + . . . ·

In the theory of continued fractions, contraction is performed with the aid of the fairly complicated formula (5.9) and analogous formulae. Here, it is performed much more easily, with the aid of squaring, cubing, etc., the matrix. We have:

3 4\ 2 3 ) · «n = 2 . 4 - 3 . 3 = - l , C O C K . bn = 3 + 3 = 6 (* £ 2).

The zero and first partial quotient of the new continued fraction have to be found directly, and not on the basis of this matrix, from a knowledge of the first convergents of the initial expansion for yfl.


We get /T 1 2 1 1

On using the cube of the initial matrix, we get

,2 3, DG K 7 Fi - i 5 1 1

V 2 - 1 + Τ 2 + Ϊ 4 + Ί 4 +

4. Hie reduction of quadratic surds to non-periodic continued fractions by means of second-order matrices with variable elements

The results of the previous section enable us to obtain as many expansions as desired of irrational square roots as non-periodic continued fractions.

EXAMPLE 21. The matrix / 1 2(/2 +1)\ V*+i i ;

satisfies conditions (5.63). In addition, it leads to the equation Pn_= Pn-l + 2(n+l)Qn-1

On (» + 1)Λ.-ι + β»-χ ' from which it follows that lim PJQn = V2. The equivalent continued

η-> · ·ο fraction (5.65) has the form

2 21 8.17 15.31 24.49 y/2 = 1 + 3 + 5 + 7 + 9 + 11 + . . . ( / i 2 - ! ) ( 2 / I 2 - ! )

. . . + 2n + l + Similarly, by starting from the matrix

/ n 2 (« f l ) \ \n + l n ) ' ction

/ 2 = 1 HI. . v 1 + 7 + . . . + 2« 2 - l + . .

we get the continued fraction ( T Î - 1 ) ( Ï Î + 2 B - 1 )


5. Extraction of the root of any rational power by means of matrices

Extraction of the cubic root with the aid of matrices. Let us consider the three sequences {Pn}, {Qn}> {Rn},the terms of which are connected by the relationships:

Pn = d V - l + ißn-l + ^ n - l . Qn = Pn-1+aQn-l + tXn-l, *n = J \ i - l + ß n - l + «*n-l

( « = 1 , 2 , . . . ) . (5.66)

These relationships may be written with the aid of matrices as

(Pn) On

{Pn, =

a t t 1 a t

y i at

(P \

Ôn-l A-iJ

(5.67)

Let x = lim PJQn and y = lim QJRn exist and be finite. Now,

EXAMPLE 22. Let a = 1, t = 2. Then

ri 1 i

2 1 1

2] 2 1

1 5 1 4 1 3

19 15 12

73 58 46

281 223 177

1081 858 681

iß approximately 1 1-67 1-583 1-5870 1-58757 1-58737

Iß approximately 1 1-33 1-250 1-2609 1-25989 1-259912

It is well known that X/Ï « 1-2599210, \fi « 1-5874011. By utilizing the square, cube or a higher power of the initial matrix,

the convergence of the process can be strengthened as desired. Extraction of the fourth root with the aid of matrices. We generalize

relationships (5.66) and consider the following equations:

Pn = aPn-l + ' ô n - l + tRn-l + tSn_lt

Qn = Λ»-1+ββη-1 + ί*π-1+ '£«-!, *n = Pn-i + Qn-i + aRn-l + tSn-i., $n — Pn-l + Qn-l + Rn-l + aSn-l·

(n - 1, 2, . . . ) .

We can write these with the aid of matrices as :


(a 1 1

U

t\ t t a)

iP*-A Qn-X

Now,

lim ^ = \J?. um

if these limits exist and are finite. When a = 1, we have the matrix fl t t t\ 1 1 + 3/ 1 + 12/+3/2 l+31/ + 31/2 + /3

1 1 t / 1 2 + 2/ 3 + 12/ + /2 4+40/+20/2

1 1 1 / 1 3 + / 6 + 10/ 10+44/ + 10/2

U 1 1 1/ 1 4 10 + 6/ 20+40/+4/2. Hence we can obtain in particular the inequalities

/2 + 12/ + 3 6/ + 10

6/2 + 10/

r 6/2 + 10/ V ' " t2 + 12/ + 3

^VT

( 1 S ( S 9),

i2 + 1 2 / + 3 ( / S 9 ) . i2 + 12/ + 3 - ^ ' - 6/ + 10 The domain in which we can apply these approximations is clear

from the following table:

c

1 2 3 4 5 6 7 8 9

10 11 12 13

f* + 12/ + 3 6/ + 10

1000 1-409 1-714 1-971 2-200 2-413 2-615 2-810 3-000 3-186 3-368 3-549 3-727

^

1-000 1-414 1-732 2-000 2-236 2-449 2-646 2-828 3-000 3-162 3-317 3-464 3-606

6/2 + 10f f2 + 12/ + 3

1-000 1-419 1-750 2030 2-273 2-486 2-676 2-847 3-000 3-139 3-266 3-381 3-488


Extraction of any rational root with the aid of matrices. It may be seen, on generalizing the method above described, that the /rth-order square matrix

fa t t ...t 1 a t .. .t

VI 1 1...J

enables approximate values to be obtained for

y?, VT\..., VF*. For example, to evaluate v4 , we only need to carry out the fol

lowing working:

1 2 2 1 ] 1 ] 1 ] 1 1 1 1 1 ]

I 2 I 1 [ 1 I 1 [ 1 I 1

2 2 2 1 1 .1 1

2 2 2 2 1 1 1

2 2 2 2 2 1 1

2 2 2 2 2 2 1.

1 13 1 12 1 11 1 10 1 9 1 8 1 7

127 115 104 94 85 77 70

In particular, Ifi « 104/85 = 1-223 or \J4 ~ 85/70 % 1-214. The accurate value of this root is 1*219... .

The convergence of Euler's algorithm. The method of extraction of roots of any rational degree with the aid of matrices was first proposed by Euler (though not in matrix form). The conditions for its convergence were recently obtained by L. D. Eskin. In particular, he showed that Euler's algorithm is convergent for any t >· 0 provided only that the zero approximation be chosen by a reliable method. Furthermore, Euler's algorithm is convergent for complex i, which enables it to be used for approximate extraction of roots of complex numbers.

6. Solution of cubic equations by means of matrices

The matrix ' a u al2 alz

021 022 023

'31 "32 *33;

eads to the equations

azlx + aZ2y + azz x = and y = 0 2 1 * + 022^ + 023

0 3 1 * + 0 3 2 ^ + 0 3 3

(5.68)

(5.69)

On eliminating Λ: from these equations, we arrive at a third degree equation in y. Consequently, the matrix (5.68) can serve, when the corresponding process is convergent, for approximate evaluation of one of the roots of a third degree equation.

We can readily obtain from (5.69) the equations

y= 1 +

x = >> +

(*21 - <?3ΐ)* + 0*22 - ato)y + 0 2 3 - 0 3 3

031^ + 032^ + 033

(011 ~ 0 2 l ) ^ + ( 0 1 2 - 0 2 2 ) ^ + 0 1 3 - 0 2 3

031^ + 032^ + 033

Let us require that the condition y2 = x be satisfied. For this, it is sufficient that the equations obtained take the form

y= 1 + 023"-0? 33 0 3 1 * + 032^ + 033

X = 7 + ( 0 2 3 - 0 3 3 ^

0 3 1 ^ + 032^ + 033 Now,

021 — 031» 022 — 032» all — a2\y alZ = û23>

012 — 022 = 023 ~~ 033·

Matrix (5.68) takes the form

011 022 + 0 1 3 - 0 3 3 0:

*11

*11

022

w22

13

*Z2)

It leads to the equation

0 1 1 ^ + ( 0 2 2 - 0 1 1 ) ^ + ( 0 3 3 - 0 2 2 ) 7 - 0 1 3 = 0 .

(5.70)

(5.71)

Let us put, in particular, an = a^ = 1 and let us write 033 — 1 = /?, alz = — q. The matrix (5.70) takes the form

(1 -p-q -q) 1 - ?

1 /> + !, (5.72)

It leads to the equation

y^+py+q = ° When/? = 0, #= — f, the matrix (5.72) becomes the matrix for extracting the cube root.

EXAMPLE 23. Let us apply matrix (5.72) for obtaining approximately one of the roots of the equation x3—x— 1 = 0 . We have

Ί 2 V 1 1 1 ,1 1 0,

1 1 1 1

4 3 2 1-5

12 9 7

1-285

37 28 21

1-333

114 86 65

1-292

351 265 200

1-325

1081 816 616

1-324 Notice that the convergence of this method is only rapid when

the equation does not possess two roots close to one another in absolute value.

7. Recurrent series. The Bernoulli—Eider method

A power series £ anxn, the coefficients of which, as from a certain

H, are connected by the same linear relationship, is described as recur-rent. The following theorem holds : the necessary and sufficient condition

00

for the sum of the series ]£ anxn, convergent for \ x | < I, to be a rational

function ofx is that the series be recurrent. D. Bernoulli and Euler applied recurrent series to the solution of

algebraic equations. Cauchy provided the basis of the method by indicating that, if xx is the root of least absolute value of the polynomial Q(x), the radius of convergence of the series

V n _ IQ


where P(x) is a polynomial having no roots in common with Q(x\ and the degree of which does not exceed the degree of Q(x), is given in accordance with d'Alembert's test by the relationship

Um fn±!*L = 1.


x± = lim n-*-°o u n + l

For instance, Euler solved the equation x3—3*+l = 0 by using the following expansion:

-T—4- 3 = 1 + 3* + 9χ2 + 26χ3 + 75χ4 + 216χ5 + 1 — 3x + x3

+ 622*e +1791*7 + 5157*8 + 14849*9 +

+ 42756A:10 + . . . s £ anxn.

Here xx = lim ajan+i, where *χ is the root of least absolute value

of the equation *3—3x+l = 0. The convergence of the process is clear from the following table:

n

1 2 3 4 5 6 7

a-*n+l

0-333 0-333 0-346 0-3467 0-34722 0-34727 0-347292

On replacing x by 1 /xy the same method can be used to find approximately the root of greatest absolute value of the equation Q(x ) = 0.


The convergence of the Bernoulli-Euler method is also only rapid when the equation does not have two roots close to one another in absolute value.

8. Connection between the Bernoulli—Eider method and matrix methods

Let us find by the Bernoulli-Euler method the root of least absolute value of the equation x3 +px -f q = 0. For this, it is sufficient to expand l/(g+px+x?) as a recurrent series. We have

-- = A0 -f ΑΛΧ -f A2x? -f . . . . q+ρχ + χ* ° λ 2 ^

On equating coefficients of powers of x9 we get

A0 = —, qAx +pA0 = 0, qA2 -\-pA1 = 0. qAz +pAz + A0 = 0.

and in the general case:

qAn +pAn_± + Λη-3 = 0 {n 3).

This relationship can be written as the matrix equation:

( 0 q 0 0

ί - i o

°i q\

-p)

(A \ I ^ n - 3

Mn-2 V ^ n - l ,

= ' ^ n - 2 Ì

î ^ n - 1

Mn J But this equation gives only one new coefficient An, while the two preceding ones are repeated. On taking the cube of the matrix composed of the coefficients, we at once obtain three new coefficients:

ί 0 q 0 0

[-1 0

0y

q -p.

3 (A Ì A n - 3

■^n-2

^η-ί,

= 'φΑη ì ?3Λη+1

k? -^n+2j

i.e. Mn-3 ^ π - 2

l^n-1 , =

'«•Λ Ì ?·Λ+ι

,#3^n+2j P2 — Pff

0 pq> -fq


On applying this method to the equation x3 — 3x— 1 = 0, we get ( - 1 0 3 | 1 26 622 14849

- 3 - 1 9 3 75 1791 42756. [ - 9 - 3 26 J 9 216 5157 123111

An even higher power of the initial matrix can be taken; this still further strengthens the convergence, though not all the coefficients of the recurrent series are utilized in this case. The elements of the matrices obtained can be divided by the same number, since we only need the ratios of neighbouring coefficients. But the series obtained here will in general differ from the recurrent series, into which l/(q+px+x?) is expanded.

On replacing x by 1/JC, the same method can be used to evaluate the root of greatest absolute value of the equation x*+px+q= 0 (if the corresponding process is convergent).

9. Solution of higher degree equations by means of matrices

We can use a method similar to the above to form nth-order matrices that can be employed for the approximate evaluation of the roots of an «th-degree equation. For example, the matrix

k 0

0 0 0 0

~lao { o

K k

0 0 0 0

-la, 0

0 . . fon··

0 . . 0 . . 0 . . 0 . .

- fo2 0 . .

. 0

. 0

• fon . k . 0 . 0 • -fon-t . 0

0 0

0 K k 0

- fo)l-4 0

0 0

0 0 fon k

- f o n -0

0 0

0 0 0 0

-3 k—

fon fon-1

0 0

0 0 0 fon

-fon-k

can serve, when the corresponding process is convergent, for approximate evaluation of one of the roots of the equation

αΛχΛ + αη-ι*η-1+ . . - -foix + öo = 0-For example, in the case of the equation


we obtain on putting k = / = 1 :

/ 1 1 0 0 1 0

- 1 1 9 V 0 0 1

The exact value of this root is 7*8873.

o 1 1 l)

1 (2) 1 1 (2) 1 1 (8)4 1 (2) 1 1 4

2 2 35 5 7

4 7

310 40

7-75

11 47

2753 350

7-866

58 397

24463 3103

7-8837

455 3500

217403 27566 7-8866

10. The idea behind Jacobfs algorithm

Euler posed and partially solved the problem of finding the solution in integers of the equation α1χ1+α2χ2+α3Λ:8 = 0· Jacobi posed the analogous problem for the equation

and developed for this purpose a special algorithm, which is a generalization of the algorithm of continued fractions. In particular, Jacobi assigned to this algorithm the following form:

Vfi+l = W n - ^ t U n , \ (5.73)

W„ hi = «n-

Here, pn and qn are positive integers, chosen in such a way that un+1 and vn+1 are the least possible positive numbers. With the aid of the numbers pn and qn, we form the following relationships:

Λη ^ ^n-^n-l +/?ny4n-2~J"^n-3> j

Bn = qnBn-l+PnBn-2+Bn^ \ (5.74) Vn — # n Q i - l + i 7 n Q i - 2 + Qi-3> /

We suppose that (A„2 A_x AQ\ ( 1 0 0Ì B_2 Β^ Β0 == 0 1 0

lc_2 c_x c0) [o o i In this case, when Jacobi's algorithm is convergent,


Jacobi's algorithm enables us to find fairly small integers An> Bn, Cn, approximately proportional to the given large numbers ul9 vl9 wv

EXAMPLE 24. Let us solve approximately the equation

jc : y : z % 49 : 59 : 75.

We have, on using Jacobi's algorithm,

n -2 -1 0 1 2 3 4 5 6

"n

49 10 6 3 1 0

Vn

59 26 9 4 0 0

Wn

75 49 10 6 3 1

Pn

1 2 1 1 0

In

1 4 1 2 3

Λ 1 0 0 1 4 5 15 49

Bn

0 1 0 1 5 6 18 59

Cn

0 0 1 1 6 8 23 75

Thus, for example, 5: 6: 8 % 15: 18: 23 % 49: 59: 75. The order of filling up this table is as follows: we first evaluate

W l=49, vx=:59, w1 = 75and^n,J5n, Cn,with τ ζ = - 2 , - 1 , 0. Next, in accordance with algorithm (5.73), we evaluate pl9 ql9 u^ va, vv2. Then, with the aid of relations (5.74), we evaluate Al9Bl9 Cv Next, in accordance with algorithm (5.73), we evaluate p2, q2, «3, v3, wz. With the aid of relationship (5.74), we evaluate A2, B2, C2, and so on.

CHAPTER VI

SOME SPECIAL CONSTANTS AND FUNCTIONS

THE present chapter contains information on real constants and some real functions commonly encountered in analysis.f

Our treatment must take into account the fact that differing definitions are current in mathematical literature. For instance, the generating function of a sequence of functions is usually defined as the function of two variables/(x, y) for which the relationship

/(*, y) = Σ Ρ«ΜΛ

is satisfied in some domain of the x9 y plane, i.e. the <pn{x) are the coefficients of the expansion of /(*, y) as a series in powers of y (cf. the definitions of the generating functions of Bessel functions and Legendre polynomials). In other cases,/(x, y) is called the generating function of a sequence if

(cf. the definitions of the generating functions of Bernoulli and Euler polynomials).

There is also a lack of uniformity as regards the notation for various functions. In such cases the different notations will be quoted, though only one of them will be utilized.

t Information concerning functions of a complex variable will be given in a later volume.

304

SOME SPECIAL CONSTANTS AND FUNCTIONS 305

§ 1. Various constants and expressions

1. Some well-known constants

1°. The number π: π = 3Ί4159 26535 89793... The number π made its appearance in connection with the calcula

tion of the length of a circumference by finding successively the perimeters of the inscribed and circumscribed regular polygons with 2n sides, π is a transcendental number.

Some approximations for the value of π: π ÄS 22/7 (Archimedes' number); π % 355/113 (Metsiev's number). Some representations ofn as series and products. On putting x= 1 in

the power series expansion of arc tan x, we get the Leibniz series

- = 1 4 + j_4 + . . . + ( _ 1 ) f t _ j_ + . . . . (6.D With x = 1/V3, we get from the same expansion

π 1 / 1 1 1 1 1 1 \ . . _ τ = -^{ιΊΊ+ΊΎ-1Ύ+-·-)· ( 6 · 2 )

Since π = 16 arc tan (1/5) —4 arc tan (1/239) (Machin's formula), we have

, J \ 1 1 1 1 1 1 1 1 \

- 4 ( 2 ΐ 9 - Τ 2 3 ^ + · · ^ · ( 6 · 3 )

Wallis1 s formula is

» , ^ Γ » 0 " 1 · 1 (6.4) 2 η->00[(2«-1)!! J 2Λ+1 ν '

As a continued fraction:

Ü-..1 il il K2

4 "" 1 + 3 -h 5 + . . . +2/2+1+ . . , (6.5)

Other relationships are:

Î = nf[l-(^W]· (6-6) - J = f a r e tan - . (6.7)

. π ~ 1 -r = > arc tan—5 —.

- = V 1 8 „to (4"+0(4« +3)·

4 „ti 2κ-1 v ; ^ W * (2n)2"»+2

0 < 0 < 1. _π _ " 2« 2π 1 ~2 ~ £ «2+r2

e2«n_! + 2 Λ +

, ^ ( - O * - 1 * ^ ι fl(-i)m i«TO+2 À (^2)2*"1 2 / i - l (2n2)m+i 2w+l

'

π 2 2 2 2 Jlsll+Jl^ 2+^2 + ^2

(6.8)

(6.9)

(6.10)

(6.11)

(6.12)

| = n n c o s 2 ^ · ( 6 · 1 3 )

π^ _ _39_ 1 __1 1 4 ~ 16 12.22.32 + 22.32.42 + 32.42.52 + · · ' · ( 6 · 1 4 ^

jt2 ~ 1

4- = Σ 4· (6-15) 6 Ä "2

/ry-2 OO 1

T-£,&+&' ( 6 · 1 6 )

π2 °° 1 Ί 2 = Σ ( - 1 ) η - 1 ^ · . (6-17)

^ . = i-an2)2 + | ^ . (6.18)

3π2 °° 1 = 1 — 2 Y -

32 ,έχί^-Ι)8' 16 ~~ 2 + 1232 + 3252 + 5272 +

3π2 1 1 1 1 256 9 123252 325272 ~ 527252

(6.19)

(6.20)

(6.21)

Inequalities connected with the number π. If a n s 0 («= 1, 2, . . . ) , but not all are zero, we have

il")' Σ«η * π« £ α» Σ Ά& (6.22)

π2 being the best constant in the sense that there exists a sequence an

for which equality is attained. If fix) ^ 0, fix) =ZÉ 0, / € l 2 (0 , oo), xf ζΖ^ (0, oo), we have

Γ/(*>&} ^π2 Γ / ^ ) Λ . Γ f\x)dx\\ x2f2(x)dx , (6.23)

where π2 is the best constant in the sense that there exist f(x) for which equality is attained.

Integral forms ofn:

T " J o tan* <£c.

H: sin x cos m* dx, m < 1.

de. _π _ p~ si T ~ J . ~ τ ντ = J 0

sin (χ2) = J 0 cos < ; 2) dx.

(6.24)

(6.25)

(6.26)

(6.27)

71 _ f °° S*n X J — f °° C0S X ri 2 Jo yfx Jo V*

• Jo 2

T

π2

l· Jo tfe-^dx.

xdx

12

e*- l

ê*+T

π2 _ Γ1 In χ </;

8 " J o ^2"1

f1 In (! + *)<& 12

2°. The number e:

e = 2-7182818284 59045

Euler's number e is defined as the limit

e = lim ( 1 + — ) = lim(l+a)"° n-*- oo \ ft / a-».o

e is a transcendental number.

(6.28)

(6.29)

(6.30)

(6.31)

(6.32)

(6.33)

(6.34)

(6.35)

(6.36)


Expressions for e, e2, e~x, (l + e)2 as series and products: 00 1

1 S nm e = — > — .

Pm n=o n\ m + (m — 1)

Λ»*ι = l + mpi+ YJ P2 + ■ · · +Pml

Λ=1» />2 = 2» Λ = 5» ^4=15> p5 = 52,...,

~ 1 \3J \5J ) \ 9.11.13.15 16 \«

= τ Σ 2n(«+l)

3 »ti

(1 + «)2 = Σ

0 n\ 2+2 n

n-o n\

e2 = lim ."V('.)(ï)(î)-C)· e-i = £ ( _ l ) n

n>»0 " ·

e - 1 = lim V«l

2e- hm p ^ - Σ («+*Ό » ft-0

_ J_ 2 _1_ 1 1 1 - 3 + 6 + 1 0 + . . . + 2(2«+ 1) + . . .

(continued fraction). Some inequalities. When 0 S ί 5 «:

(6.37)

I (6.38)

(6.39)

(6.40)

(6.41)

(6.42)

(6.43)

(6.44)

(6.45)

(6.46)

1 ^ e n. n

1+— s en. /I

(6.47)

(6.48)

When w„ > 0 n t\n t2

0 ^ l -e«( 1 - - i ) ^ — . (6.49) 1 « i n

(6.50)

(6.51)

Minima: (6.52)

(6.53)

The number e is the base of logarithms with which the inequality is satisfied:

log ax ·£ x - 1 (Ostrogradskii). (6.54)

The functions ex and e~x are defined as the limits

£ V«l"2 · · · "n n~l __

e o

hm sup — ^^

min Xe = e' 0 ^ Ä < o o

i n i TÌ — ^ — — Il i ILL — —

o<x<oo in x

^

^

- 1

e.

€·

e.

= lim(l-*Y. e· = lim ( 1+^·Γ, «_* = Km (!-■=■) . (6.55)

3°. The Euler-Mascheroni constant C:

C = 0-5772156649 015325 . . .

The Euler-Mascheroni constant C is defined as the limit

C = Hm ( £ — - Inm), (6.56)

the existence of which follows from the convergence of the series with general term

g(n) =

It is unknown whether C is rational or irrational.

Integral forms. Since

, 1 i + 7 + . . +

and

we have

— ln/w = — y1 dy

from which we find, on passing to the limit, that

C = Γ (1 - e-%-* dy - ["e-vy-1 dy (6.57)

or

C = r f l - e - ' - e " ~ j r 1 A .

It follows from (6.58) that

C = f°[(l - e-')"1 - i"1] e-* di.

Other integral forms.

(6.58)

(6.59)

(6.60) C= - PV'lnf<ft.

C " - J " ( C 0 S ' _ W ) T · <6·62» C = I - J ; ( ^ - ^ ) T · <«*>


C = - In In— du (6.67)

arc cot t - e-pt \ _L _ ιη ^. (6#68) c = i : ( ; C = ( — arccot/ — cosp/ Wi — In/?. (6.69

2 f~ ώ C = sin ί Ini —. (6.70)

The asymptotic expansion is

m-i i 1 1 1 n—! « 2m 12m2 120m4

- + : f e - Ì + , I S 5 B « . 0-.-L(MI) Infinite products expressible in terms of e and C are:

en ^\ 2 3 m /

Π —r= lim ~(—i

c lim .. , W . N = e°. (6.72)

(6.73) 4°. Catalan's constant Cf:

C? = 0-91596 55941 77219 0

(6.65)

(6.66)

Catalans constant G is defined as the number

:tanx G

__ f1 arc ta:

"Jo~ Γ ■ dx. (6.74)

An expression for Catalan's constant as a series can be obtained by expanding x"1 arc tan x in (6.74) as a power series:

β - Σ - I m

Other integral forms are :

G

„£0(2m+l)2·

_ 1 p»/» r<ft ~ 2 J o SÌQ'

G? = j la 2 + 2 •»/4

t tan t dt.

6 ? = - - j l n 2 + 2 tcottdt.

±G--îln2+r' 1 4 J o (cosi+s sin t) sin ί

^ π i o , Γπ/4 * Λ G = — In 2 + —, — —

8 I (sin / + cos t) sin t π G = - — In 2 - 2

'π/4 In sin t dt.

±6? =

G = — In 2 + 2 In cos ί dt

1 Λ»/2

f*/2

e - - J . - ï + F * Vi In ί Λ

( l ± c o s i ) A + - j l n 2 .

ln( l + tani) A - — In 2.

G= - - I n 2 - 2 V i - î 2

(6.75)

(6.76)

(6.77)

(6.78)

(6.79)

(6.80)

(6.81)

(6.82)

(6.83)

(6.84)

(6.85)

(6.86)


G = p l n d + O-jAy-fLlni (6.87)

G "toa-if1!"^*. (6.88) 4 2Jo sll-t*

^> π i ^ 1 f1 arc cos / . G = ± - l n 2 + -j 1 ± r dt. (6.92)

^ f°° arc cot t j __ π t ,_ „Λ Λ λ G = _ _ _ Λ τ - 1 η 2 . (6.93) ^ f1 arc tan t , π , ^ „ Λ,χ G = j o ^ Î T 7 r * + - 8 - l n 2 . (6.94)

β._|-«^£Α (695).

G = I In 2 + j 1 ^ - arctan Λ - ^ L . (6.96)

G = 2 J ^ - a r c t a n ^ - ^ - J In 2. (6.97)

(arc tant ) 2 —r -—· (6.98)

G = "5"+T Γ(arCCOt °2 "^=5* (6,99) tdt

Expressions in terms of complete elliptic integrals (see (6.380) and (6.381)):

G = j \ K(k) dk. (6.100)

G = [lE(k)dk-\. (6.101)

(6.89)

(6.90)

(6.91)


2. Soumne merical expressions

1°. Factorials. When n is a positive integer:

n\ = 1 ,2 .3 . . . n. (6.102) By definition,

0! = 1.

The gamma function is a generalization of the factorial (see § 4, sec. 5).

When m is a positive integer:

(2m)!! = 2 . 4 . 6 . . . 2m. (6.103)

( 2 m + l ) ! ! = 1 .3 .5 . . . (2m+ 1). (6.104)

(2m) ! = 2™m !(2m - 1 ) ! ! (6.105)

The number nil is called the double factorial of n. Expressions for factorials, inverse factorials and their sums:

1 °° 1 ~m\ = n S i m(m+l)...(m + n) * ( 6 , 1 0 6 )

(n+l)! = 1+ £*fc! (6.107)

f _ L = g + g = cosh 1 = 1-54308 . . . . (6.108) n-i(272)! 2

Ä * e~* = sinh 1 = 1-175201 . . . . (6.109) ntio(2if-l)! 2

Σ feif = cos 1 = 0-54030.... (6.110)

Σ ( - 1 ) " - 1 , , ' n , = sin 1 = 0-84147.... (6.111) n - 0 (Z/I—1J!

Σ 7 Ì 2 = Jo(2) = 2-27958530 . . . . (6.112) n « l V*'·}

Ä ! = /j.(2) = 1-590636855... . (6.113) nt-o » ! ( « + ! ) !

£ 1 »ΐθ »!(»+*)! 4(2). (6.114)

Σ ^TzS- = Λ(2) = 0-22389078 . . . . (6.115) n«o \η·)

l „ ^ f e ì % r = y'(2) = 0 ' 5 7 6 7 2 4 8 1 · · · - <«-1Ιδ> | „ Ί & = «2>· <6·1 Ι 7>

Inequalities and asymptotic formulae:

^.v:(i)'-..<^(i)·. («-»s) (6.119) ■' ~ (τ)"-

^ - 4. (6.120) («1)2

»! ~ V2^ (7ÌY 1 + a + · · ·) (Stirling). (6.121)

- n + 0 ' (2/j - 1 ) ! ! = V2 (2n)ne "", 10n | < 1. (6.122) 1 , n + i

, n + T | 0 „! = V 2 ^ — J 724^+12 (Gauss). (6.123)

A formula for the logarithm as a factorial:

to"!=,L1°'(ft+[F]+[>]+-··} (6·124)

where the summation is over all primes p not exceeding n. ( [x] is the integral part of x. See § 3, sec. 1, 3°.)

Divisors of a factorial. The greatest index with which a prime p appears as a factor in w! is

\î]+[?Y-+[ï] where /?fe ^ w, but pk+1 > «.

For example, the highest power of the number 2 by which 50! is divisible is

50 Γ50Ί Γ50Ί Γ50Ί Γ50Ί An

Thus 50! is divisible by 247. 2°. The Kronecker delta <5™ (or <5mn) is defined by

n " 10, n = m •Φ m.

(6.126)

The integral form is:

% = [*9m(x>Pn(x)dx (6.127)

where {<pk(x)} is a system of functions orthonormal in [a, b]. For instance,

°*-K. cos nx cos mx dx. (6.128)

3°. The binomial coefficients ( ) or CT\ where n is any real W number and m is a positive integer, are defined by

i - l ) . . . ( « - m + l ) in\ _ n(n-m\

When « is a positive integer:

( : ) m ! (n — m) !

(6.129)

(6.130)

The beta function is a generalization of the binomial coefficient (see § 4, sec. 5, 2°).

Some relationships between binomial coefficients (72, m integers):

( : ) 71 ' = 0 for >i> 0, m > n. (6.131)

r, m - ( 6 · 1 3 2 )

( ;K,r)+-+C*t)-n+ i ) ·("ο)-(")+(^)-·-+(-ι>'(-)=(-ι)*("ΐ1)·

(?H?)+-+(.-i)-»-<"··**

(?) + (τ) + · · + ( η ^ ΐ ) = 2i"_'(»iseven).

= (3 - l)»+i -?L sin ^ - .

(;H3)+(6)+-^(2"+2cos^)·(")+(7)+(")+-=Κ2·+2οο5^·(6-( Ô ) + ( : ) + 0 ) + - ^ ( 2 " - I + 2 Î - S T ) ·(ϊΜ")+(?)+···-:Κ2"-'^Τ>( ; Η Σ ) + ( . Ο ) + · - Κ 2 Μ - 2 Ϊ - Τ ) ·

(6.134)

(6.135)

(6.136)

(6.137)

(6.138)

(6.139)

(6.140)

(6.141)

(6.142)

(6.143)

(6.144)

(6.145)

(6.146)

(6.147)

(;)+(")+(û)+-=K2"--2ÎsinÎ>·(;)-(;Η0-(Σ)+-- ΐ ΐ β -τ·

(S) + 2 ( " ) + 3 (2) + - + < " + 1 ) (») = <" + 2>2""'·

(ϊ)-ί(τ)+3(τ)-···+<-,>"·(:)-0

( ; ) 4 ( ; κ α ) + - - · + ^ ) = ^ ·

(ΪΗ(;)4(;)-·+^00=' = 1 + τ + · · · + 7 ·= 12 + 22+ . . . + «*.

(;)(")+(ΐ)(^)+···+

(;)CH0(.:.)+-+(^)(Î)-- ( ,?* ) - {d§SU·

319

(6.148)

(6.149)

(6.150)

(6.151)

(6.152)

(6.153)

(6.154)

(6.155)

(6.156)

(6.157)

(6.158)

(6.159)

(6.160)

(?)'-(ΐ),+···+(*),-(-,Κ*)-( - 1 ) nw· (6·161)

(^>j-(^'j+ . . . -(^;j=o. <,,*>

OMO 2 + . . . +71

/ 2 » - 2 \ (2/j-l)! „ , „ N

-^ -^(„ - i j - Ï^Tïy f · · (6·163)

Binomial and polynomial formulae:

(a1 + a2r= Σ τέτΦΦ- (6.164)

fa + fli» . . . +flp)n =

Σ „,„,"' , , W · · · » (6.165)

Some identities in connection with binomial coefficients:

ί,-ίΚ*)1**1-""-1'*· (6ΐ66)

â( î - ' ) , ( :h-^- i a r â - (6·ΐ68)

The Bernshtein polynomials Bn{f) of a function f(x) are defined by the relationship

W ) = | o ( ^ ) A l - ^ - / ( | ) . (6.169)

If/(*) is continuous in the interval 0 x < 1, the sequence{2?n(/)} is uniformly convergent to/(*) in this interval:

lim 2?n(/) =/(*). (6.170)

Asymptotic formulae.

2n\ 2*n 1*.-)~jTneQn | 9 » I < L

w \ 1 « 2

tó = - ^ ^ Γ Λι, ωη - 0 as n — .x / v m 2(n — m) 2

fe-l

£(^~*(=)ν

("^)-7WiT+Ä^-,+0<"'-,>·

«. = .t i t ')

Fibonacci numbers and the golden section. The sums

[—1

are connected by the relationships

Wn = "η-1 + "π-2> Λ > 2 , UX = «2 = I·

The numbers wnare called Fibonacci numbers. Binet9 s formula holds:

The following are consecutive values of the Fibonacci numbers:

2 ../ \ 2___./_. (6.178)

, , , , , , , ,

The Fibonacci numbers appear in the expansion as a continued

•action of the number— (v5 -1) =

have the form un/un+l(n = 1, 2, . . .)

fraction of the number— (V5 -1) =— -r . The convergents Δ 1 " " Γ ~ 1 τ ~ · · ·

321

(6.171)

(6.172)

(6.173)

(6.174)

(6.175)

(6.176)

(6.177)

(6.178)

The number a = -— (yß-l) is occasionally called the number of

the golden section. It is defined as the mean proportional between 1 and its complement to 1 :

a2 = 1.(1-a).

It was supposed in classical aesthetics that, for instance, the rectangle with ratio of sides a, or the ellipse with ratio of axes a, were particularly pleasing to the eye. Hence the name golden section.

§ 2. Bernoulli and Euler numbers and polynomials

1. Bernoulli numbers and polynomials

Special number sequences, e.g. Bernoulli and Euler numbers, prove useful for obtaining power expansions of numerous functions.

1°. The Bernoulli numbers Bn (n = 0, 1, 2, . . . ) are defined as the coefficients of the expansion as a power series of the function t{el -1 )" 1

f o r | / | < 2 w : t °° tn

1ΠΓ7=Σ**7Γ· (6-179) e — ι na=0 m

These numbers were first mentioned by Johann Bernoulli (Ars Conjectandi, 1713), in connection with the problem of summing powers of the natural numbers. The numbers B figure in the power expansions of the functions tan i, tanh t, t cot t, t cotanh t, etc., in the Euler-Maclaurin summation formula, in the asymptotic form of Euler's gamma function.

There is no one accepted notation for the Bernoulli numbers. In certain works, Bn is used to denote what we denote in the present section by | Bn |, or B2n9 or | B^ |.

Recurrence relations. Using (6.179), rewritten in the form

71=0 n\

cross-multiplying the series on the right-hand side and comparing

coefficients of powers of t, we find that

C5UA+<2+i£i - i+ · · · +CLiBn-k+i+ ■ · · · · · + Q + i - 8 i + 50 = 0 (6.180)

or, in symbolic form: (l + 5)"+i-5»+i = 0, (6.181)

where, after raising to a power in accordance with the binomial formula, all the exponents of the powers are replaced by subscripts.

Since i C e ' - l ^ - C - i X e - ' - l ) - 1 = -t =

= 2 t B2k+1 -~—^- (6.182) kTo (2fc+l)! v '

all the Bernoulli numbers with odd subscripts, n s 3 , are zero. All the Bernoulli numbers with even subscripts are non-zero, B2n being positive for n odd and negative for n even (see (6.194)).

Laplace's formula

Bn = (-1)»«!

J_ 2! J_ 3! J_ 4!

1

_1_ 2! J_ 3!

0

_1_ 2!

1 1

0

0

0

1

(6.183)

(n+1)! n! ( « - ! ) ! " ' 2 !

gives an explicit expression for the value of a Bernoulli number in terms of its subscript.

The Bernoulli numbers are rational. This follows, for instance, from the previous formula.

STAUDT'S THEOREM. Every Bernoulli number Bn can be written in the form

5n = Cn-E-£TT> (6-184)


where Cn is an integer, and the summation is over all fc > 0 such that k+l is prime, k being a factor ofn.

For example, when n = 6, the factors of n are k = 1, 2, 3, 6. On adding unity to each of these, we get the numbers 2, 3,4, 7, of which the primes are 2, 3, 7. Consequently, B6= 1—(1/2)—(1/3)—(1/7) = = 1/42.

Some Bernoulli numbers:

B0 = 1 , Bx = - - , B2= τ , Bi = - — ,

1 ß - 1 B - 5 42 ' ΰ* - ~ 30' Bl° ~ "66

n _ 6 9 1 » _ 7 p _ 3 6 1 7 R _ 4 3 8 6 7

^12 ~ ~~~273Ö~' ßu * T ' 16 ~ 5ÏÔ~' 18 ~ ~79Γ'

174611 „ 854513 ■#20 — 5 ™ — > ^22 — 330 * Ά 123 '

236364091 „ 8553103 #24 — ôrîri » ^2· ~~ 2730 * " 6 _ 23749461029 _ 8615841276005

^28 - 8 7 0 . -»so - 1 4 3 2 2

7709321041217 „ 2577687858367 510 ■832 ^TÔ ' 5 3 4 -

_ 26315271553053477373 36 ~ 1919190

_ 2929993913841559 _ 261082718196449122051 Bzs - -e , Bi0 - j ^ .

Some power expansions with coefficients expressible in terms of

Bernoulli numbers. Since — coth — t — {t){é—1)} + — t, it follows

from (6.179), after replacing t by 2t, that

~> 72η fi fin t cotanh t = 1 + £ Z™. , M -= ». (6.185)

we get

// cotanh it = t cot /, tan t = cot t — 2 cot It, t t cosec t = ί cot t +1 tan —,

tanh t = 2 cotanh 2i—cotanh f,

cosech ί = — cotanh t — cotanh —,

^ c o t r = l + E n\\ **"> ΙΊ < π ·n« 1 ( 2 Λ ) !

ί cosec * = 1 + Σ V , **η'2η, I 'I < *.n - i (2Λ)!

tanh t = £ 2 2 η ( ^ 7 υ ^η'2"-1, I /1 < T ·n t i (2/1) I 2

/ cosech t = 1 + £ 4 = ^ £ . 52n^n-i, I /1 <: π.n - i (2Λ)!

-~ ' - ! ( - ' rX-'*- ' - · '«'-f

Expressions for the sums of some numerical series in terms of Bernoulli numbers. If we compare expansion (6.186) with

o© * 2 00 *2n 00 1

it follows that I L . - l Z . ^ _ ( - l ) - i 2 ( 2 i i ) l Ä 1 ^» = Tôvûïï λ -^Γ·

(6.186)

(6.187)

(6.188)

(6.189)

(6.190)

(6.191)

(6.192)

(6.193)

(6.194)

The equations follow from (6.194): V 1 (-1)η-ΐ(2π)2* D Σ —zr = ^Γ-η #<

τη«ι m ι2η 2(2«)! '2n· (6.195)

£ . ( - i^ ' Î Î - ( - ' ) -g^ - 1 ^^ *>*> m - l w 2 n

(-1)»-ΐ(22η-1)π2* ^ ( 2 / 7 1 - 1 ) ^ 2(2«)!

In particular, 00 1 π2

„£i m2 6

«V i} m* 2 - 12· S 1 352π2 π2

m è x ( 2 m - l ) 2 - 4 - 8 ' S 1 _ π2

mèi »I4 90 * S 1 π

Jixnfi "945* Ä 1 π8

mèx m8 ~ 9450 '

yj.iy 1 )

(6.198)

(6.199)

(6.200)

(6.201)

(6.202)

(6.203)

Some integrals expressible in terms of Bernoulli numbers. Integral forms of Bernoulli numbers. The relationship

1 1 £-mt;2n-ljft T W * * (2«-l)!

leads in conjunction with (6.194) to the following relationships: Γοο * 2 n - l

-δ2η = ( -1)»- 1 4" | ο -ρ 5 Γ Γ Τ Λ.

C°° t2ndt

Bm = ( - l )n _ 1 ^ 2 ^ , - ^ f°°i2n-2 In (1 - e-2«') A.

(6.204)

(6.205)

(6.206)

B2n =

B2n =

B2n =

^271 =

B2n =

B2n =

B2n =

^2n+2 —


/ 2 7 1 - 1

' \π) 22"-1 J 0 sinh/>/ w ft (lnx)g*-i ^

^ 22n-%2nJ0 1-je ,N , 2/ι Γ- (In ί ) 2 η _ 1 .

- ììn-i i — ' „ dt. ' ( l -2»>i*»J0 1-ί2

_l)n-x *, ρ θ η θ ^ ' (1-2»)«»«J0 1- i 2

- l ) n " i

1_)»· - 1

(22n_ 1)π2η

2n (1-22")π2η

fi I + ί2

fin t)2n l Τί

( m t ) ( 1 - i 2 ) 2 0 '*/2

(In tan ί ) 2 η _ 1

dt:

dt cos 2/

327

(6.207)

(6.208)

(6.209)

(6.210)

(6.211)

(6.212)

(6.213)

-1> n+1 2(n+l)(2n+l) fi p a n O ^ l n a - O - y . (6.214)

π2η+2

2°. The Bernoulli polynomials are given by the formula

BJx) = (x + 5 ) n = j ^ / ^ 2?ftx"-*. (6.215)

A number of authors consider, in addition to the polynomials Bn(x), the polynomials

φη(χ) = Bn(x)-Bn.) (6.216) It follows from (6.215) that

2?n(0) = Bn, 9>n(0) = 0. (6.217)

The generating function for the Bernoulli polynomials is te^iê— — I ) - 1 . Its expansion as a power series, convergent for | /1 < 2π, has the form

tt*v-D-i = !**(*)-5-· ( 6 · 2 1 8 > n=o "·

The difference equation

/ (*+! ) - / (* ) ^rnx»-1 (6.219)


is satisfied by the Bernoulli polynomial. This follows from (6.218) and the relationship

te(x+Dt(et_ i)-i _ te*{é-1)-1 = té*.

Recurrence formulae. Differentiation of (6.215) gives *;(*) = nBn^(x) (6.220)


Bn(x) = Bn(0) + n (XBn^(x) dx. (6.221)

Some Bernoulli polynomials:

B0(x) = 1.

^(Jf) = X—j.

£2(x) = x*-x + j .

B3(x) = x*-jx*+jx.

Bt(x) = x 4 - 2 ^ + x 2 - ^ - .

Bs(x) = xS—2xi + J^—ß χ·

B6(x) = *»-3*»+!**-l** + -L.

7 7 7 1 £7(x) = X1-jXe + — Xi-jX3 + jX.

B6(x) = Λ · - 4 Χ * + ^ - * · - 1 Χ * + j * « -

59(x) = J C 9 - | - X 8 + 6 X 7 - - ^ - X 5 + 2 X 3 -

£ioW = x10 - 5x* + - y ** - 7xe + 5x4 -

1 30' 3

ΊΟ*·

2 * + 66 ·

(6.222)

(6.223)

(6.224)

(6.225)

(6.226)

(6.227)

(6.228)

(6.229)

(6.230)

(6.231)

(6.232)

Zeros of Bernoulli polynomials. The following formula holds:

* n ( l - x ) = (-l)»j;(*), (6.233)

from which, letting x = 0, we get

Βη(1) = (-ψΒη (6.234)

and, since B2k+X = 0, we have

Bn(l) - Bn = Bn(l) - *n(0) = 0 (6.235) for all n.

It follows from (6.216) and (6.235) that

<Pn(0) = <Pn(V = 0 . (6.236)

It follows from (6.233) with « = 2fc+ 1 and x = ~- that

B*k+i(X) = -**k+i(j) (6·237)

or

<P2ft+i (j) = 0. (6.238)

The polynomials φη(χ) with even subscripts vanish on the segment [0, 1] if and only if x= 0 or x = 1; the <pn(x) with odd subscripts

vanish if and only if x = 0, x = -—, or x = 1.

The polynomials (p2k(x) with even subscripts are of constant sign, the same as that of (—1)\ on the interval (0, 1); the polynomial

9>2fc+i(*) has the sign of (— l)*""1 on the interval ( 0, ■— ), and the (\ \ \ 2J sign of (— 1)Ä on the interval I -- , 1 J.

A multiplication theorem for Bernoulli polynomials: m - l / « \

s=o \ m J


This relationship follows from the fact that

^ Bn(mx)tn emxtt 1 éms*mt(l + ét+ . . . + *0»-i>*) î i " 0 n\ ex—\ m emt—l

(X+-L·) mt Bn ( x + — ) mntn

Trigonometric expansions of Bernoulli polynomials in the interval (0,1):

Βλ{χ) = x — = - Σ ™ · (6·241)

τ, / Ν 1 ^ cos27rmx ,,**«>, *tâ = "ΖΓ Σ =ä · (6·242)

^ m-i m

( _ 1 ) π - 1 1 ~ COS2JTWX ,r*A^

( - ί)""1 D ω _ 1 y sin2rcms ί 6 244Ì

Integral forms of the Bernoulli polynomials in the interval (0, 1) :

B*nW = ( - l)n+12« Γ GOa?tX~e~*? ' 2 n - 1 dt (6.245) 2nV ' v ' J0 cosh 2πί-cos 2πχ v y

(n = 1, 2, . . . ).

W * ) = (- D"M2»+ « J" c o s h 2 ^ s 2 T O C '* * (6.246) (w = 0, 1, 2, . . . ) .

Application to the summation of powers of the natural numbers. On assigning to x in (6.219) the values 0,1,2,. . . , p and summing, we get

t J71"1 = Bn(p+i)-Bn

«-1

or, on replacing n — 1 by n,

f sn = *n+lQ>+!)-*»+!. ( 6 - 2 4 7 ) s=i n+l


(6.248)

For example,

p(p+l)(2p+\) 6

^1..j&M)=A.^_v+>,.aç2. («49)

= -^P(p+l)(2p+ l)(3p*+3p+ 1). (6.250)

= J2P2(P+ D W + 2^ + D- (6·251)

The Euler-Maclaurin formula establishes the connection between the integrals and the sums:

- % m [ / ' - - ' Μ . - / ' " - ' ^ ) ] - ^ - " ^ 5 " . (6.252)

If f(x) and all its derivatives encountered in (6.252) tend to zero, while the derivatives of even order differ from zero and all have the same sign, the summation formula holds for m = «>, if the sum and integral appearing in it are convergent.

Stirling's formula is obtained from the Euler-Maclaurin formula if we put/(x) = In x in the latter and make the lower limit of integration equal to unity: m-i i Σ Ink = mln/n-m — — ln(tfi+l) +

+ r t , - n - ^ ( 53) r - 1 2r(2r-l)

When n = 2, it follows from (6.253) that the asymptotic Stirling formula for the factorial (see (6.121)) is

m+1-m\ ^ yßnm 2e~™. (6.254)

2. Euler numbers and polynomials

1°. The Euler numbers Eh (k= 0, 1, 2 , . . . ) are defined as the coefficients in the expansion as a power series of the function

sechi = Σ Λ τ τ . Ι Ί ^ Τ · <6·255)

Since the function sech / = 2/(e'-fe~t) is even, it follows from this that the Euler numbers with odd subscripts are zero :

£2m+i = 0 (m= 1, 2, . . . ) . (6.256)

Recurrence relations. It follows from (6.255) that eo f2n eo *fe

1 -£Ϊ5Γ£*ΪΓ· ("5 7 )

whence £0 = 1 and £o + QnE2 + C*nEA + ...+ Cgt^E^-* + E2n = 0. (6.258)

In symbolic form:

(E+l)n+(E-l)n = 0, E0 = 1. (6.259)

Some Euler numbers:

E0 = 1, £2 = - 1 , ^ 4 = 5 , £6 = - 61, £8 = 1385, £ o = -50251, E^ = 2702765

£14 = -199360981, ^ = 19391512145, £18 = -2404879675441, £^0 = 370371188237525.

All the Euler numbers are integers. This follows from the recurrence formulae for the Ek. Two adjacent numbers with even subscripts have opposite signs.

SYLVESTER'S THEOREM. If α, β, γ9 . . . are factors of the number n—m, the difference E2n—E2m is divisible by those of the numbers 2a + 1, 2/3+ 1, 2y+ 1, . . . that areprime.

The connection with Bernoulli numbers. Starting from the identity Ax 2x __ Ixe-*

e*x— 1 e2x— 1 ~~* ~~ e* + e~x * we obtain

n«0 /Ζί i=0 l' m=o W !

from which it follows that n(E-l)n~l

B»= - y ( y - 1) ' ( 6 ' 2 6 0 )

Starting from the identity

■{é*x-ex) = e 4 x - l y e*-f-e-

we get

which gives _ (4l + 3)*-(42î+ l)k

±<k-i = 2k ' (0.261)

It follows from the last relationship that

2Ak+\ iR V 4 / \ 4

which gives, on taking (6.215) and (6.233) into account,

2MElk = - 2 ^ + 1 ( 1 ) . (6.263)

The sign of the Euler numbers. As follows from the discussion on page 329, B2k+1(x) has the sign of (-1)*"1 in the interval (0, 1/2), so that E2k has the sign of (—1)Ä.

Some power expansions with coefficients expressible in terms of Euler numbers. Observing that sec t = sech it, we get

s e c '^ | ( - 1 ) f t ( l ï ï i 2* '"-Y· (6·264)

On integrating the last series, we get

In "" {hi)\ - I.(- »"TO'"" '" - 7' ("65)

Integration of series (6.255) gives

m»*<-i+&2E>W+W· ( 6 · 2 6 6 )

7%e expressions for the sums of certain numerical series in terms of Euler numbers. On putting x = 1/4 in (6.244) and taking (6.263) into account, we get

1 32/1+1^52/1+1 72Ä+1 "*" * · · "~ 4*+i(2Ä:)! * ι ° · ^ / ;

For example,

Integral forms of Euler numbers. It follows from (6.246) and (6.263) that

£>n = ( - 1)η22λ+ι r2w sech {nt)dt (n = 0, 1, 2, . . . ). (6.269)

2°. The £i/fer polynomials are given by

*„(*) = (*4 + f ) n = |o(^)2->£ft(*4)n-\ (6.270) where £Ä are Euler numbers.

The generating function for the Euler polynomials is 2eact(ei+ l)"1. Its expansion as a power series, convergent for 11 | < π, is

The difference equation /(χ+1)+/(χ) = 2*" (6.272)

is satisfied by the Euler polynomial En(x). This follows from the relationship

2e(*+iV+l)- 1 + 2^(e*+l)-1 = 2e*<.


Recurrence formulae. On diflFerentiating (6.270), we get Efa) = nEn^(x) (6.273)


en{x) = En (j\+* Γ En-i(x) àx. (6.274)

Since £n(l/2) = 2"nEn (see (6.270)), we have

En(x) = 2-nEn + n i* 4 - i W àx. (6.275) Jl/2

In view of the relationship

2^+iv+D-1 - ΣΕτ(χ)-τ Σ £r= Σ *«(*+n-^ we have the recurrence formula

Ux+l) =j£(jjlUx), or, on taking (6.272) into account,

-£;(*) = 2*»-J;o("W). Some Euler polynomials:

E0(x) = 1.

£i(*) = * — 2 ·

i^(*) = X 2 - * .

W = ^-J^H-î· £4(;c) = χ 4 - 2 χ 3 + χ.

£5ω=^-4^+4^-γ. £·,(*) = xt-ìxt+Sxt-lx. irr Λ 7 7 β 35 , 21 . 17

£·8(Λ:) = χ8 - 4χ7 + 14.x5 - 2%χ* - 17*.

(6.276)

(6.277)

(6.278)

(6.279)

(6.280)

(6.281)

(6.282)

(6.283)

(6.284)

(6.285)

(6.286)

The connection between Euler and Bernoulli polynomials. Starting from the relationship

2 [e2 +l) = teX V - U - i - t e V - l ) - 1 , te

we get

= n-i2ÎBn(x)-2"Bn(j\{. (6.287)

Trigonometric expansions of Euler polynomials. On using the trigonometric expansions of Bernoulli polynomials in the interval 0 < x < 1, we get

E2k(x) = ( - 1)* 4(2fc) ! £ [(2Λ + 1)π]-Λ-ι sin (2« + 1)π*

(« = 1, 2, 3, . . . ). (6.288)

4 + 1 ( x ) = ( - l)h+14(2k +1)! £ [(2« +1)»] -»-» cos (2« + l)roc

(n = 0, 1, 2, 3, . . . ) . (6.289) A multiplication theorem:

τη-1 / r \ En(mx) = m» £ ( - l ) r £ n ( * + — I , m-=2fc + l. (6.290)

r«=o V w y m-i / r \

En(TOc) = -Im^n+l)-1 X ( - i ) ^ n + 1 ( x + — ) , m = 2k. (6.291) r*=o y m )

Integral forms of Euler polynomials may be obtained in the same way as those of Bernoulli polynomials:

2n(*) = (-l)M Γ i- / N / IN« A i f2n sin π * cosh πί , , , ~Λ~χ ^2nW = ( - 1)Λ4 — — — Ä (6.292) 2nv J v ' cosh 2 π / - c o s 2πχ v '

( 0 < x < 1; TI = 0, 1, 2, . . . ) , ^ 2 η + 1 0 Ο δ π Λ : δ ΐ η 1 ]

csoh 2nt - cos 2πχ / x / , Ν ^ , . Γ°° t2n+1cosnxsmhnt , ,r _Λ_

3m+i(*) = ( - 1 ) " + 1 4 j o ^ ■ , , , _ _ „ . „ Λ (6.293) I 0

(0 < x < 1; w = 0, 1, 2, . . . ) .

§ 3. Elementary piecewise linear functions and delta-shaped functions

1. Piecewise linear functions

1°. The absolute value of x (written as \x\) is:

[ —x, x < 0, x = 0, x = 0,

x, x > 0.

\x+y\ ^ \x\ + \y\, \x-y\ ^ ll*i-l>>l|.

(6.294)

(6.295)

The binomial expansion of |JC| = vl—(1— x2), for \x\ < 1, is

Ι-Ι = 1 - ^ - | 2 - ^ ^ - ( 1 - ^ · (6-296)

The approximation, for | JC | ^ 1, by Bernshtein polynomials (see § 1, sec. 2) of even degree is

«■LffiC l+x\ft fl-xY l-x l+x

(6.297) The approximation by Fejer polynomials, for | x | ^ π, is

. . . . .. Γπ 8n^n-k cos (2/c - l)x Ί

1*1 - t a ^ , ^ . [ τ — , Σ ^ - ! (2fe_1)2 j -(6.298)

The expansion in Legendre polynomials Pn(x) (see Chapter IV, §4, sec. 5), for \x\ < 1, is

1*1 = Σ *»*«(*). n=»o

1 5 / , Μ Ι , Μ Ι 1 ν ( 2 Λ - 3 ) ϋ flîft+i = 0, α„ = - j , fla = -g-, a2ft=(-nft+1(4fc + l) ^ + 2)!!

Fourier's expansion, for | ;c | < π, is

l i — π _ ^ V cos(2&+l)x | X | - "4" f t èb (2k+lf '

(6.299)

(6.300)

The integral form is _ 2 f~ 1-cc " " J o <2 1*1 = e i ' C?Xt dt. (6.301)

Polygonal functions ; the approximation of continuous functions by polygonal functions. Let A^x^ yj (ι = 0, 1, 2, . . . , w, . . . ) be given points. The function

k fai*) = yo+-Y[x-*o + \x-Xo\] +

+Vt fo+i-*dK*-*i)+l*--*ill (6.302)

where &{ = Cvi+1—}>i)/(*i+i—*i)> is described as polygonal; its graph is the polygon Α0ΑλΑ2 . . . ^4n. It must be borne in mind that

* (JC —a for x > a. If f{x) is a continuous function in [a, b] , there corresponds to

the subdivision a = x0 < JCX < x2 < · · · < *n ^ * a polygona function whose graph is a polygon inscribed in the graph of f(x), with vertices at the points (xi9 y^), y{ = / ( ^ ) (/ = 0, 1, 2, . . . , n).

Every continuous function is the uniform limit of polygonal functions.

On taking expansions (6.296)-(6.301) into account, we find that any polygonal function, defined in an arbitrary segment [a, b], is the limit of a polynomial sequence, uniformly convergent in [a, b] (Weierstrass's theorem).

Schauders basis. The polygonal function

Faß(x) = j ^ ( l * - a M 2 x - a - j S | + |x- j8 | ) , α < β, (6.303)

is equal to zero outside the interval (a, ß); in (a, β) its graph is an isosceles triangle of height 1 :

Faß(x) = 0, x < a, x > ß;

Foß(x) = 2x-2a, x < Z±L; (6.304)

Faß{x)= - 2 x + 2j8, x>2±L.

Let y = f(x) be a continuous function in (α, β). We introduce the numbers

Schauder*s basis is defined as the sequence of functions in [0, 1]:

1, λ% F01(x), F χ(χ), Fj. (x), F !(*), °· T T'1 °· 7

^L λ^ F± m W . · · · 4' 2 2k' 2k

(6.306)

(/ = 0, 1, . . . , 2 f c - l ; * = 0, 1,2, . . . ) .

Any function/(jc), continuous in the interval [0,1], can be expanded (uniquely) as the series

/ (* )= / (0 ) + [/-(l)-/(0)]* +

+ Σ Ύ<*± izi(f)F± m(x). (6.307) Ä - 0 t - 0 2*' 2* 2*' 2*

The partial sum of the last series

Sn(x)=ÄO) + [fO)-f(0)]x +

+ Σ Ύα± i±i(/) *1 i±i(*) (6·308) Â - 0 i - 0 2*' 2* 2*' 2*

has a graph consisting of a step-function, inscribed in the curve y = /(*), with vertices at the points (z/2n,/(//2n)) (i = 0, 1, . . . , 2n).

The derivatives of the functions of Schauder's basis form a system of Haar functions (see (4.24) and (4.25)).

2°. Sign x (signature x) is

- 1 for x < 0, sign x = <J 0 for x = 0, (6.309)

1 for x > 0

or

sign x = |* | f ° r * * ° ' (6.310) 0 for x = 0

ΟΓ 2 sign x = Urn — arc tan nx. (6.311)

n—> » ft Fourier's expansion is

signx = - - £ L , / , | * | < π . (6.312) π fc« o 2/c + 1

The expansion in Hermitian polynomials (see Chapter IV, § 4, sec. 10) is

sign * = g ( - 1)« - ^ ^ 1 i l tf2n+1(x). (6.313)

The integral form is

» w J o ' sign x = lim ^ ^ Λ (6.314)

The sequence of signs of sine (Rademacher functions) is

pk(t) = sign (sin 2knt), (6.315) rk(t) = sign (sin 2fori). (6.316)

3°. The integral part x (written as [x], entière x, E(x)). If x = n + r, « is an integer, 0 ^ r < 1, then [x] = n.

The following relationships hold:

[x+y]^[x]+\y\; (6.317)

M = i - I 72 is an integer; (6.318)

M + Γ* + 1 1 + . . . + Γ* + - ^ 1 = [nx]. (6.319)

If p and # are mutually prime integers, we have

= J ! + J 2 + . . . + ** + . . . +■ sn9 (6.321) where sk is the number of factors of k.

Fourier's expansion is

1 1 v2! sin 2ττηχ , Λ „ . „ . „Λ Λ Χ x-[x] = -?-- Σ (* - 0, ± 1 , ±2, ±3 , . . .). (6.322)

The function {*} = x— [χ] is called the fractional part of x. It is a periodic function with period equal to unity.

4°. The distance to the nearest integer [denoted by (x)] is

(*) = min {x- [x], 1 + [JC] - x} . (6.323)

The function (x) is periodic with period equal to unity. The Fourier expansion is

1 2 y, cos2rz(n + l)x T " " ^ n è o (2/2+ 1)2 w - 4-4 Σ ^:::». («*>

5°. The jump function l(x) (Heaviside's unit function) is

„ . ί θ for ^ < 0, „ _„_x l(x) = J (6.325) [ 1 for x ^ 0.

The rectangular pulse is

i/ m i/ x f 0 for A; < a, ^ > 3, Q ,. __ .x l ( x - / 0 = l ( * - « ) = <! K' i 3 > a . (6.326) ( 1 for oc ^ x ^ p,

If /(*) is a continuous function in [a, b], the sequence of functions

/»(*) = ΣΛ**)[1(*-**)- l(*-**-i)] (6.327)

tends uniformly to /(x) on indefinite subdivision of [a, b] by the points xfe:

a = x 0 ·< Xi -< x2 < . . . ·< Λ:η = è.

6°. The representation of certain piecewise linear functions with the aid of integrals and series:

i r ^ c o s ^ P ** O ^ a , ( 6 J 2 8 ) π J o I ( 0 for x ^ a.

Γ°° x , ^ r •* x if. f 1 for 0 < x < a, „ „ΛΛΧ aJi(rt)J0(Sx)dS = J (6.329) Jo [ 0 for x > A,

2 Ρχ 2a . ηπ ηπχ > — sm -^- cos — = a £ΖΛ πη

+ 1 for 0 < x < ~ Û,

— 1 for — a < x < a. (6.330)

O oo 2

- Σ ( - l ) n + 1 — sin — = x, 0 ^ x ^ a. (6.331)

2 ^ a . /rex , x _ ,, „„^ — y — s i n — = 1 , 0 ^ x ^ a. (6.332) 0 n«i nn a a

2 £, 2a2 . im . τζπχ — Σ ΤΛ s i n r s i n

a η^! π V 2 Ö

x for 0 ^ ^ ^ —Û.

a—x for -r-a ^ x ^ a. (6.333)

1 - 0 K c o $ _ _ „ - C o s - w

2 s cos -rrdu = ί πκ' 0

1 for |f| =£ 1 - 0 ,

■ i ( l - | i | ) for l - i s | < | S l . (6.334)

0 for |f| s 1.

i: sin2 —

πχ2 cos tx dx = to" I for M ^ 1, for | ; | ÉÊ 1.

(6.335)

_2 sin az cos zxdz = <

1 for 0 ^ x < Û,

y for x = ö,

0 for x > Û.

(6.336)

sin ax dx = < -1 for a < 0, 0 for a = 0, 1 for a > 0.

(6.337)

,. 2 ÇT sin lim — — T - ^ J O t

ht cos (x—a)tdt = I

a—h,

for x = 0—A,

( 0 for jc _1_ 2 1 for a—A < x < fl + A,

— for x = 0 + A,

0 for x > a+A. (6.338)

2. The ô (delta)-function

A sequence of functions {uv(x)} is said to be weakly convergent in the interval (a, b) if, given any continuous function /(*), the following limit exists:

uv(x)dx. lim / (*>„ (6.339)

If the sequence {uv(x)} satisfies the conditions: I. For any M > 0, we have for | a | ^ M, | b | ^ M :

uv(x)dx\ < K(M); 1 a ; .

(6.340)

II. Given fixed a and b, | a | > 0, |6| > 0,

lim n / x ^ f0 for a < b < 0 or 0 < a < *, „ „,1Χ t/v(*) rfx = 1 (6.341) a (1 for Ö < 0 < Ä .

then the limit (6.339) exists, does not depend on the choice of sequence {uv(x)} and is equal to /(0). The limiting element of the (weakly) convergent sequence {uv(x)} is referred to here as a delta function: δ(χ). Thus

ί f(x)ô(x)dx=f(0). (6.342)

Sequences satisfying conditions I and II are described as delta-shaped.

Examples of delta-shaped sequences are:

1 ue(x) = π χ 2 +ε 2 ·, ε > 0, ut(x) -* ò(x)\ (6.343)

i - Ê ut(x) = -—7=· e « , * > 0, rç(x) - δ(χ); (6.344)

1 sin vx uv(x) = - — j A 0 < v < », H„(X) - ί ( χ ) . (6.345)

The elementary properties of the delta function are:

δ ( -χ) = Ò(x); (6.346)

/(χ')δ(χ' - je) = f(x)ô(x' - x) ; (6.347)

χδ(χ) = 0; (6.348)

where xs are the simple roots of the equation φ(χ) = 0;

δ(αχ) = M ; (6.350)

δ ( χ 2 _ α 2 ) = * * - « ) + * * + « ) . (6.351) Ζ | X J

\x\ò(x2) = <5(χ). (6.352)

The convergence of the series expansions and the integrals quoted below is understood as weak convergence.

The Fourier expansion, \x\ ^ 1, is:

c, ,Λ Ι Ι ^ ηπ(χ — χ') . , . _

à(x-x') = -JÌ+J Σ cos — ^ — ~ · (6·353) The expansion in Bsssel functions (see §4, sec. 6), 0 ^ r ^ R, is:

p ^ m ^ n ^ j / m ^ n ^ J

ν-*-ΐΨ ν w x " - / · <6·354> The expansion in Legendre polynomials, \x\ ^ 1, is:

M 2 δ(χ' -χ) = Σ ^ψ-Ρη(χ')ΡηΜ- (6.355)

The expansion in Hermite polynomials is:

n-o <sjn2nn\

Integral forms. Fourier's integral is:

a(x' — *) = -^- cos k(x' — x) dL· 271 J-~ The Fourier-Bessel integral is:

ò(r'-r) = Jï? Jm(kr)Jm(kr') dk,

5(r') = r' [°*kJ0(kr')dk.

The involution is:

ô(t - μ^)ο(ΐ - μ^άί = δ(μχ - μ2).

The derivatives of the delta function are given by:

Γ /(x)<5'(*-*)</* = -/'(A);

f~ f(x)òW(x-h)dx = (-1)V<*>(A).

Series of delta functions are: cosx4-cös2x+ . . . -f cos«x+ . . . =

1 °° = ~ γ + ^ Σ <5(χ-2π«);

sin x+2 sin 2x+ . . . 4- η sin TI* + . . . =

= -πΣ<5'(χ-2π/ζ);

cos χ +4 cos 2x+ . . . +η2 cos nx+ . . . =

= - nf,o"(x-2mi).

(6.356)

(6.357)

(6.358)

(6.359)

(6.360)

(6.361)

(6.362)

(6.363)

(6.364)

(6.365)

§ 4. Elementary special functions

1. Elliptic integrals

DEFINITION. Any integral of the form

Jo y/W) where R (x) is a rational function of x, and P(x) is a polynomial of the third or fourth degree, can be transformed to a sum of integrals, reducible to elementary functions, and to integrals known as elliptic integrals in Legendre's normal form. The latter include the following:

the elliptic integral of the first kind,

fein φ dx

m Ψ) = ,,. Ζλ . . - î (6.366)

the elliptic integral of the second kind,

E{K Ψ) = r n ' Λ / 1Γ Ζ ^ 2 dx; (6.367)

Jo V T ^ the elliptic integral of the third kind,

fsin φ dx n(k9 λ, φ) = . = - . (6.368)

Jo (i+;^2)V(i^2)(i-^2) The notations: F((p, k), Ε(φ, k) and Π(φ, λ, k) are sometimes

employed. The parameter k is called the modulus of the integral, and λ the

parameter of the integral of the third kind, k2 < 1. The number k' = \l\—k2 is called the complementary modulus. The quantity a is often employed, instead of k, where k = sin a. If we put x = sin ψ, the elliptic integrals reduce to the normal

trigonometric form:

Ç9 f* £(fc,<p) = ^l-k2 sin2 \pdy> = Δφαψ, (6.370)

Jo Jo

n(k, κ φ) _ p * r à , J 0 ( 1+λ sin2^) VI — k2 sin2 y J 0 (1 + λsin2rp)A\p

(6.371) where zfy = Vl—&2sin2y>.

The following notation is used for a commonly encountered combination of these integrals:

m 9) = £5*. *>-*<* , φ) __ f * sin2 γ> k*

'in 9 ^2 dx

yJil-X^il-kW)

Representations as series:

(6.372)

F{k, φ) = A0+jA1k*+±^.A2k* + -±^-A3k«+ .... (6.373)

E(k, ψ) = ^-y^-Ii-^-là"^"

- Τ Ϊ 6 Χ ^ 8 - · · - (6·374> where

Λη = I ' sin*»* dz ( « » 0 , 1 , 2 , . . . ) . (6.375) '" = J> The coefficients A n can be evaluated successively in accordance

with the formulae: 2 M 1 1

A0 = <p, An = - ^ - ^ Λ - Ι y c o s ^ s^271"1 <P- (6.376)

If φ is close to π/2, the following formulae are convenient:

F(k, φ) = FÎk,ï\-F (k, <pj. (6.377)

E(k, φ) = EÎk, y ) -E(kt φτ) + k2 sin φ sin φν (6.378)

9?! being given by the equation

S m 9 ? 1 = /T—Î2 2 ' ( 6 ' 3 7 9 )

y ! — Z:2 sm2<p If the upper limit of the integrals is equal to π/2 (φ = π/2)9 the

complete elliptic integrals are obtained. The complete elliptic integral of the first kind is

T ; Jo " κ-*(*ίΙ-Ι f dx !0ν(1-*2)(1-/:2χ2)

The complete elliptic integral of the second kind is

(6.380)

Here, *-*H)-J>*-J!wf c ("S1)

D.D[k,î)~Zj£: (6.382)

Representations as series and products:

* - τ { > * £ [ 2 £ ? ! ! Η <6-383>

Κ = τ Π ( 1 +Λ„); where fcn = j " ^ ' " * * - * , *o = *.

(6.386)


2.0

1.5

1.0

0.5

a=90°8Q°70°60Q

FIG. 5. Elliptic integral of the first kind F(k, φ); £=sina-const. 0.0 V

Λ ■*—

A y

ψ\

6 é

1

1

ra n II \JY/ //A wA

/ / / m m

| · 1

i ' !

1 1

[7Π Y Â ΓΤ

"Γ ! i

l

I -

i |

Τ Π i

Ώ A

0 10 20 30 40 50 60 70 60 90° — + - φ

Fio. 6. Elliptic integral of the second kind E(k,(p); & = sin a »const.

0 10 20 30 60 50 60 70 SO 90°

2.0

1.5

LO

0.5

0.0 E—

if. ΤΓ Ψ !*-£> G0°

v^ ΛΟ

%,

S -5UC

-UOc-

-30'. 3^ -20° !"Ί 1 — ι —

0°\

- 1 -

— —t-_L

—

——

Π

Ì

*Π

6? /0 20 30 40 50 60 70 60 30°

FIG. 7. Elliptic integral of the first kind F(k, φ);φ** const, &=*sin a.

£0:

/5

/ 0

0.5

0.0

¥ **sJ^ _#?<

-Έ3 p·^ OU"

-50°-—t— 40°

— i — 1 !

-20° |

-/03

?σ

—■■

0 /0 20 30 40 50 00 70 80 90°

FIG. 8. Elliptic integral of the second kind E(k, φ); φ«const., k=sin a.

The following formulae are employed when k is close to 1 :

^ ^ ' ( - ' - Ε Γ - έ - ) * · + · · · (6·387)

, 12.32.4 / 2 2 1 \ , + 22.42.6 \ m 1.2 3.4 5.6 J * + - (6.388)

Here, m = In 4/fc'. Some integrals are:

ί J *2 "' (6.390) jfc

Kkdk = E- k'2K = ^ ( K - D); (6.391)

f 1 -t- A-2 7r'2

m <ft = i ± i L j B - ^ - B : . (6.392)

Graphs of the elliptic integrals are shown in Fig. 5, 6, 7 and 8.

2. Integral functions

DEFINITION. The following definite integrals are known as integral functions:

The integral exponential function Γ* e%

ï i ( * ) = | i - *

J o l n <

Ei(x) = | 4- * (* -< °)· (6·393)

The integral logarithm

liW = I τΞτ (0 < x < 1). (6.394)

The integral sine

û(x) = (x)=-j-JZELä (siO = f ) . (6.395)

The integral cosine

ci (x) = - 'COS t dt (x < 0).

In addition, the following notations are used:

Si(*) = j + sKx) = Γ* sin ί , . „., Ν — — Λ | Si(oo) = ΐ·

Ci (x) = ci (x).

When x > 1, the function 1Ì(JC) is defined as

(6.396)

(6.397)

(6.398)

. j r ^ r ■£!■ When jc > 0, the function Ei(*) takes complex values; the real

part of Ei(x) is denoted by Ei(x). The bar above the Ei is occasionally omitted.

Relationships between the integral functions are

Ei (In x) = li (*) (x <

Ei (In x) = li (x) (x >

H (e*) = Ei (x) (x <

Ei (ix) = ci (x) + i si (x) ;

Ei(x ± iO) = ËT(JC) T ni

Si ( -x ) = -S i (x) ;

s i ( - x ) = — si(x) — n\

i);

i);

0);

»

Ci(-;c) = Ci{x) ± m (x>0);

(6.399)

(6.400)

(6.401)

(6.402)

(6.403)

(6.404)

(6.405)

(6.406)

Si(2,) = ^ + £ ( i ^ J i / i . (6.407)

Representations as series are:

Ei (x) = C fin ( - * ) + £ J£T (X< ° ) ; (6,408)

l i (x) = C + ln Χ + f ΤΓΓ (* =** °)' i6·409)

li (*) = C + ln (-In x) + £ - ^ i (0 < x < 1); (6.410)

li (x) = C + lnln jc+ £ - 2 ^ 1 (* > i); (6.411)

s i W = - f H - | i ( - l ) ^ ( 2 f c _ Q 1 . 1 ) ! ; (6.412)

d W = C + l n , + | i ( - l ) ^ . (6.413)

Here, C = 0-57721 56649... is Euler's constant (see p. 310). Approximate formulae (for small values of x) are:

li (x) % — In — (6.414)

Si(;c) * x; (6.415)

si(x) ^ x ~ y ; (6.416)

Ci(x) ^ ΕΪ(ΛΓ) ^ Ei ( -x ) % C+lnx. (6.417)

Asymptotic formulae (for large values of x) are:

(6.418)

(6.419)

s i n W , 2! 4! \ Cl(*) = — ( ^ 1 — + - - · · · ) -

cos x ( 1 ! 3 ! _5! \ sin x x I x x? xb · · · I ^ χ

(6.420)

Ei(x) ~ — ; (6.421)

E( -JC)«4T" · (6·4 2 2)

Some numerical values are:

Ei ( -1) = -0-21938 3934 . . . . (6.423)

E i ( l )= 1-895117816... . (6.424)

li (1-45136 92346 . . . ) = 0. (6.425)

Some limits are:

lim si(x) = - π ; (6.426)

lim ci (x) = ± ni; (6.427)

lim [*esi(*)] = 0 (ρ < 1); (6.428)

Um [xe ci (*)] = 0 (ρ<1) ; (6.429)

lim e-* ΕΤ(χ) = 0; (6.430)

lim xe"* l ì (x) = 1. (6.431) x-»- + ~

Some integrals are: f x 1 _ fi-mx

Ei (—mt) dt = x Ei ( - mx) - — ; (6.432) Jo m

f~ e-vi Ci for) dt = - l i n Λ + ^ . Υ (6.433)

+2.0

HO

too

-LO

-10

V

Y P'f\

ù ifl/vr

V -)/ \

f\

u \

\

JL

—Lij W V<7

-T I v

^+^ S/fxì-

tfux

,? L f q A yL

/

7

Fio. 9. Integral functions: the sine Si(;t), cosine Ci(x), logarithm li (x) and expo

nential functions Ei(—x) and Ei(;c).

100

500

4 )[/

Ivi x\

1000 2000

1000

10.&

FIG. 10. The integral exponential function Ei(x) and the integral logarithm li(;t).

ao: a 000/

Γλ?°'5Λ / 1

°·*noi

nn

Λ'Λ" , νΛ Λ Α

»rsa£t\1nn/\υΛnm\

nnn\UUL\

iInn/

0 I 2 3 U 5 6 7 8 9 I 0 I I

FIG. 11. The integral exponential function —Ei(x).

er& si (qt) dt = arc tan — ; <1 ß,

for oc < ß. (6.438)

The graphs of the integral functions are shown in Fig. 9, 10 and 11.

3. The error function

DEFINITION. The error function is defined as the function

erf x = -?= f* e-t2 dt [erf (~) = 1]. (6.439) ν π J o

It is often defined as

φ(χ) = JÏ Γ e 2 dt (6.440)

or (by S.N. Bernshtein) as

*BW=MY^ *=* * w = T e r f ( ^ ) ■ <s-44l) The following notations are also encountered:

erfc x = 1 - erf x = 4 = Γ *"'* * . (6-442) V** J*

LOc) = Γ *-<* dt = ^ erfc *. (6.443)

Relationships between the functions erf x and ç>(x) are:

Φ(χ) = erf f - ^ Λ ,(6.444)

erfx = 0(xj2). (6.445)

The derivatives of the error function are:

- j - (erf x) = - = e~** ; (6.446)

The integral of the error function is:

f erf x </x = x erf * + — e~xt + C. (6.448)


Other integral forms are:

erfx = -?= τ ά ;

LO

0.9

Οδ

07

0.6

0.5

ΟΛ

0.3

0.2

0.1

tn{xy) = 2y_Cx

V^Jo e~v2t2dt;

«^-Μπ*

1.5 2.0 2.5

FIG. 12. The error function erf x * (2/^/π) e-^ dt.

The representation as a series is

erf x

(6.449)

(6.450)

(6.451)

l

3.0 ■x

2 - x**-1

φ£ι(~ 1)k+1 (2*-!)(*-!)! " ^ ά0 (2Λ + 1)!! ' 2 . ^ 2f t^+i

= e-xt Y

Asymptotic formulae are:

rik+i, erf(V ) = I - -«"* Σ (- Dfc V x . „. π k-o ft+4 π

1

(6.452)

+— R,,, (6.453)

where

Ή) | R | < ^ /—, x = xé* and φ2 < π2; (6.454)

he 2 cos ~

g-x* Γ i 13 13 5 Ί eTicx^^[l-^+Wf-li^F+'··]· (6<455)

The graph of the function erf x is shown is Fig. 12, which also gives the graphs of tanh x and 1— e* for comparison.

4. Fresnel integrals

DEFINITION. The following functions are called Fresnel integrals: the Fresnel sine integral,

S(x) = G | sin*2* S ( + o o ) = i , (6.456)

the Fresnel cosine integral,

C(x) = / i . cos ί2 <ft C(+ oo) = y . (6.457)

The following definitions of the Fresnel integrals are also occasionally encountered:

S*(x) = Γ sin ί2 Λ = 5 ( 5 * j , (6.458)

C*(*) = Γ cos ~ ί2Λ = C ( / | Λ (6.459)


. , Λ 1 Çxt sin t S{x) = —=z\ —=rdt; ν 2 π . / ο \M

_. , 1 f*1 cos f , C(x) = - = — = ^ Λ,

(6.460)

(6.461)


S(xy) = -jL f* sin (j?t2) dt; (6.462) 2y rx

Ç{xy) = -^L f * cos {ft2) dt. (6.463)

The representations as series are:

The asymptotic formulae are:

-=r—p=- cos^ + O —; 2 V^x \ *2 5(x) = ±—^— cos x* + 0 ( -^- ) ( * - - ) ; (6.466)

c(x) = - j + T ^ " s in χ2+0 (le) (* - "^ ( 6 , 4 6 7 )

The connections between the Fresnel integrals, Bessel functions and the error function are:

* ) = τΓ4 ( ί ) Λ ; ( 6 · 4 6 8 )

C(*) = 1 Γ 7 - 1 ( 0 Λ ; (6.469)

C(2) + /5(z) = Jl vf^yjljydt; (6.470)

C(z) - iS(z) = -1= erf (zJl)=M Γ «-«»A. (6-471)

Some limits are:

lim S(x) = lim C(x) = j - , (6.472) χ - > + oo Ä->- + oo

J i m {Χ·ΓΑ(*) - i " | j = J j n ^ |*e|"c(*) - y ] | = 0 (ρ * 1). (6.473)

(6.464)

(6.465)


1.0

0.9

ae

OJ\

0.6

0.5\

OA

0.3

0.2

01

° 10 20 30 UO

FIG. 13. The Fresnel integrals S(x) and C(x), x2=*z.

Some integrals are:

l

50

S(ocx) dx = pS(ocp) + cos (a2/?2) — 1

I o «>/2π

C(ocx) dx = pC(xp) 7 = ^ ; o oc \J2TI

r°° r i Ί v ~ — S(x) sin 2px dx =

Jo L 2 J π

^ rr· π . p2

2 V 2 cos — sin ~ -o 2

j ;[ i-cW]s i sin 2/»x dx = — 2 y/ 2 sin -5- sin-^r-

o 2

(6.474)

(6.475)

(P > 0);

(6.476)

(P > 0).

(6.477) The graphs of the Fresnel integrals are shown in Fig. 13.


5. Gamma and beta functions of Euler

The so-called Euler B- and Γ-functions are important non-elementary functions. They are defined by improper integrals dependent on a parameter; they have been carefully studied, and fairly detailed tables have been compiled for them.

The fundamental properties of the functions are described below. All the parameters appearing in the formulae are real. The operations employed, of diflFerentiation, integration, change of the order of integrations, summation, etc., are valid, inasmuch as the integrals are absolutely and uniformly convergent for the values of the parameters considered.

1°. Definitions, functional equations and elementary properties of the Γ (gamma)-function of Euler. For a >- 0, Euler's gamma function is defined by the integral

Γ(α)= *«-i*-*dc. (6.478)

Change of the variables leads to the formulae: j;

s te Γ(α)

= Γ An l Y " 1 dx; (6.479)

Γ(«) = Γ~ e*er** dx. (6.480)

Integration by parts of (6.478) gives Γ(χ) = (a - 1) Γ(α - 1) (6.481)

and in general, for integral n and a ==- 0, we have Γ(«) = ( α - 1) (a-2) . . . (α-/! + 1)Γ(α-η+1). (6.482)

On putting a = n (integer ) in (6.482), we get Γ(«) = («-1)! (6.483)

Continuation of the gamma function into the negative semi-axis is accomplished in accordance with the formula

Γ(α-1) = ^ ^ - (6.484) OC— 1

Γ(χ) = Γ ( * + 1 ) (6.485)

first into the interval (— 1, 0), then into ( — 2, — 1), so that, for — « < < x < — (Λ—1),

Γ(χ) = —, J{X\n) TV* (6-486) v ' x(x + l)...(x+n-l) v '

whence, for 0 < a < 1,

Γ(α - n) = ( - l ) n Ti v o Γ ( ^ , T . (6.487)

(1 —a)(2 — a) . . . (n — a)

The function /fa-f-1) is often denoted by the symbol Π(χ). The gamma function is discontinuous for all integral negative

values of the argument, since it follows from (6.485) that

lim | χ^Γ(χ +1) | = co. (6.488) x-* - n x > — n

It follows from (6.478) that, at points where the gamma function is continuous, its derivatives may be evaluated from the formulae

Γ'(α) = x«- 1 *-* In x dxy

f(ft)(a) = Γ χ α - ν - * On *)*</*. (6.489)

REMARK. An equation of the form

F(x9 y(x)9 y'(x), . . . y™(x)) = 0 where F is a polynomial in its arguments x, y(x), y'(x)9 . . . , y{n)(x), is called an algebraic differential equation of the nth order. A function y(x) is described as hypertranscendental if it is not a solution of any algebraic differential equation with polynomial coefficients. The function Γ(χ) is hypertranscendental.

For instance, y(x) = sin x is a solution of the algebraic equation y'2+y2 = 1, so that it is not a hypertrascendental function.

An elementary asymptotic formula. It follows from the inequality with 0 < a < 1)

ηαΓ(η)-β~ηηα+η-ι < Γ(α + «) =

= P + f~ < Λβ-ΐΓ(/ι + 1) + β-η/ ιβ + η-1 (6.490) Jo Jn

that

. S . ^ - ' · <M"> The representation of the gamma function as a series is mentioned

in Chapter III, § 3. sec. 16. Expressions for Γ(μ) and Γ~\<χ) as infinite products. If we use

(6.491) and replace Γ(<ζ + η) in accordance with (6.482), we arrive at the Euler-Gauss formula

Γ(α) = lim na —, ^ - — ^ 7 r , (6.492) w n-^oo χ(α+1)(α + 2 ) . . . ( α + «)

or J » = lim ?7α . (6.493)

n"~ Π(«+»)

We can obtain from these last formulae, on making use of the transformation

& ^iL'n/i+i^ n\n« na JJ·^ v)

-α 1 η η+ Σ 7 n = e »»«1

·>*=!

a \ «Π I + T F

the Weierstrass formula

L=n.)^n((u i )^) , (6.494) A'

where C is the Euler-Mascheroni constant (6.56). The formula for the complement

Γ(α)Γ(1-α) = - ^ (6.495) sin ττα

follows from (6.493) and the formula

sinrca πα

-lim Π(ΐ-ί)-


In particular, it follows from (6.495) that

r ( i ) = JZ. (6.496)

Since we have, from the formula for the complement,

. .. . 2π . « - I ' sin — sin — . . . sin π n n (6.497)

we obtain, in view of the relationship Tr1 · kn n 11 s i n — = 2 ^ ΐ

theEuler product:

The multiplication formula fe"x / v \ — -& +— Π Γ α + τ = (2π) 2 fc 2 T(/ca), a>0 , k is an integers 1, *«o \ KJ ■ (6.499) follows from (6.493). When k = 2, we get the (Legendre) doubling formula

r(a)rL + Ì Ì = V2^2"2 o +^r(2a). (6-500)

TAe logarithm of the gamma function. On taking logarithms of (6.493) and passing to the integrals, we arrive at the integral form of the logarithm of the gamma function:

1ηαΓ(α) = Y Λη — a l n — ^ ì = η-ι\ 7 ΐ + α " + 1 /

oo Γ0 / enx^ g(n+a)x enx_e(n+l)x \ = n?J..( 5

" Γ—)*-

1ηΓ(«)= P f f ! ! ^ _ ( a - i ) e * 1 Ì 2 L . (6.501)



1.W = Jt"jV ■)*-+ " +'l'-1\; ' ) " ] 4 , «>0, (6.502)

InA«)= Γ Γ - ^ - Χ « - » ] - ^ - , «»0.(6.503)

Raabe's formula. On taking logarithms of the Euler product (6.498), we get

k

and if £1 \n ) n \2 2nJ 2n

k 1 — = a — = aa, w — a «

we have fi

In Γ(α) da = In ^/H. (6.504)

The itaróe integral is obtained by differentiation with respect to the parameter:

/ = ΙηΓ(οι + χ)αχ = α(1ηα-/+1η 72π). (6.505)

Another expression for the Raabe integral is obtained by integration of (6.501):

It follows from (6.501), (6.506) that

1 η Γ ( α ) - / + ^ = f° / ( χ > α χ ^ (6.507)

where

On introducing the Binet function

ω(α) = f° f(x)e«xdx, (6.508) j>:

we get an asymptotic expression for the logarithm of the gamma-function:

In Γ(α) = In V ^ + ( * — i ) In α-α+ω(α). (6-509)

It also follows from (6.501) and (6.506) that

/ - I n Γ foe + λ \ = f° F(x)e«* dx% (6.510)

where

From this we obtain an asymptotic formula for -Γ( α + —-1:

In Γ ( a + ~ ) = a ( Ina-1) + In y/to-ϋφ), (6.511)

where

ωι(α) = f° .Ρ(Λ:)^ dx. (6.512)

The following relationship holds:

ωχ(α) = ω(α)-ω(2α); (6.513)

it is obtained by taking logarithms of the doubling formula (6.500) and substituting for In Γ(<χ) and In -Γ(2α) in accordance with (6.509)

and for In -Γ(α + -— ) in accordance with (6.511).

The following formulae are due to Schaar:

eaxdx ω ( α ) = ! f{x)**dx = ^ \ ^ Ak2n2

1 f-~ ocdx aP + x2 ln(l-é?2**). (6.514)

ω π J 0 a2 -f x2 π

Λ-°° 2a dx , „ . 2 t 2 l n ( l - e 2 * * ) = 4a2-fx2 y

_ ^ l n ( l + e 2 * * ) . a2 + xz

(6.515)

The StirUngand Gauss asymptotic formulae. By using (6.179) and (6.508), we can obtain the Stirling series for the Binet function:

B2 1 B5 1 ω(α) = z - « 5

- + 1.2 a ' 3.4 a3 f . ..

Ä 2n-2

where

Ä = Θ Ä 272

(2n- l ) (2*-2)a*»-*

1 (2n-l)2n a2»-*' 0 < 0 < 1

— + *, (6.516)

(6.517)

When ?7 = 1,

^)=9έ=Ε and Stirling's formula follows from (6.509):

1ηΓ(«) = In φπ + iœ-- j In α - α + — , (6.518)

(6.519) Γ(α) = 72π« 2« 1 2 α ·

Stirling's formula is again obtained when a is equal to an integer m:

(6.520) m\ = V^wf-y ) *12m·

An asymptotic formula for ωχ(α) follows from (6.513) and (6.516):

B2 (. 1 \ 1 S4

In particular,

l~i)i-+ B, 5τ(ι-Ίη^+··'·^21)

. Λ fl B2 Λ 1 \ 1 Θ

«1(«) = ö_^i - T J- = —, Gauss's formula follows from (6.522):

0 < 0 < 1. (6.522)

1ηΓ( α + 1 J = a (Ina- 1)+ In V ^ - ^ ·

1

(6.523)

When n is an integer and a = « +—, a formula is obtained for a

factorial that gives a better approximation than Stirling's formula:

n\ — -J2jz ( 1 \ » + T

\ 24JI+12 (6.524)

The power expansion of the logarithm of a gamma function. It follows from (6.481) and (6.493) that 1ο§Γ(1+α) =

"JiL «M*2 " ^ ^ *· </a =

= — Ca + 2 a 2 _ A a 3 + ^ a + e e > j (6.525) 4

where 1 1

SP = 1+-^r+ · · · + — + 2P 72


The trigonometric expansion of the logarithm of the gamma function (Kummer*s series). If, when χξ, (0, 1),

In Γ(οί) = -^- + Σ (αη c o s 2nna + bn sin 2ηπχ\ (6.526) 2 η=ι

where

Γ (JC) cos 2ηπχ dx (n = 0, 1, 2, . . . ) ,

) sin 2ηπχ dx (n = 1, 2, 3, . . .),

an = 2 In

fcn = 2 PlnTtf

we have by the formula for the complement: oc

In Γ(χ) + In Γ(1 - a ) = a0 + £ 2an cos 2κπα =

> (6.527)

whence

, Λ , i Λ ^ cos Iwix ,, „ „ = 1η2π— Ιηβιηπχ = 1η2π + V , (6.528)

n- i «

j a0 = In VSÏ, an = JL (« = 1, 2, . . . ) . (6.529)

We have from (6.502) and (6.64):

Z>„ = 2 o^Jo( '^^- X + 1 ) S m 2 r o r ^ =

+ 1

-\ im

' (e-u _ g-2n«u) J*l _. JL ( c + i n 2ηπ). (6.530)

Further expansions of the logarithm of the gamma function are

l n r ( l + a ) = l - ^ - C a - ^ « 3 - Ì 5 - a S - . . . , ( 6 . 5 3 1 ) 2 smrca J j

i r . / , N \ nx 1 1 + a 1ηΓ(1 f a ) = — - -T l n - l + 2 sin πα 2 1 - a

+ ( l - C ) a - ( j e - l ) ^ ~ ( 5 5 - l ) - y ~ . . . . (6.532)

The logarithmic derivatives of the gamma function (the psi or digamma function). By definition,

, Λ Γ'(α) d ί _ , , ^(α) = Τ ( ά ) = ^ 1 η Γ ( α ) ;

ir// N /"(a-fi) d , π / , .

so that

!F(a) = tfaO+i.

x )=-c + lo(^T-^)·

(6.533)

The function ψ(α) satisfies the functional equations :

ψ(1+χ)-ψ(α) = 1 . (6.534) a

y(a) —φ(1—a) = — jrtanTra. (6.535)

y(a)-y(—a) = —Trtanrca . (6.536)

V<l+«)-V(l-a) = - TrtanrcaH—. (6.537) a

^ί-r-4-a ) ~ v ( y - a j = ntennoc. (6.538)

y(ma) = m-1 £ y ( a + — + In m. (6.539)


^ _ ™ _ - J . _ C + £ ( ± - - L . Y (6.5*)) Γ(α) α ψ-ι\ν v+x J

or, in View of the fact that

we have

* « ) » - C + £ ( — ^ - - - i — ) . (6.

φ(α) = -C+jtifrdt- Γ tv+a-xdt\ = Λΐ i __ t«-i

= - C + J i-t dL <6 ·5 4 2) On carrying out the change of variable 1 + j = t~\ and taking

account of (6.63), Cauchy*s formula is obtained:

« = £[«-»-y(a) =


φ) = -C +

y(<

1 1*1 0+>>)aJ 7 '

a > 0.

—I——r- dt, a > 0; 1 —e l

tat

a =~ 0;

(6.543)

(6.544)

(6.545)

(6.546)

(6.547)

H; dt ¥>(*)=! K l + O ^ - O + O - " ] — - C , « > 0 . (6.548)

Some particular values of the gamma function and its derivatives are:

/ (1 ) = Γ(2) = 1;

/χο·5) = v»; Γ(-0·5) = -2yß;

Γ ( ' 7 + Τ ) = # ( 2 Λ - 1 ) ! ! ;

(6.549)

(6.550)

(6.551)

(6.552)

^ T - » j = (-i)"(2„_I)!r ψ(1) = - C;

(6.553)

(6.554)

V>(j)= -C-21n2:

^(i:)=-c-f-31n2;

V ( T ) - -C + f -3In2; π2

v'O) = -y;

*(4)-ΐ-

373

(6.555)

(6.556)

(6.557)

(6.558)

(6.559)

The-graphs of Γ(χ), Π(χ), 1/Γ(χ) and 1/Π(χ) are shown in Fig. 14 and 15.

75 -U -3 -2 -/

"

'.

4ΊjΛ

m~X + / 0\ 1 2 3 4

.-J - 4 - 3 -2 - / 0 I 2 3

^-x

FIG. 14. The gamma functions Γ(χ) and Γ(χ+1) = Π(χ\


13


.-3 -2 - /

ΠΩ U.O

π7

+ππ\•LU. L/fc

PA\

-.,o\ M l

0 / 2 J

1 I ' 1 1 1 1 ί 1 1 1 1 1 1 1 1

à W

"-4 - 3 - 2 - / 0 1 2 J ■ 4

FIG. 15. The functions ΐ/Γ(χ) and \jll(x).

Properties uniquely defining the gamma function. A function F(x)9 continuous along with its derivative for x > 0, is a gamma function Γ(χ) if it satisfies any one of the groups of conditions enumerated below:

I. (a) F(x+1) = xF(x),

(b) F(x)F(l-x) π suurjc

\X+2)~ 2»-i (c) F(x)F[

Π. (a) F(x+1) = JCF(X),

F(2x).

(c) F(x) ?± 0 for x =- 0.

ΠΙ. (a) f(l) = 1, (b) F(x + l) = xF(x), (c) f(x) is a logarithmically convex function for x =► 0 (see

Chapter I, § 3, sec. 17).

IV. (a) F(l) = 1, (b) ^(χ+1) = χ/·(χ), (c) (e/x)xF(x) is decreasing for x >- 0 (or x =► Af).

2°. Definitions, functional equations and elementary properties of the B (beta) function of Euler. Euler9s beta function is defined by the integral

B (α, ß) = p * « - i ( l - * y - i i f e , a > 0, ß > 0. (6.560)

Integration by parts leads to the functional relationships

B ("> ß β T ^ - T * (*> ^ - *)' 0 ^ 1> (6·561> α-t-p — ι

5 <α' Ä = Ι ^ 1 ι Ä ( « - 1, ft, a > 1. (6.562) α+ρ— 1

The -5-function is symmetrical:

Β(?9β) = Β(β,*). (6.563)

The connection between the binomial coefficients and their generalization. Since, when m and n are positive integers,

(; i- l)!(m-l)!

we have

m5(

whence, when a > 0 , 0>O,

1 _/m + n-l\ m, n) ~ I m — 1 y ' 1 fm + n-l\

(6.565)

(Î7') = râ· τι expression in terms of the gamma function. On replacing JC in

(6.560) by y/(I +y\ we arrive at the formula

5 ^ ) = | ~ 7 Π ^ ' ( 6 · 5 6 8 )

(6.564)

(6.566)

(6.567)

whence to° r(oc+ß)ya-idy =

0 0 + j ) a + '

g-«(i+y)^o+^-iyi-i dx dy = o Jo

= Γ e-*xß-i dx f°° e-*u(xyy*-id(xy)9 (6.569) Jo Jo

and finally,

Β(χ,β) = Γ(*)Γ(β)


S(*,j8) = 2 ^ / 2

δίη2α-ΐφ cos 2 ^ _ 1 c)^;

Λπ/2 1 1 £(α,β) = 2 sin2a * " Τ ' ^15 ν

Λοο ^α— 1 Γ«> £2α—1 Β (α, /3) = - - — ^ Λ = 2 ~j ÖT-TÄ ώ,

Jo (1+0 α + * Jo (l-t- i2)r t+"

*(«.Ä Jo d + 0 " + ' (ß+l

dt, α ^ 0, /? > 0;

B (α, α) = 1

22α-1 0 V7

Representations as a series and infinite product:

~ fc(« + /g + Â:) *<*.«= H («+*)os+fc)·

(6.570)

(6.571)

(6.572)

(6.573)

(6.574)

(6.575)

(6.576)

(6.577)

6. Bessel functions

The Bessel functions of the first kind Jn(x) are defined as the coefficients of the power expansion in the variable t of the generating

function e2V lJ :

ht-rl)x 7< -™*-1

e* = e2 e 2 =

= Σ Σ ( J+..,., = Σ W . (6-578) eo (_ iywi+2s

w - . S 2 ^ + J ) l j r — * - + - . (6·579) ^-nW = ( - DVn(x). (6.580

The Bessel functions of an imaginary argument In(x) are defined from the relationship

e\(t+t-i)x= ~ / η ( χ ) ί η ; ( 6 5 8 1 )

n = —oo

«> rtt+2·

«*> = 8Σ ^ , , + , ) , , , . (6.582)

Jn(x) = i-n/n(ix), (6.583)

LnW = iV_n(&) = i"(- l)Vn(fr) = /-Vn(/x) = 7n(x). (6.584)

Tfte trigonometric forms of the generating functions. It follows from (6.578) with / = ei) = /0(x)-j-2 Y 72/i(*) COS 2&Φ, (6.586) Ä«=l

oo sin (x sin <ρ) = 2 £ ΛΑ+Ι(*) s i n (2fc+ 1)φ, (6.587) fe=0

and similarly,

e<*cos9 = /0(*) + 2 £ /·/,(*) cos JÇ>, (6.588) 8 * 1

oo

cos (x COS φ) = / 0 ( * Η 2 Σ (-l)fe«/2ft(*)cos2fop, (6.589)

sin (x cos φ) = 2 £ ( - l)fe*Wi(*) cos (2fc + 1)φ. (6.590) ft

BesseVs integral. On regarding (6.586), (6.587), (6.589), (6.590) as Fourier expansions, we get

f* / · \ j f πΛι(*) f o r n — 2k, n = 0, , , _π1λ cos «99 cos (x sin φ) dp = ^ nx (6.591) Jo [ 0 for n = 2fc+l. f* . . / . w ί ° for n = 0f n, =2fc, , , __„

sin 72<p sm (x sin φ)αφ = 1 (6.592) Jo

{nJn(x) for n = 2*+l .

It follows from these formulae that

1 f * /n(;c) = — cos (ηφ — χ sin <p) dtp (Bessel) (6.593) 7 r J o

where n is zero or a positive integer. Addition theorem. Since

Σ Λ(" + ν)' = * = e e =

= Σ Λ(")«8 Σ W r ,

we have

4»(« + ») = Σ W«- . («0 , (6.594) S - i — β ο

or

Λι(« + «0 = Σ Λ(«ΟΛι-·(«0 +

+ Σ ( - l)s {/.(«)/»+.(») + /„+.(«)/.(«)}. (6.595)


Similarly,

379

s«o

+ Σ {hWnUv) + W«)/.(iO}· (6.596)

FIG. 16.

TTie addition theorem in Neumann's form. It follows from Fig. 16 that

R = yfrZ + Qp — lrQ cosy),

Rcos(cp+oc) = t cosa-f-ρ cos /? = r cosa -f-ρ cos (π —ψ-τα) = = r cos a + ρ cos (a — φ),

whence

Σ inJn(R)ein,peina = Σ imJm{r)ëma X ( - 1)ψ,/Ί(ρ)β«*β*ία =

= Σ ^ ί Λ β Σ ( - l^ feVn-^)«- 4 1 » , (6.597) n [——«

and

/ n ( i ? y ^ = Σ (-VlJi(Q)Jn-i(r)e-il*, I - -» .

/n(Ä) cos πφ = £ ( - l)lJi(Q)Jn-i(r) cos fy,

/»(Ä) sin n9> = X ( - ìy+Vjfe)/«.^) sin Ιψ. ι

(6.598)

(6.599)

(6.600)

When n = 0, J0(R) = Ja(R)Jo(Q)Ji(r) cos Ixp (6.601)

When r = ρ, R = 2ρ sin —- y = z sin 0,

J0(z sin 0) = Jg ( i A + 2 £ /? ^ cos 2/Θ. (6.602)

Bessel's differential equation, which is satisfied by the Bessel functions Jn(x), is

X2 ζζ. + * * L + (X2 _ = 0 (6 603)

Correspondingly, the In(x) satisfy the equation

*2 Έ?+* ί* ~(jc2+"2)>; = °- (6,604)

Recurrence relations are:

/ n + iW = - 4 W - (6.605)

/n-i(*) = ~ Λι (*) + TO ; (6.606)

TO = y [Λι-ι (*) - Λι+iWl ; (6.607)

2/2 Λι+ι W + Λι-ι(*) = — TO ; (6.608)

^ (xVn) = x» Jn_lt ^ (*-«/„) = - * -V n + 1 . (6.609)

The Bessel functions of the first kind Jn(x), n not an integer,

~ C—IV / x \ n + 2 s

W - £ A . + I ) W + . + I ) ( T ) <6·6 1 0> satisfy the differential equation (6.603).

A second linearly independent solution of this equation is provided by the function J_n{x) (n not an integer), so that the general solution has the form

y= Ci^i(*) + C2/_n(x),

where Cx and C2 are arbitrary constants. The recurrence formulae (6.605)-(6.609) also hold for Jn(x), where n is not an integer.

Bessel functions of the first kind J x (x)9 where n is an integer,

are expressible in terms of elementary functions; in particular,

T W - ° Γ ( * + Ι ) / Υ - Ι - Μ + Λ 1 2

- {Ξ-\ϊ fi V r - 1 ^ χ28 - fi* smx - fi- -V2/ \^Ji>( } (2^-hl)! "" V ^ x V * * S m * ;

(6.611)

/ . i w = J E c o s x - (6*6i2)

Expressions for / x (x), n any integer, can be obtained by successive η+Ί

applications of the recurrence formulae. , Bessel functions of the second kind Yn(x) (Weber functions) are

expressible in terms of Bessel functions of the first kind with the aid of the equations :

v , Λ v Jv(x)cosvn-J_p(x) r ,ccA*>\ YJx) = hm v : ^-^ for n an integer (6.613) v-+n sinwr o v /

, . , N Jn(x) cos ηπ— JJx) „ . Yn(x) = - ^ — : =2i-i for n not an integer. (6.614) sinmr v

Like /n(*), the function Yn(x) satisfies the differential equation (6.603), the general solution of which has the form, when n is an integer, y = ^ / η (χ ) + <72Γη(*).

The Macdonald function is usually taken as the second solution of equation (6.604):

„ , , π Ι_η(χ) — Ιη(χ) Γ . A Kn{x) = -r- . — ^ ^ for w not an integer. 2 sin wr (6.615)

The recurrence formulae are:

W * ) = 7 *"»(*) - Yn(x) : (6-616)

r»-i(*) = 7 y»W+nW; (6.617)

r; (*) = i [Γη_χ (x) - rn+1 (*)] ; (6.6I8)

r»+i(x)+y.-i <*) = Y r»(*); (6.619)

i , « W - U * ) = -^/»(*); (6.620)

**+i(*) - *»-i(*) = ^ *„(*)· (6.621)

Relationships between Bessel functions of the first and second kinds are:

4 .WW*) - Yr, W/«+i(x) = - J- ; (6.622)

/ n ( Ä « W + Kn(x)In+1(x) = 1 . (6.623)

Representations as series are:

Jo(X) = i-(fJ+(è(f)4-w(T)e + ··· : ( 6 · 6 2 4 )

,lW = £-^*J+^L(£j__^^J+ ... ; (6.625) - » J Î W ( T J £ * · (6·626) _2

π \ " "" 2 7 0 ( * ) = £ C + ln ·=-/„(*)


-Lli^(iT+\\U+ai)-' (6·627) /oW - l + ( | ) 2 + ( ^ ( f ) 4 + ( 3 V 2 ( f J + ··· ; (6-628)

m = y + 5r(j)3 + 2!T!(f )S + 3!Vi(f)7+ · · · ; ( 6 · 6 2 9 )

JWx) - - ( C + I n | J / o W + n | ^ - 2 ( f ) 2 \ Σ j ; (6.630)

" Ι . Τ Π Ϊ Γ Π 5 Γ ( Τ Γ 1 ( Ι Τ - Ϊ Γ 3 ) Ϊ (6·631)

where C is Euler's constant (see § 1, sec. 1, 3°). Asymptotic formulae (for large values of x) are:

Jo(x) * J J j MOW sin U + j j + δο(*) cos ί x +j)J ; (6.632)

Λ(*) * ^ [ w s i n ( x - j j + ßi(x) cos ( * - T ) ] ; (6-633)

Y0(x) * J^ I - Λ)(*) cos I x+j J+δο(*) sin ί x+-J J J ; (6.634).

Ux)s* Jèil~Piix)C0S{x~T)+Qiix)nn\x~j)y (6·635)

where _R32_ 12.32.52.72

o W " 2!(8x)2 + 4!(8x)* Λ / s 1 12·32·52

ôoW = -TTÖZ + -USx 3!(8JC)3 ' · " 12.3.5 12.32.52.7.9

lW * + 2!(8JC)2 4!(8X)4 + , -

1.3 12.32.5.7 Ql{X)~TÎSx 3!(8x)3 + - * · ;

■ /„ ( * ) * ^2πχ

^/2πχ

1 + 12.32 12.32.52

+ .... ,. + I! 8x ' 2!(8x)2 ' 3!(8x)3

1.3 12.3.5 12.32.5.7

8λ- 2!(8x)2 3!(8x)3

ΰ I

0.5

.y*<Jo(x)

Λ \

' \ \

(6.636)

(6.637)

/ 2 y &\ 5/ e /?■ a \ /o\///2 &' k

-0.5 \

FIG. 17. Bessel functions of the first kind /„(*) and Jx{x).

ηη\^

Λ,ΛΙαΊ-

H

-U. J

I

-n5\ , . ,IO 15 20

FIG. 18. Bessel functions of the first kind/2(x), /3W,/8(^),/16(jc).

HO

FIG. 19. Bessel functions of the second kind Y0(x) and YJx).

/ H -2

-3

- 4

-3 -2 -fa I 2 3 'x

y

1.5

■1.0

0.5

0

1

\ \U~K,M

/ _

2 ; Fio. 20. Bessel functions of the

first kind Iu(x) and /,(*). Fio. 21. Bessel functions of the second kind Κ„(χ) and £,(*).

ττ< ^ -* Ιπ Λ I2 l2-32 12.32.53 \ U*)~e ^ _ ^ l - _ + ^ _ _ _ _ + . . . j ; (6.638)

1 W \J2x\ 8x 2\(Hx?^ 3! (8A:)3 " V " (6.639)

Graphs of Bessel functions are shown in Fig. 17, 18, 19, 20 and 21.

NOMENCLATURE

(a b) — The interval a ·< x b [a, b] — The segment a&x^b (a,b] — Tne semi-interval a^x^b [a,b) - The semi-interval asx<b (— ~,ß] - The infinite semi-interval xna [by + oo) — The infinite semi-interval x^b {xn} — The set of elements xn x € X — The element * belongs to the set Γ x~ZX οτ x i X — The element x does not belong to the set X XcY - The set Ar is a subset of the set Y ΧΞ Y or X<t Y - The set * is not a subset of the set Y Au B or A+B — The union (sum) of sets A and 5 /4 n By or 4 xi?, or A · i? or Λ.Ο— The intersection (product) of sets A and B B\A or B-A - The complement of the set A with respect to the set B x — The strict upper bound of the set X

XÇ.X sup/(jc) — The strict upper bound of the function / o n the set X

XÇ.X inf x - The strict lower bound of the set X

xiX inf/(x) - The strict lower bound of the function / o n the set X

xex max {αλί a2, . . . , an} — The greatest of the numbers alf a2> . . . , an

n min {alt a2, . . . , an} - The least of the numbers ai9 a2, . . . , an

n anj «= [aXy a2y . . . , ani . . . } - The sequence with the general term an anmj — A double sequence

o(a») — a* is an infinitesimal of lower order with respect to βη if βη**0(αη) 0(an) - an has a rate of decrease not faster than βη o(x)y 0(x) - Orders of the function x

lim f(x) - The limit of the function f(x) as x - x0, x € X

Ei — One-dimensional coordinate space (numerical axis) En — «-dimensional coordinate space Ek+En_k - The direct sum of manifolds ρ(Χ,Υ) - A distance

386

NOMENCLATURE 387

θ (0, 0, . . . , 0) - The origin of coordinates f(X)~f(xi, x2, . . . , xn) — A linear function, or a function of a vector (point) j | X11 - The norm of the vector X ^ΣλΣι... x„ — Gram's determinant for vectors

Ln — /i-dimensional linear system X _L Ek - The vector X of En is orthogonal to Ek pTSkX — The projection of the vector X on to Ek prUQX— The projection of the vector X on to the vector UQ

lim fiX) - The limit of the function f(X), when XtM tends to Λ

Q4 — The exterior domain Q(- The interior domain ^ - (Λ> /2> · · ·> / , . ) ~An operator K=* F(J0 - The vector form of writing an operator y{ =« ftiX) = ftixu x29 . . . , jcj - The coordinate form of writing· an operator y{ =■ ailxl+at2x»+ . . . +ainxn (/ - 1, 2, . . . , n) — The coordinate form of

writing a linear operator £"ηφ — The space with norm q>iX) EHV =*Ε* — The space Εηψ is conjugate to the space ΕΛφ Pm

β P\Pt- · · Pm ~ The w t n partial product {/>„,} - The sequence of partial products

oo

m - Pm+iPm+2 · · · - f][ Λ - The remainder product

oo

Y % - A double series

Λ — The radius of convergence of a power series rjx) - The remainder term of the Taylor series

oo

Fix) ~ Y Ai/x* — The asymptotic expansion of the function Fix)

if S) - I f(x)g(x) daix) - The scalar (inner) product of the functions /(*) -i: i a and #(*)

11/H -The norm of a function

1/-'5

if—g)2 daix) - The root square deviation of functions / and g i a

rn fix) - The /zth-order inverse derivative of fix) ax a* an ό0+-— — — —A non-terminating continued fraction bx +b2 + . . . +bn + . . .

<P,t/ßn — The /7th convergent of the fraction K, K — The ordinary and singular valuss of the same continued fraction C — The Euler-Mascheroni constant G - Catalan's constant

388 NOMENCLATURE

n\ — Factorial n n\\ - Double factorial n ônm or δ™ Kronecker delta. C) or Q - The binomial coefficients Pn(x), Pn(x), Pn(x) - Arbitrary orthogonal polynomials Bn — Bernoulli numbers 5„W — Bernoulli polynomials <pn(x) = Bn(x)-Bn

BJJ) — Bernshtein polynomials L*(x), La

n(x). Lan(x) — Laguerre polynomials

LH(x), Ln(x), Ln(x), ln — Legendre polynomials Tn(x), Tn(x), fn(x) - Chebyshev polynomials of the first kind Un(x)> Un(x\ Un(x) — Chebyshev polynomials of the second kind Λ, Αχ) — Chebyshev polynomials with respect to a system of points En — Euler numbers En(x) — Euler polynomials Hn(x), Hn(x)y Hn(x) — Hermite polynomials 4ίλ' μΚχ\ If* μ)(χ), 4Κ μ)(χ), J'LK μ)Μ - Jacobi polynomials 1 x I — The absolute value of x sign x — The sign of x [x] or E(x) — The integral part of x {x} — The fractional part of x (x) — The distance to the nearest integer l(x) — Heaviside's unit function δ(χ) —The delta function ·δ(α» β) - Euler's beta function Γ(<χ) — Euler's gamma function Ei(*) - The integral exponential function Ei(jc) - The real part of Ei(x) li x — The integral logarithm si (JC) or Si x - The integral sine ci x or Cix —The integral cosine erf x, φ{χ), φΒ{χ), erfc x, L(x) — The error function. F(k, φ) - The elliptic integral of the first kind E(k, <p) — The elliptic integral of the second kind Π (k, λ, φ) - The elliptic integral of the third kind w,?»)-[f».rt-%rt]/*1

K — The complete elliptic integral of the first kind E — The complete elliptic integral of the second kind D - (K-E)/k2

ψ(α), Ψ(μ) — The logarithmic derivative of the gamma function (the psi functions) S(x), S*(x) - The Fresnel sine integral C(x), C*(x) — The Fresnel cosine integral

NOMENCLATURE 389

JH(x) — Bessel function of the first kind ΥΛ(χ) — Bessel function of the second kind In(x) — Bessel function of an imaginary argument Kn(x) — Macdonald function

REFERENCES CHAPTER I

1. I. N. BRONSHTEIN and K. A. SEMENDYAYEV, A Guide-Book to Mathematics for Technologists and Engineers (Spravochnik po matematike dlya inzhenerov i uchashchikhsya vtuzov), Fizmatgiz, Moscow (1959). English translation by Pergamon Press,(1963).

2. CH. DE LA VALLÉE-POUSSIN, Cours d'Analyse Infinitésimale, Gauthier-Villars, Paris-Louvain (1937-47).

3. E. GOURSAT, Cours d'Analyse Mathématique, 5th ed., Gauthier-Villars, Paris (1942).

4. A. YA. DuBOvrrsfai, Axiomatic construction of the real numbers, MatematU cheskoye prosveshchenie, No. 2 (1957).

5. M.A. KARTSEV, The Arithmetical Principles of Electronic Digital Computers (Arifmeticheskiye ustroistva elektronnykh tsifrovykh mashin), Fizmatgiz, Moscow (1958).

6. A. N. KOLMOGOROV, A note on the foundations of the theory of real numbers, Matematicheskoye prosveshchenie, No. 2 (1957).

7. A. N. KOLMOGOROV, The basis of the method of least squares, Usp. Mat. Naukl, No. 1 (1946).

8 M. G. KRASNOSEL'SKII, Convex Functions and Orlicz Spaces ( Vypuklyye funktsii i prostranstva Orlicha), Fizmatgiz, Moscow (1958).

9. N. N. LUZIN, The Theory of Functions of a Real Variable (Teoriya funktsii deistviteVnogo peremennogo), Uchpedgiz, Moscow-Leningrad (1940).

10. V. NEMYTSKII, M. SLUDSKAYA and A. CHERKASOV, A Course of Mathematical Analysis (Kurs matematicheskogo analiza), Vol. I and II, Gostekhizdat, Moscow (1957).

11. G. POLYA and G. SZEGO, Aufgaben und Lehrsatze aus der Analysis, 2 volumes. Springer, Berlin (1925); reprinted, Dover, N.Y. (1945).

12. V. I. SMIRNOV,/4 Course of Higher Mathematics (Kurs vysshei matematiki), Vol. I, Gostekhizdat, Moscow-Leningrad (1948). English translation by Pergamon Press, (1964).

13. E. T. WHITTAKER and G. N. WATSON, A Course in Modern Analysis, 4th ed., Cambridge Univ. Press (1940).

14. G. M. FIKHTENGOL'TS, A Course of Differential and Integral Calculus (Kurs differentsiaVnogo i integrarnogo ischisleniya), 3 volumes, Gostekhizdat, Moscow-Leningrad (1951, 1951, 1949).

390

REFERENCES 391

15. A. Y A . KHINCHIN, A Short Course of Mathematical Analysis (Kratkii kurs matematicheskogo analiza), Fizmatgiz, Moscow (1959).

16. A. Y A . KHINCHIN, The elementary linear continuum. Usp. Mat. Nauk. 4 t

No. 2 (1949).

CHAPTER II

1. A. D. ALEKSANDROV, The Interior Geometry of Convex Surfaces (Vnutrennyaya geometriya vypuklykh poverkhnostei), Gostekhizdat, Moscow-Leningrad (1948).

2. A. D. ALEKSANDROV, Convex Polyhedra (Vypuklyye mnogogranniki), Moscow-Leningrad (1950).

3. D . HILBERT and S. COHN-VOSSEN, Anschauliche Geometrie, Springer, Berlin (1932).

4. M. A. LAVRENT'EV and L. A. LYUSTERNIK, Foundations of the Calculus of Variations (Osnovy variatsionnogo ischisleniya), Vol. I, Part I, United Scientific and Technical Press, Moscow-Leningrad (1935).

5. L. A. LYUSTERNIK, Convex Figures and Polyhedra ( Vypuklyye figury i mnogo-granniki), Gostekhizdat, Moscow (1956).

6. L. A. LYUSTERNIK, Applications of the Brunno-Minkovskii inequality to extremal problems, Usp. Mat. Nauk, No II (1936).

7. L. A. LYUSTERNIK, The Brunno-Minkovskii inequality for arbitrary me asurable sets, Dokl. Akad. Nauk, 3, pp. 55-8 (1935).

8. L. V. KANTOROVICH and V. V. NOVOZHILOV, (Editors), Linear Inequalities -and Associated Problems (Lineinyye neravenstva i smezhnyye voprosy); a collection of translations (Sb. perevodov), Foreign Literature Publishing House, Moscow (1959).

9. G. MINKOVSKII, General theorems on convex polyhedra, Usp. Mat. Nauk, No. II (1936).

10. V. I. SMIRNOV, A Course of Higher Mathematics (Kurs vysshei matematiki) Vol. I, Gostekhizdat, Moscow-Leningrad (1948) English translation by Pergamon Press, (1964).

11. G. M. FIKHTENGOL'TS, A Course of differential and Integral Calculus (Kurs differentsiaVnogo i integralnogo ischisleniya), Vol. I. Gostekhizdat, Moscow-Leningrad (1951).

12. E. HELLY, A set of convex bodies with common points, Usp. Mat. Nauk, No. II (1936).

13. G. E. SHILOV, An Introduction to the Theory of Linear Spaces (Vvedenie v -; teoriyu lineinykh prostramtv)* Gostekhizdat, Moscow (1952).

14. D. O. SHKLYARSKII, Conditionally convergent vector series, Usp. Mat. Nauk, No. X , pp. 51-9 (1944).

15. O. SCHREIER and E. SPERNER, Einführung in die Analytische Geometrie und Algebra, 2 volumes, Vandenhoeck and Ruprecht, Göttingen (1948-51).

16. I. M. YAGLOM and V. G. BOLTYANSKII, CV nvex Figures (Vypuklyye figury), Gostekhizdat, Moscow-Leningrad (1951).

392 REFERENCES

CHAPTER III

1. YA. I. ALIKHASHKIN, A method of computing the debit for the head flow to an imperfect bore-hole, Vychislit. matem. No. 1 (1957).

2. E. GOURSAT, Coun d'Analyse Mathématique, Vol. I, Gauthier-Villars, Paris. (1942).

3. L. V. KANTOROVICH and V. I. KRYLOV, Approximation Methods of Higher . Analysis (Priblizhennyye metody vysshego analiza), 4th ed., Gostekhizdat,

Moscow-Leningrad (1952). 4. I. I. PRIVALOV, Fourier Series (Ryady FurVe), United Scientific and Technica

Press, Moscow-Leningrad (1934). 5. V. I. ROMANOVSKII, An Introduction to Analysis (Vvedeniye v analiz), Gosuch

pedizdat, Tashkent, UzSSR (1939). 6. I. M. RYZHIK and I. S. GRADSHTEIN, Tables of Integrals, Sums, Series and

Products (Tablitsy integralov, summ, ryadov i proizvedenii), Gostekhizdat, Moscow-Leningrad (1951).

7. G.S. SALEKHOV, The Computation of Series ( Vychisleniye ryadov), Gostekhizdat, Moscow (1955).

8. G. S. SALEKHOV, The application of Laplace transforms to the summation of series expanded in special functions, Uch. zap. Kazansk. ped. in-ta, No. 10 (1955).

9. G. P. TOLSTOV, Fourier Series (Ryady FurVe), Gostekhizdat, Moscow-Leningrad (1951).

10. G. M. FIKHTENGOL'TS, A Course of Differential and Integral Calculus (Kurs differentsiaVnogo i integrar nego ischisleniya), Vol. II and III »Gostekhizdat, Moscow-Leningrad (1951, 1949).

11. G. H. HARDY, Divergent Series, Clarendon Press, Oxford (1949). 12. K. KNOPP, Theorie und Anwendungen der unendlichen Reihen, Berlin (1924).

CHAPTER IV

1. YA. L. GERONIMUS, The Theory of Orthogonal Polynomials (Teoriya orthogonal* nykh mnogochlenov), Gostekhizdat, Moscow-Leningrad (1950).

2. D. JACKSON, Fourier Series and Orthogonal Polynomials, Cams Monograph of Math. Assoc. of Am., Oberlin, Ohio (1941).

3. L. N. KARMAZINA, Tables of Jacobi Polynomials, (Tablitsy mnogochlenov Jakobi), Izd. Akad. Nauk SSSR, Moscow (1954).

4. S. KACZMARZ and H. STEINHAUS, Theorie der Orthogonalreihen, Chelsea, N.Y. (1951).

5. M. A. LAVRENT'EV and B. V. SHABAT, Methods of the Theory of Functions of a Complex Variable (Metody teorii funktsii kompleksnogo peremennogo), Fizmatgiz, Moscow (1958).

6. L. A. LYUSTERNIK, On the computation of the values of functions of one variable, Matematicheskoye prosveshcheniye, No. 3 and No. 4 (1958).

REFERENCES 393

7. 1. P. NATANSON, The Constructive Theory of Functions (Konstruktivnaya teoriya funkt sii), Gostekhizdat, Moscow-Leningrad (1947).

8. I. M. STESIN, The conversion of orthogonal expansions into continued fractions, Vychisl. matem., No. 1 (1957).

9. A. S. HOUSEHOLDER, Principles of Numerical Analysis, McGraw-Hill, N.Y. (1953).

10. P. L. CHEBYSHEV, Collected papers, (Sobraniye sochinenii), Vol. I, Izd. Akad. Nauk SSSR Moscow (1944).

11. O. PERRON, Die Lehre von der Ketterbrücken, Leipzig und Berlin (1928).

CHAPTER V

1. I.V.ARNOLD, The Theory of Numbers (Teoriya chisel), Uchpedgiz, Moscow (1939).

2. J. L. F. BERTRAND, Traité d'Algebre, Hachette, Paris (1866-7). 3. I. M. VINOGRADOV, Foundations of the Theory of Numbers (Osnovy teorii

chisel), Gostekhizdat, Moscow-Leningrad (1949). 4. N. N. VOROB'EV, Fibonacci Numbers (Chisla Fibonachchi), Gostekhizdat,

Moscow (1951). English translation by Pergamon Press (1961). 5. A. CAUCHY, Cours d'Analyse Algébraique, Leipzig (1864). 6. N. I. LOBACHEVSKII, Algebra or the Evaluation of Finîtes (Algebra Hi vychis-

leniye konechnykh), Vol. IV of the complete works, Gostekhizdat, Moscow-Leningrad (1948).

7. N. N. MARAXUYEV, Elementary Algebra (Elementarnaya algebra), Vol. I, Theory, (Teoriya), Moscow (1903).

8. A. A. MARKOV, Lectures on Continued Fractions (Lektsii o nepreryvnykh drobyakh). See Collected works on The Theory of Continued Fractions and the Theory of Functions that Least Deviate from Zero (Izbrannye trudy po teorii nepreryvnykh drobei i teorii funktsii, naimenee uklonyayushchikhsya ot nulya) Gostekhizdat, Moscow-Leningrad (1948).

9. A. MESHKOV, A Course of Higher Algebra (Kurs vysshei algebry), Part I, SPb (1862).

10. M. V. OSTROGRADSKII, Lectures on Algebraic and Transcendental Analysis (Lektsii algebraicheskogo i transcendentnogo analiza), First Year, SPb (1837).

11. P. ROSHCHIN, Notes on the differential and Integral Calculus (Zapiski po differential'nomu i integral'nomu ischisleniyam), Part I, SPb (1888).

12. B. I. SEGAL, Continued fractions, Matematicheskoye prosveshcheniye, No. 7 (1936).

13. T.I. STŒLTJES, Recherches sur les fractions continues oeuvres, 2, (1894). 14. A. K. SUSHKEVICH, The Theory of Numbers ("Teoriya chisel), Khar'kov,

Izd. Kliar'k. un-ta (1954). 15. K. FERBER, Arithmetic ( Arifmetika) (1914). 16. A.YA. KHINCHIN, Continued Fractions (Tsepnyye drobi), Gostekhizdat, Mos

cow-Leningrad (1949).

394 REFERENCES

17. A. YA. KHINCHIN, Elements of the theory of numbers (Elementy teorii chisel) Encyclopedia of Elementary Mathematics, Book I {Entsiklopediya elementarnoi matematiki, Kniga pervaya), Gostekhizdat, Moscow-Leningrad (1951).

18. A. N. KHOVANSKII, The Application of Continued Fractions and Generalizations of them to Problems of Approximate Analysis (Prilozheniye tsepnykh drobei i ikh obobshchenii k voprosam priblizhennogo analiza), Gostekhizdat, Moscow (1956).

19. A. N. KHOVANSKII, Euler's works on the theory of continued fractions, Istor. matem. issled. No. 10, 305-326 (1957).

20. N. G. CHEBOTAREV, The Theory of Continued Fractions (Teoriyanepreryvnykh drobei), Kazan' (1938).

21. L. D. ESKIN, A note on Euler's algorithm for the extraction of roots (Ob algoritme Eilera dlya izvlecheniya kornei), Uchen. zap. Kazansk. un-ta 115, book 14, pp. 139-43 (1955).

22. L. D. ESKIN, The problem of the nature of the convergence of Euler's algorithm for the extraction of roots, Iz. vyssh. uchebn. zaved., Matem. 4 (5), pp. 275-8 (1958).

CHAPTER VI

1. CH.DE LA VALLÉE-POUSSIN, Cours d9Analyse Infinitésimale, Gauthier-Villars Paris-Louvain (1937-47).

2. I. M. GEL'FAND and G. E. SHILOV, Generalized Functions and Operations on them (Obobshchennyye funktsii i deistviya nod nimi), Fizmatgiz, Moscow (1958).

3. A. O. GEL'FOND, The Calculus of Finite Differences (Ischisleniye konechnykh raznostei), Gostekhizdat, Moscow-Leningrad (1952).

4. I. S. GRADSHTEIN and I. M. RYZHIK, Tables of Integrals, Sums, Series and Products (Tablitsy integralov, summ, ryadov i proizvedenii), Gostekhizdat, Moscow-Leningrad (1951).

5. D. IVANENKO and A. SOKOLOV, The Classical Theory of Fields ( Klassicheskaya teoriya poly a), Gostekhizdat, Moscow-Leningrad (1949).

6. V. A. KUDRIAVTSEV, The Summation of Powers of Natural Numbers and Bernoulli Numbers (Summirovanie stepenei chisel naturalnogo ryada i chisla Bernoulli), United Scientific and Technical Press, Moscow-Leningrad (1938).

7. E. LANDAU, Einführung in die Differentialrechnung und Integralrechnung, NoordhofT, Groningen (1934). English translation: Chelsea, N.Y. (1951).

8. E. T. WHITTAKER and G. N. WATSON, A Course in Modern Analysis, 4th ed., Cambridge Univ. Press (1940).

9. G. M. FIKHTENGOL'TS, A Course of Differential and Integral Calculus (Kurs differentsiaVnogo i integralnogo ischisleniya), Vol. I, Π, ΠΙ, Gostekhizdat, Moscow-Leningrad (1951, 1951, 1949).

10. G. H. HARDY, Divergent Series, Clarendon Press, Oxford (1949). 11. E. CESÀRO, Elementares Lehrbuch der algebraischen Analysis und der Infinitesi

malrechnung, Part I, Leipzig (1904).

http://Ch.de

REFERENCES 395

12. E. CESÀRO, Elementares Lehrbuch der algebraischen Analysis und der Infinitesimalrechnung, Part II, Leipzig (1904).

13. YA. N. SPIELREIN, Tables of Special Functions (Tablitsy spetsiaVnykh funktsii), Parts I and II, GITI, Moscow-Leningrad (1933).

14. E. JAHNKE and F. EMDE, Funktionentafeln, Chelsea, N.Y. (1951). 15. H. BATEMAN, Higher Transcendental Functions, N.Y. (1953). 16. N. E. NÖRLUND, Vorlesungen über Differenzenrechnung, Springer, Berlin (1924).

INDEX

Abel's test for convergence 104 Abscissa 2 Absolute value 2 Argument of a function 10

Basis of Euclidean space 172 of a linear system 43 orthogonal 50, 173 orthonormal 50 Schauder's 339

Bernoulli continued fractions 245 numbers 322—32 polynomials 153, 327-32

Bernshtein polynomials 320 Bessel

function 181, 377-85 inequality 173, 189 integral 378

Beta function 317, 375-6 Binet

formula 321 function 367

Body 55 convex 72-84

reciprocal 80 Bolzano-Cauchy test for conver

gence 21 Boundary 55 Bounds, upper and lower

of a function 11, 60 of a sequence 17 of a set 6

Brunno-Minkovskii inequality 83

Catalan's constant 312 Cauchy

convergence test 88, 89 formula 372

inequality 41 Cauchy-Bunyakovskii inequality 177 Cesàro summation 145 Chebyshev

polynomials 151, 223-40 weight 210

Chebyshev-Laguerre weight 211 Chebyshev-Hermite weight 211 Christoffel's formula 207 Circle of convergence 130 Coefficient

Fourier 136, 173, 189-91 binomial 317-20

Complement of a set 6 Completion 9 Component of an operator 66 Cone 81

tangent 82 Constant

Catalan's 312 Euler-Mascheroni 310

Contraction of a continued fraction 245

Convergence absolute 57, 90 conditional 56, 90 of continued fractions 261-9 of double series 109-17 of Fourier series 133-6 of infinite products 105-9 of power series 123 of sequences, tests for 88-104 of series, tests for

Abel's 104 Bolzano-Cauchy 21 Cauchy's 88, 96 d'Alembert's 89, 95 Birichlet's 104 Ermakov's 102 Gauss's 99 Lobachevskii's 102 Raabe's 97

397

398 INDEX

improvement of 157-61 mean 33, 191 uniform 31, 61, 261

tests for 118-9 Convergent 204, 242 Coordinate 2

of a vector 40

ô (delta) function 343 d'Alambert's convergence test 89 Dedekind section 7 Denominator, partial 244 Dependence, linear 43 Difference of two sets 6 Dini-Lipschitz condition 219 Dirichlet's convergence test 104 Domain 55

convex 72 of definition of a function 10 of definition of an operator 65

Envelope convex 73 linear 48

passage to the limit 65 Ermakov's convergence test 102 Error function 357-9

approximate expansion 281 Estimate, integral 153-6 Euler

constant (Euler-Mascheroni) 310 formula 308 numbers 332-6 product 365

Exponential function 273, 277, 351, 352, 354, 355, 356

Extension of continued fraction 245

Factorial 315 divisors of 316 double 315

Fibonacci numbers 231 Form, linear 45 Formula

Binet's 321 Cauchy's 372

Euler's 308 Gauss's 316, 369 Laplace's 323 Machin's 305 quadrative 204 Raabe's 366 Rodrigue's 213, 220 Schaar's 368 Stirling's 316, 368 Wallis's 305 Weierstrass's 364

Fourier coefficient 136, 173, 189-91 series, trigonometric 132-9

Fractions, continued 201-4, 241-303

and matrices 287-98 arithmetical 252 associated with a series 203 contraction of 245 convergent 261 corresponding to a series 255 equivalent to a series 255 essentially divergent 261 even 13 expansion of power series in 271-2 extension of 245 non-essentially divergent 261 of Daniel Bernoulli 245 of Stieltjes type 203 ordinary 244 ordinary value of 246 periodic 254

in the limit 268 regular 252 singular value of 247 tests for convergence of 261-9 transformation of 244-5, 247-52 uniformly convergent 262

Fresnel integrals 359 Function

Bessel 181, 377-85 of first kind 380 of second kind 381

beta 317, 375-6 bilinear 46 Binet 367 concave 36 continuous 29, 5S

INDEX 399

convex 35, 73-6 delta 343 . discontinuous 29 error 357-9

approximate expansion as 281 exponential, integral 277, 351, 352,

354, 355, 356 expansion as continued fraction

273 gamma 362-74

continued fraction approximation for 282

incomplete 278 generating 151

for Bernoulli polynomials 327 for Chebyshev polynomials 225 for Euler polynomials 334 for Jacobi polynomials 221 for Laguerre polynomials 235 for Legendre polynomials 216

Haar 181 Heaviside 341 hypertranscendental 363 integral 351-6 inverse 13 jump 60 Laplace 185 left continuous 29 logarithmically convex 36 linear 45 Macdonald 382 monotonie 34 odd 13 of complex variable 130 of several variables 57 orthogonal 177, 184 Pearson 210 periodic 14, 62-4 piecewise

linear 337-45 smooth 133

polygonal 338 Rademacher 340 right continuous 29 smooth 133 square integrable 177 uniformly continuous 30, 59 vector 57, 67 Weber 381

weight 178 Functional 31

Gamma function 362-74 continued fraction approximation

282 incomplete 278

Gauss convergence test 99 formula 316, 369

Gram determinant 46, 186

Haar function 181 Hahn-Banach theorem 79 Half-space 72 Heaviside function 341 Hermite polynomials 236-7 Hyperplane 48

of support 76, 79

Image 14 Independence, linear 43, 187 Inequality

Bessel's 173, 189 Brunno-Minkovskii's 83 Cauchy's 41 Cauchy-Bunyakovskii's 177 for e 309 for π 307 triangle 40, 76

Infinitesimal 33 Integral

Bessel 378 elliptic 346-51

of first kind 346 of second kind 346 of third kind 346

Fresnel cosine 359 sine 359

Raabe 366 Intersection of sets 5 Interval 4

closed 5 infinite 5 semi-closed 5

400

Jacobi polynomials 219-23 weight 210

Laguerre polynomials 233-5 Laplace

formula 323 function 185 transform 150

Lebesgue function of orthonormal system

209 integral 171

Legendre polynomials 151, 214-9 weight 211

Limit lower 24 of a convex body 73 of a function 28 point 19, 55 upper 24

Macdonald function 382 Machin's formula 305 Majorant of a series 89, 90, 118 Manifold

linear 47 of constancy 62-4

Mapping 13, 66 contraction 70 symmetric 83

Metric 39 Euclidean 39

Metrization 54 Modulus 2

complementary 346

Norm 33 of a function 179 of a vector 41, 172

Number(s) algebraic 1 Bernoulli 322-32 Euler 332-6 Fibonacci 321

INDEX

irrational 1 natural 4 « transcendental 1

Numerator, partial 241

Operator 65 continuous 66 linear 67

Order of Fourier coefficients 136 of an infinitesimal 33

Origin 2

Parseval's equation 173, 190 Pearson

equation 210 function 210 weight 213

Period of a function 15 fundamental 63

Point boundary 55, 72 of condensation 54 fixed 69 interior 72 limit 19, 55 of sharpening 82 singular 131

Poisson summation 143 Polyhedron

convex 73 reciprocal 81

Polynomials Bernoulli 153, 327-32 Berashtein 320 Chebyshev 151, 237-40

of first kind 223-30 of second kind 230-3

Hermite 236-7 Jacobi 219-23

ultraspherical 220 Laguerre 233-5 Legendre 151, 214-9 of second kind 202 orthogonal systems of 197-240

convergence of Fourier series in

INDEX 401

207-10, 218, 228, 232 recurrence relations for 199,

221, 224, 231, 234, 236 Power moments 200-1 Power series 120-9

differentiation of 123 integration of 123

Pre-image 150 Process

iterative 69 Schmidt orthogonalization 188

Product Euler 365 infinite 105

convergence of 105-9 inner 41, 176-9 of sets 5 scalar 41, 176-9

Quotient, partial 241

Raabe formula 366 integral 366

Rademacher function 340 Radius of convergence 121 Range of a function 10 Relation, recurrence

for Bernoulli numbers 322 for Bernoulli polynomials 328 for Chebyshev polynomials 224

231 for continued fractions 243 for Euler numbers 332 for Euler polynomials 335 for Fourier series 199-200, 221,

224, 231, 234, 236 for Hermite polynomials 236 for Jacobi polynomials 221 for Laguerre polynomials 234 for Legendre polynomials 215

Remainder term of a double series 114 estimates of 93-4 of a series 92 of Taylor series 123-6

Rodrigue's formula 213, 220 Root square deviation 179

Schaar's formula 368 Schauder's basis 339 Schmidt orthogonalization process

188 Section

Dedekind 7 golden 321

Segment 5 Sequence

confinal 8 convergent 20 delta-shaped 34c divergent 21 double 11 equivalent 7 functional 30 fundamental 7, 21, 54 iterative 26, 67 monotonie 18 numerical 16 recurrent 26 uniformly distributed 25

Series 86-169 absolutely convergent 90 alternating 90, 103 asymptotic 140-2 conditionally convergent 90 convergent 87, 99 divergent 87 double 109-117 Fourier 132-9, 191-4 of functions 117-46 Harmonic 89 Kummer's 370 Leibniz's 305 negative 90 orthogonal 170-240 positive 90 recurrent 298

use in solution of equations 298-300

semi-normal 203 Stirling's 282, 316 trigonometric 132-40,161-9 vector 56

Set 4 bounded 6 closed 55 complete 8, 9 convex 72

402

Lebesgue 12 open 55

Space conjugate 78 Euclidean 39 Hilbert 170 n-dimensional 39

linear 54 vector 40

Stirling's formula 316, 368 Stoltz's theorem 247 Subset 5 Sum

generalized 142-6 of series 86 of sets 5 partial 56, 86

Summation of series Cesàro 143 generalized 142-6 numerical 146-69 Poisson 143

Support function 77 hyperplane 76, 79

System binary 3 biorthogonal system of functions

51, 194-7 closed 190 decimal 2 linear 42 orthogonal systems

of function 172-240 of polynomials 197-240 of trigonometric functions 174

INDEX

/7-adic 2

Transformation 70 Abel's 161-2 Rummer's 156-7 of continual fractions 244 orthogonal 50 similitude 83

Uniform convergence 51 test for 32

Union of sets 5

Value ordinary 246 singular 246

Vector space 40

Wallis's formula 305 Weber function 381 Weierstrass's formula 364 Weight 176

Chebyshev's 210 Chebyshev-Hermite's 211 Chebyshev-Laguerre's 211 function 176 integral 178 Jacobi's 210 Legendre's 210 Pearson's 213

Zero, asymptotic 142

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