legendre polynomials
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Lecture slidesTRANSCRIPT
Some Special Functions of Some Special Functions of Mathematical Physics
1 Manoj Kumar Pandey, Dept. of Mathematics
Special Functions
� Legendre Polynomials
� Bessel Functions and Gamma Function
2 Manoj Kumar Pandey, Dept. of Mathematics
Legendre Polynomials� The Legendre Equation is
wheren is a non negative integer.
We have found solution of this equation near the ordinary pointx = 0.
,0)1(2)1( 2 =++′−′′− ynnyxyx
Manoj Kumar Pandey, Dept. of Mathematics3
which is valid for -1<x< 1.
y(x) = a0
1− n(n +1)2!
x + n(n − 2)(n +1)(n + 3)3!
x4 +⋯
+ a1
x − (n −1)(n + 2)3!
x3 + (n −1)(n − 3)(n + 2)(n + 4)5!
x5 +⋯
� Howeverthe solutions most useful in the applications arethose bounded near x =1.
Manoj Kumar Pandey, Dept. of Mathematics4
� Here we will solve the Legendre’s equation nearx =1, interms of hypergeometric functionF(a, b, c, x).
We change the independent variable fromx to t by
In terms oft the equation becomes
(1 ) (1 2 ) ( 1) 0 .....(*)t t y t y n n y′′ ′− + − + + =
1(1 )
2t x= −
Manoj Kumar Pandey, Dept. of Mathematics5
Heret = 0 corresponds tox = 1 of the Legendre equation.(*) is a hypergeometric equation with
⇒ a = -n, b = n+1, c = 1
(1 ) (1 2 ) ( 1) 0 .....(*)t t y t y n n y′′ ′− + − + + =
1, 1 2, ( 1)c a b ab n n= + + = = − +
Therefore, the solution of (*) near t = 0 is
( ) ( , 1,1, )y t F n n t= − +
Since c=1 which implies the exponentsm1= 0 andm2 = 1- c = 0henceonly onesolutionis possibleat thismoment.
Manoj Kumar Pandey, Dept. of Mathematics6
henceonly onesolutionis possibleat thismoment.
To find a second linearly independent solution we take
� y2 = vy1, where
−=
−=
∫=∫=′ −−−−
)1(
11
)1(
1
11
22
)1(/)12(
21
21
tyttty
ey
ey
vdttttdxP
Manoj Kumar Pandey, Dept. of Mathematics7
� Sincey12 is a polynomial with constant term 1, the bracketed
expression on the right is an analytic function of the form
−− )1()1( 2
12
1 tyttty
⋯+++ 2211 tata
We can write
This yields
Hence the general solution of (*) near t=0 is
⋯+++=′ taat
v 21
1
⋯++= tatv 1log )(log 112 ⋯++= tatyy⇒
Manoj Kumar Pandey, Dept. of Mathematics8
The solution will bebounded near t =0 if and only if c2 = 0.
.2211 ycycy +=
If we replacet by (½)(1-x), we obtain the solution of Legendre’s
equation nearx=1.
The solution of the Legendre’s equation which is bounded near
Manoj Kumar Pandey, Dept. of Mathematics9
x = 1 are precisely constant multiplies of the polynomial
F(-n, n+1, 1, ½(1-x)).
Legendre Polynomials� Thenth Legendre polynomialdenoted byPn(x) is defined by
Pn(x) = F −n,n +1,1,
1− x
2
(−n)(n +1) 1− x (−n)(−n +1)(n +1)(n + 2) 1− x 2
Manoj Kumar Pandey, Dept. of Mathematics10
=1+ (−n)(n +1)
(1!)2
1− x
2
+
(−n)(−n +1)(n +1)(n + 2)
(2!)2
1− x
2
+⋯
+ (−n)(−n +1)⋯[−n + (n −1)](n +1)(n + 2)⋯(2n)
(n!)2
1− x
2
n
Which gives
Pn(x) =1+ n(n +1)
(1!)22(x −1)+ n(n −1)(n +1)(n + 2)
(2!)222(x −1)2 +⋯
+ (2n)!
(n!)22n(x −1)n.
Manoj Kumar Pandey, Dept. of Mathematics11
with Pn(1)=1.
This is not a suitable form and very inconvenient tool. So we
look for a simple form.
+(n!)22n
(x −1) .
Rodrigues FormulaThe nth Legendre polynomial is a polynomial of degree n
satisfying the Legendre’s equation with Pn(1) =1.
This is given by the Rodrigues formula
21( ) ( 1) , 0,1,2,....
2 !
nn
n n n
dP x x n
n dx= − =
Manoj Kumar Pandey, Dept. of Mathematics12
Remark: Any polynomial of degree n satisfying Legendre equation
with Pn(1) = 1 is called Legendre polynomial .
( ) ( 1) , 0,1,2,....2 !n n n
P x x nn dx
= − =
2(1 ) 2 ( 1) 0x y xy n n y′′ ′− − + + =
� To show which is a polynomial of degreen, is a solution of the Legendre equation.
� Leibnitz’s rule for higher order derivative
( )2( ) 1n
n
n
dy x x
dx= −
Manoj Kumar Pandey, Dept. of Mathematics13
� Leibnitz’s rule for higher order derivative
( ) ( )
( ) ( ) ( 1) (1) ( 1) (1) (1) ( 1) ( )
0
( 1)( )
2!n n n n n n
nn r r
r
n nuv u v nu v u v nu v uv
nu v
r
− − −
−
=
−= + + + + +
=
∑
⋯⋯
Legendre Polynomials
� Rodrigues formula provides a
relatively easy method for computing the successive Legendre
polynomials.
21( ) ( 1)
2 !
nn
n n n
dP x x
n dx= −
Manoj Kumar Pandey, Dept. of Mathematics14
� Some of the first few Legendre Polynomials
( ) ( )
0 1
2 32 3
4 2 5 34 5
( ) 1, ( ) ,
1 1( ) (3 1), ( ) (5 3 ),
2 21 1
( ) 35 30 3 , ( ) 63 70 15 ,....8 8
P x P x x
P x x P x x x
P x x x P x x x x
= =
= − = −
= − + = − +
Properties of Legendre PolynomialsOrthogonalityOrthogonality
The most important property of the Legendre polynomials
is
0 1( ), ( ), , ( ),nP x P x P x⋯ ⋯
Manoj Kumar Pandey, Dept. of Mathematics15
is
that is Legendre polynomials forms an orthogonal sequence offunctions in the interval [-1, 1].
∫−
=+
≠=
1
1 .12
2,0
)()(nmif
n
nmifdxxPxP nm
ProofProof:: Let f(x) be any function with at leastn continuousderivatives on the interval , and consider the integral
By Rodriguesformula
11 ≤≤− x1
1( ) ( )nI f x P x dx
−= ∫
Manoj Kumar Pandey, Dept. of Mathematics16
By Rodriguesformula
after integration by parts, we get
1 2
1
1( ) ( 1)
2 !
nn
n n
dI f x x dx
n dx−= −∫
11 112 21 11
1
1 1( ) ( 1) ( ) ( 1)
2 ! 2 !
n nn n
n n n n
d dI f x x f x x dx
n dx n dx
− −
− −−−
′= − − −
∫0
Therefore, we get
continuing to integrate by parts, we obtain
11 2
11
( 1)( ) ( 1)
2 !
nn
n n
dI f x x dx
n dx
−
−−
− ′= −∫
1 ( ) 2( 1)( )( 1)
nn nI f x x dx
−= −∫
Manoj Kumar Pandey, Dept. of Mathematics17
If f(x) = Pm(x) with m < n, thenf(n)(x) = 0 and consequentlyI = 0.
which proves the first part i.e. the case whenm ≠ n (????)
For the second part lets takef(x) = Pn(x).
1 ( ) 2
1
( 1)( )( 1)
2 !n n
nI f x x dx
n −
−= −∫
Since we get,!2
)!2()()(
n
nxP
n
nn =
)!2(2
)1()!(2
)!2(
1
1
1
222
n
dxxn
nI n
n
∫
∫ −=−
Manoj Kumar Pandey, Dept. of Mathematics18
If we change the variable by , and recall the formula
We conclude that in this case , hence the proof.
.)1()!(2
)!2(2 1
0
222
dxxn
n nn ∫ −=
θsin=x
.)12()!2(
)!(2cos
222/
0
12
+=∫
+
nn
nd
nnπ
θθ
)12/(2 += nI
Every polynomial of degree k can be expressed as a linear
combination of first k+1 Legendre polynomials:
That is
0 1( ) , ( ) , , ( )kP x P x P x⋯
Manoj Kumar Pandey, Dept. of Mathematics19
.)()(0∑
=
=k
nnn xPaxp
For example let
( )0 11 , ( ),P x x P x= =
p(x)=1+2x+3x2+5x3
Manoj Kumar Pandey, Dept. of Mathematics20
2 22 2 0 2
3 33 3 1 3
1 1 2 1 2( ) (3 1) ( ) ( ) ( ) ,
2 3 3 3 31 3 2 3 2
( ) (5 3 ) ( ) ( ) ( )2 5 5 5 5
P x x x P x P x P x
P x x x x x P x P x P x
= − ⇒ = + = +
= − ⇒ = + = +
Therefore
0 1 0 2 1 3
0 1 2 3
3
1 2 3 2( ) ( ) 2 ( ) 3 ( ) ( ) 5 ( ) ( )
3 3 5 5
2 ( ) 5 ( ) 2 ( ) 2 ( )
( ).
p x P x P x P x P x P x P x
P x P x P x P x
a P x
= + + + + +
= + + +
=∑
Manoj Kumar Pandey, Dept. of Mathematics21
Therefore, any polynomialp(x) of degreek can be written as
0
( ).n nn
a P x=
=∑
.)()(0∑
=
=k
nnn xPaxp
Legendre SeriesWhat about an arbitrary function?
An arbitrary functionf(x) can be expressed asLegendre series
Need to calculate coefficientsan for the above expression.0
( ) ( )n nn
f x a P x∞
=
=∑
Manoj Kumar Pandey, Dept. of Mathematics22
n
Multiply Pm(x) on both sides and integrate from –1 to 1, we get
Using orthogonality of Legendre polynomials
1 1
1 10
( ) ( ) ( )m n m nn
f x P dx a P x P x dx∞
− −=
=∑∫ ∫
1
1
2( )
2 1m
m
af x P dx
m−=
+∫
Therefore, we have
.)(2
1 1
1dxPxfna nn ∫−
+=
Manoj Kumar Pandey, Dept. of Mathematics23
Problem: Find the first three terms of the Legendre Series of
0 1 0( ) ( )
0 1.
if xi f x
x if x
− ≤ <= ≤ ≤
Let f(x) be a function defined in [-1, 1], now consider the problem
of approximating f(x) as closely as possible in the sense of least
square by polynomials p(x) of degree ≤ n.1 2
1[ ( ) ( )] .I f x p x dx
−= −∫
Least Square Approximation
Manoj Kumar Pandey, Dept. of Mathematics24
The Problem is to minimize I:
It turns out that the minimizing polynomial is:
With
0 0 1 1( ) ( ) ( ) ( )n np x a P x a P x a P x= + + +…
.)(2
1 1
1dxPxfna nn ∫−
+=
is called the generating function of the Legendre polynomials.
20
1( )
1 2
nn
n
P x txt t
∞
=
=− +
∑
Generating Function and Recursion Formula
Manoj Kumar Pandey, Dept. of Mathematics25
Recursion Formula :
1 1( 1) ( ) (2 1) ( ) ( ), 1,2,....n n nn P x n xP x nP x n+ −+ = + − =
Show that
Recursion Formula
2 1
2
( ) (1) 1, ( ) ( 1) ( 1) ( ) (0) 0,
1 3 (2 1)( ) (0) ( 1) .
2 !
nn n n
nn n
i P ii P iii P
niv P
n
+= − = − =⋅ − = − ⋅
⋯
Manoj Kumar Pandey, Dept. of Mathematics26
2( ) (0) ( 1) .2 !n n
iv Pn
= − ⋅
� If p(x) is a polynomial of degree such that1
1≥n
Manoj Kumar Pandey, Dept. of Mathematics27
Show thatp(x) = c Pn(x) for some constant c.
1
1( ) 0 for 0,1, , 1.kx p x dx k n
−= = −∫ ⋯