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ADDIS ABABA UNIVERSITY
COLLEGE OF NATURAL SCIENCE
DEPARTMENT OF MATHIMATICS
PROJECT TITLE: Fourier series
Adviser: Ato Bizuneh M.
Prepared by: Getnet Bikis
ID NO NSR/4324/05
MiliyonNew Stamp
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TABLE OF CONTENTS PAGES
Acknowledgement .... 2
OBJECTIVE ... 2
CHAPTER ONE: SERIES
1.1. INTRODUCTION TO SERIES . 3
1.2. CONVERGENCE OF SERIES ....................................... 3
1.3. POWER SERIES .... 5
1.4. TAYLOR AND MACLAURIN SERIES.. 6
CHAPTER TWO: FOURIER SERIES
2.1. INTRODUCTION... 7
2.2. PERIODIC FUNCTIONS ... 8
2.3. DIRICHLET CONDITIONS . 8
2.4. FOURIER SERIES OF FUNCTIONS
WITH PERIOD 2 9
2.5. 2L-periodic functions 10
2.6. FOURIER SERIES OF EVEN
AND ODD FUNCTIONS . 12
2.7. FOURIER CONVERGENCE
THEOREM .. 15
References 16
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ACKNOWLEDGEMENT
In the accomplishment of this project successfully, many people have best owned
upon me their blessing and the heart pledged support. This time I am utilizing to
thank all the people who have been concerned with the project.Primarly I would like
to thank God for being able to complete this project successfully. Then I would like
thank my advisor Ato Bizuneh whose valuable guidance has been the one that
helped me path this project and make it full proof success his suggestions and his
instructions has served as the major contributor towards the completion of the
project.
Then I would like to thank my parents and my friends by their financial support .Last
but not least I would like to thank my classmates who have helped me a lot.
OBJECTIVE:
To define basic Fourier series and to establish some elementary
facts about them.
To have a better understanding about series.
To understand how the Fourier coefficients are calculated.
To know about how series converges and diverges.
To investigate the validity of Fouriers claim and derive the basic
properties of Fourier series.
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CHAPTER ONE: SERIES
1.1. Introduction to series:
Definition: Let {an} be a sequence of real numbers, then the expression of the
form a1+a2+ denoted by is called series.
Example: 1 +1
2+
1
3+ =
1
=1
1.2. Convergence of series
Definition: If sn s for some S, then we say that the series
converges to S .If (Sn) doesnt converge, then we say that the series
diverges.
Example: 1
() converges because = 1
1
2 log
diverges because = log( + 1) .
Necessary condition for convergence
Theorem If converges, then an 0.
Proof: Sn+1-Sn =an+1 S-S=0.The condition given above result is necessary
but not sufficient condition.
Example: If ||, then, diverges because an doesnt tend to
zero.
Necessary and sufficient condition for convergence
Theorem: suppose an 0 for all n, then converge if and only if (Sn)
is bounded above.
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Example: The harmonic series
diverges because
S2K 1+
+2(
)+4(
)++2K-1(
)=1+
for all k.
Theorem: If || converges, then converges.
Remark: note that converges if and only if converges
for all p.
Test for convergence
Theorem (comparison test): suppose 0 an bn for all n , for
some k, then
1. The convergence of implies the convergence of
2. The divergence of implies the divergence of .
Example: 1.
() Converges because
()
()
()
Converges.
2
diverge because
.
Theorem. (Ratio test): consider the series ,an 0for
all n.
1. If
eventually for some 0
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Example.1.
! converges because
0
2
! diverges because
=(1+
)n 0
Theorem (Root test): 0 an xn for some 0
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Radius of convergences of power series
Let () = ( ) be the function defined by this power series.
Note that f(x) is only defined if the power series converges. So we
consider the domain of the function f to be the set of x values for which
the series converges. There are three possible cases.
1. The power series converges at x=c(note f(c)=a0)
2. The power series converges for all x. i.e. (- , ).
3. There is a number R called the radius of convergence such that the
series converges for all c-R
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c=0 it is called the Maclaurin series).if f(x) is represented by a power
series centered at c,then f(x)=()()()
!
Example: find the maclaurin series for f(x)=ex centered at x=0.
Now, f(x) =ex f (0)=1
(x)=ex ,(0)= 1
(x)=ex ,(0)= 1
: :
: :
()()= ,()(0)= 1
Therefore ex =1+x+
!+
!+=
!
Example: find the Taylor series for f(x) =lnx centered at x=1
Solution: f(x) =lnx , f(1)=0
()=
, (1)=1
()= ,(1)= -1
:
()(1)= (1)! Hence the Taylor series for lnx
ln = (1)()( 1)
!
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CHAPTER TWO:
2. FOURIER SERIES
2.1. INTRODUCTION
A Fourier series is a specific type of an infinite mathematical series involving
trigonometric functions. The series gets its name from a French mathematician and
physicist named Jean Baptist Joseph, Baron de Fourier. Fourier series are used in
applied mathematics and especially in physics and electronics to express periodic
functions such as those that comprise communication signal wave forms.
Fourier series simply states that periodic signals can be represented into sums of
sines and cosines when multiplied with certain weight.
The power series and Taylor series is based on the idea that you can write general
function as an infinite series of power. The idea of Fourier series is that you can
write a function as an infinite series of sines and cosines. You can also use functions
other than trigonometric ones, but I will leave that generalization aside for now,
except to say that Legendre polynomials are an important example of functions
used for such more general explanation.
2.2. Periodic functions
Definition: A function f(x) is said to be periodic if there exists a number p>0 such
that f(x+p) =f(x) for every x.The smallest such p is called the period of f(x).
Example: sinx and cosx are periodic with period 2, because sin(x+2)=sinx and
cos(x+) =cosx.
Sin (x) and cos (x) are periodic with period 2.
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If L is fixed number, then sin (
) and cos (
) have period L.
2.3. Dirichlet conditions
The particular conditions that a function f(x) must fulfill in order that it may be
expanded as a Fourier series are known as the Dirichlet conditions, and may be
summarized as:
1). the function must be periodic.
2). It must be single valued and continuous except possibly at a finite number of
finite discontinuous.
3). It must have only a finite number of maxima and minima within one periodic.
4). the integral over one period of f(x) must converge.
2.4. Fourier series of functions with
period 2
In this section we will confine our attention to function of period 2.We want to
determine what the coefficients in the Fourier series
+ +
must be if it is converge to agiven function f(x) of period 2.For this
purpose we need the following integrals in which m and n denote positive
integers. 0, , =
,
=
0, , =
,
= 0 ,
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These formulas imply that the functions cosnx and sinnx for n=1, 2,2, constitute a
mutually orthogonal set of functions on the interval[,]. Two real valued
functions u(x) and v(x) are said to be orthogonal on the interval [,] provided
that: ()() = 0
.
Definition: Let f(x) be a piecewise continuous function of period 2 that is defined
for all x,then the Fourier series of f(x) is the series:
+ + . Where an=
(), 0
bn=
(), 1
Example. The functions 1, cosx, sinx, cos2x, sin2x...cosnx, sinnx have period 2 .
Example. Find the Fourier series for .
Solution.
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2.5. -
We have computed the Fourier series for a functions of different periods. Well, fear not, the computation is a simple case of change of variables. We can just rescale the independent axis. Suppose that we have a -periodic function (
Is 2-periodic. We want to also rescale all our sines and cosines. We want to write
If we change variables to us
We compute and as before. After we write down the integrals we change variables from back to .
The two most common half periods that show up in examples are the simplicity. We should stress that we have done no new mathematics, we have only changed variables. If you understand the Fourier series for functions, you understand it for moving some constants around, but all the mathematics is the same.
Example. Let
-periodic functions
We have computed the Fourier series for a -periodic function, but what about functions of different periods. Well, fear not, the computation is a simple case of change of variables. We can just rescale the independent axis. Suppose that we have
is called the half period). Let . Then the function
periodic. We want to also rescale all our sines and cosines. We want to write
us see that
as before. After we write down the integrals we change
The two most common half periods that show up in examples are and 1 because of the simplicity. We should stress that we have done no new mathematics, we have only changed variables. If you understand the Fourier series for -periodic
erstand it for -periodic functions. All that we are doing is moving some constants around, but all the mathematics is the same.
periodic function, but what about functions of different periods. Well, fear not, the computation is a simple case of change of variables. We can just rescale the independent axis. Suppose that we have
. Then the function
periodic. We want to also rescale all our sines and cosines. We want to write
as before. After we write down the integrals we change
and 1 because of the simplicity. We should stress that we have done no new mathematics, we have
periodic periodic functions. All that we are doing is
moving some constants around, but all the mathematics is the same.
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Extended periodically. Compute the Fourier
we can writeeven and hence
Next we find
You should be able to find this integral by thinking about the integral as the area under the graph without doing any computation at all. Finally we can we notice that is odd and, therefore,
Hence, the series is
Let us explicitly write down the first few terms of the series up to the
.2.6.Fourier series of even and odd functions
periodically. Compute the Fourier series:
. For we note that
You should be able to find this integral by thinking about the integral as the area under the graph without doing any computation at all. Finally we can
is odd and, therefore,
Let us explicitly write down the first few terms of the series up to the
Fourier series of even and odd functions
we note that is
You should be able to find this integral by thinking about the integral as the area under the graph without doing any computation at all. Finally we can find . Here,
Let us explicitly write down the first few terms of the series up to the harmonic.
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The function of defined far all x is said to be even if f(-x) = f(x) for all x and of is odd if
f(-x) = -f(x) for all x. the first condition implies that the graph of y=f(x) is symmetric
with respect to the y-axis, where as the second condition implies the graph of an
odd function is symmetric with respect to the origin.
Example f(x) = x2n and g(x) = cosx are even functions
f(x) = x2n+1 and g(x) = sinx are odd functions.
If f is even ()
= 2 ()
If f is odd ()
= 0
Definition: - Fourier cosine and sine series
Suppose that the function f(x) is piece wise continuous on the interval [0, L], then
the Fourier cosine series of f is the series.
F(x) =
+ cos
with an=
()cos
, a0 =
()
,
And the Fourier sine series f is the series
()= sin
With bn =
()sin
Note: - the coefficients are called Fourier coefficients.
Note: - even function of period 2
If f is even and L = , then f(x) = a0 + cos with coefficients
a0 =
()
an =
()cos
, n1
Odd functions of period 2
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If f is odd and L = then f(x) = sin with coefficients
bn=
()sin
, n1
Example: - let f(x) = x2 be a 2 periodic function x [-, ]. Find the Fourier series of the parabolic wave.
Solution: - since f(x) = x2 is even function, the coefficients of bn = 0
a0 =
()
=
=
=
a0 =
()cos
=
f(x)cos
=
cos
apply integration b part twice to find.
an =
cos
. Let u = x2, du = 2x, du = cosnx dx
u= cos =
an =
[(
)]0 -
]
=
[2 sin n - (-2) sin (-n ) - 2
=
sin
Again applying integration by part we get
an =
[(-x
)] -
)dx]
=
[cos(n ) -
()
]
Since sin n = 0 and cos n = 1n for integer n, we have
an =
(-1)n =
(-1)n
The Fourier series expansion for the parabolic wave is
f(x)=
+
(-1)
n cosnx
Example: - suppose that f(x) = x for 0
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Solution:- a0 =
dx =
[
x2] = L
an =
dx (integrating by parts)
Let u =
=> x =
, dv =
dx
Thus
dx =
.
(Again using by parts)
We have
dx =
.
[U sin u + cos u]
=>an==
,
0,
Therefore the Fourier cosine series of f is
F(x) =
-
(cos
+
cos
+
cos
+ ., for 0
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2.7. Fourier convergence theorem
Here is a theorem that states a sufficient condition for convergence of a given
Fourier series. It also tells us to what valued does the Fourier series converge to at
each point on the real line.
Theorem: -suppose f and f1 are piecewise continuous on the interval L x L. further suppose that f is defined elsewhere so that it is periodic with period 2L. Then
f has a Fourier series as stated previously whose coefficients are given by Eulers
formulas. The Fourier series converge to f(x) at all points where f is continuous and
to [ () ()
] at every point c where f is discontinuous.
References:
1. R.C.Mcowen, partial differential equations, methods and applications, Pearson education, INC, 2003.
2. Advanced engineering mathematics, Erwin Kreyszig,9 edition.
3. H.M.Lieberstein, theory of partial differential equations, academic press, 1972.
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