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.