fourier series

ADVANCED

ENGINEERING

MATHEMATICS

GUJARAT POWER ENGINEERINGAND

RESEARCH INSTITUTE

FOURIER SERIES

GROUP MEMBERS

Pinky Chaudhari (131040109006)

Harwinder Kaur(131040109015)

Vibha Patel (131040109044)

Samia Zehra (131040109052)

Guided By : Prof. Nirav S. Modi

INDEX

Fourier Series

General Fourier

Discontinuous Functions

Change Of Interval Method

Even And Odd Functions

Half Range Fourier Cosine & Sine Series

FOURIER SERIES

A Fourier series is an expansion of a periodic

function in terms of an infinite sum

of sines and cosines.

General Formula For Fourier Series

Where,

Formulas To Solve Examples

2SC = S + S

2CS = S – S

2CC = C + C

2SS = cos(α-β) –cos(α+β)

Even*Odd = Odd

Even*Even = Even

Odd*Odd = Even

Odd*Even = Odd

1 2 3 4' '' ''' ________uv uv u v u v u v Where,

u, u’, u”, u’’’,_ _ _ _ are denoted by derivatives.

And

V1,v2,v3,v4,_ _ _ _ _ are denoted by integral.

Discontinuous Type Functions

In the interval

The function is discontinuous at x =x0

2C X C

1 0

2 0

( ),

( ), 2

f x C x x

f x x x C

f(x)

0 00

( 0) ( 0)( )

2

f x f xf x

So Fourier series formula is

0

0

2

0 1 2

1( ) ( )

x C

C x

a f x dx f x dx

0

0

2

1 2

1( )*sin( ) ( )*sin( )

x C

n

C x

b f x nx dx f x nx dx

0

0

2

1 2

1( )*cos( ) ( )*cos( )

x C

n

C x

a f x nx dx f x nx dx

Change Of Interval Method

In this method , function has period P=2L ,

where L is any integer number.

In interval 0<x<2L Then l = L/2

When interval starts from 0 then l = L/2

In the interval –L < X < L Then l = L

For discontinuous function , Take l = C where

C is constant.

General Fourier series formula in interval

2C x C L

21

( )*sin( )

C L

n

C

n xb f x dx

l l

21

( )*cos( )

C L

n

C

n xa f x dx

l l

2

0

1( )

C L

C

a f x dxl

0

1

( ) cos( ) sin( )2

n n

n

a n x n xf x a b

l l

Where,

Even Function

The graph of even function is symmetrical

about Y – axis.

Examples :

( ) ( )f x f x

2 2, ,cos , cos , sinx x x x x x x

Fourier series for even function

1. In the interval x

0

1

0

0

0

( ) cos( )2

2( )

2( )*cos( )

n

n

n

af x a nx

a f x dx

a f x nx dx

Fourier series for even function (conti.)

2. In the interval l x l

0

1

0

0

0

( ) cos( )2

2( )

2( )*cos( )

n

n

l

l

n

a n xf x a

l

a f x dxl

n xa f x dx

l l

Odd Function

The graph of odd function is passing through

origin.

Examples:-

( ) ( )f x f x

3 3, , cos ,sin , cosx x x x x x x

Fourier series for odd function

1. In the interval x

1

0

( ) sin( )

2( )sin( )

n

n

n

f x b nx

b f x nx dx

Fourier series for odd function (conti.)

In the interval l x l

1

0

( ) sin( )

2( )sin( )

n

n

n

nf x b x

l

nb f x x dx

l

Half Range Fourier Cosine Series

In this method , we have 0 < x < π or 0 < x < l

type interval.

In this method , we find only a0 and an .

bn = 0

Half Range Fourier Cosine Series

1.In the interval 0 < x < π

0

1

0

0

0

( ) cos( )2

2( )

2( )*cos( )

n

n

n

af x a nx

a f x dx

a f x nx dx

Half Range Fourier Cosine Series(conti.)

2. In the interval 0 < x < l

Take l = L

0

1

0

0

0

( ) cos( )2

2( )

2( )*cos( )

n

n

l

l

n

a n xf x a

l

a f x dxl

n xa f x dx

l l

Half Range Fourier Sine Series

In this method , we find only bn

an =0

a0 =0

Half Range Fourier Sine Series

1. In interval 0 < x < π

1

0

( ) sin( )

2( )sin( )

n

n

n

f x b nx

b f x nx dx

Half Range Fourier Sine Series (conti.)

2. In the interval 0 < x < l

1

0

( ) sin( )

2( )sin( )

n

n

n

nf x b x

l

nb f x x dx

l

Thank you!!!