fourier series
DESCRIPTION
fourier series easy to learn ..it will help you to understandTRANSCRIPT
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
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