GEC PATANADVANCED ENGINEERING

MATHEMATICS

GUIDED BY:

Prof:- V.C.MAKWANA

Prepared By

FOURIER SERIES& INTEGRAL

• Aadarsh Desai 140220119001• Ashwin Sivadas 140220119004• Modi Darshan 140220119045• Patel Mitul 140220119077

INTRODUCTIONFourier series is used in the analysis of Periodic function e.g., heat wave formula, sound wave formula, full wave rectifier. The Fourier series makes use of orthogonality relationships of sine and cosine functions. It decomposes the function into sum of sine-cosine functions. The computation and study of Fourier series is known as harmonic analysis.

Fourier series and Its Representation

Representation of a function over a certain interval by linear combination of mutual orthogonal function is called Fourier series representation.

This series is known as Trigonometric Fourier series .

ADVANTAGES, USES OF FOURIER SERIES

Discontinuous Function

One of the advantages of a Fourier representation over some other representation, such as a Taylor series, is that it may represent a discontinuous function. An example id the sawtooth wave in the preceding section.

Periodic Functions

Related to this advantage is the usefulness of a Fourier series representing a periodic functions . If f(x) has a period of 2 , perhaps it is only natural that we expand it ina series of functions with period 2 22, , 32

This guarantees that if,

our periodic f(x) is represented over one interval 2,0 or , the representation holds for all finite x.

DIRICHLET’S CONDITON

The Fourier series of function f(x) exists only if the following conditions are satisfied :-

• f(x) is periodic i.e. f(x)=f(x+2l) where 2l is the period of the function

• F(x) and its integrals are finite and single valued • F(x) has finite no. of discontinuities i.e. it is piecewise

continuous• F(x) has finite no of maxima and minima

• From figure , so• Clearly x(t) satisfies the Dirichlet conditions and thus has a Fourier series representation

EXAMPLE: THE RECTANGULAR PULSE TRAIN

2T 0 2 / 2

even function: odd function: some characteristic: even + even=even

even * even=evenodd + odd=oddodd*odd=eveneven*odd=oddIf f is even, then If f is odd, then

EVEN AND ODD

PERIODIC FUNCTION

Examples:Even function : Cosine function i.e. cos(θ)

Odd function: Sine function i.e. sin(θ)

Periodic Function: Both sine and cosine functions are periodic with a period of 2 π

FOURIER SERIES OF A PERIODIC FUNCTIONIf f(x) is periodic, of period 2l, then we define the Fourier series of f, say FS f, as𝐹𝑆𝑓 (𝑥 )=𝑎0+∑

𝑛=1

∞

(𝑎𝑛cos 𝑛𝜋 𝑥𝐿 +𝑏𝑛sin 𝑛𝜋 𝑥𝐿 )Where the coefficients are given by the Euler formulas,

Example: The function given by:

y = -x -п<x<0=y = x - 0<x< п π≤x≤0y=x 0≤x≤ π

The period of the above function is 2π.Thus 2l = 2π Therefore l= π

=

= 0

+

n=0 FS f

n=1 FS f

f is symmetric about x=0 and also about x=L. Because of its symmetry about x=0, f is an even function, and its Fourier series will contain only cosines, no sines. Further, its period is 2L, so L is half the period.

HALF RANGE COSINES

, ( 0< x < L)

𝑎𝑜=1𝐿∫

0

𝐿

𝑓 (𝑥 )𝑑𝑥𝑎𝑛=2𝐿∫0

𝐿

𝑓 (𝑥 ) cos 𝑛𝜋 𝑥𝐿 𝑑𝑥

Proof:For the half-range cosine case the period is 2L,

,

,

.

HALF RANGE SINESf is antisymmetric about x=0 and x=L, the period is 2L, and we have the half-range sine extension.

, ( 0< x < L)

𝑏𝑛=2𝐿∫0

𝐿

𝑓 (𝑥 ) sin 𝑛𝜋 𝑥𝐿 𝑑𝑥

Example:- F(x)= sin(x) 0<x<π HRC L= π

=

( 0< x < )

, ( 0< x < L)

= 0 for n>1;

( 0< x < )

HRS:

FOURIER INTEGRALFourier series apply on finite interval but the Fourier integral is apply on infinite interval and does not apply on the periodic function i.e. -∞<x< ∞,0<x< ∞,

In this there are three type of integral

1) General Fourier integral2) Fourier cosine integral3) Fourier sine integral

∫ ∫

∫

dvwvvfwBdvwvvfwA

dwwxwBwxwAxf

)sin()(1)(,)cos()(1)( where

)sin()()cos()()(0

GENERAL FOURIER INTEGRAL

Fourier integral of f(x):-

Integral SineFourier :)sin()()(,0)(

)sin()(2odd is f(x)function theIf

Integral CosineFourier :)cos()()(

0)(

)cos()(2)(1)(1

)cos()(1A(w)even is f(x)function theIf

0

0

00

0

∫

∫

∫

∫∫∫

∫

dwwxwBxfwA

dvwvvfB(w)

dwwxwAxf

wB

dvwvvfdvdv

dvwvvf

FOURIER SINE & COSINE ITEGRAL

dwwxwwdwwxwAf(x)

dvwvdvwvvfwB

wwdvwvdvwvvfwA

f(x)

)cos()sin(2)cos()(

is f of integralFourier The

0)sin(1)sin()(1)(

)sin(2)cos(1)cos()(1)(

1xfor 01x1-for 1

Let

00

1

1

1

1

1

1

∫∫

∫∫

∫∫

EXAMPLE

THANK YOU