1Gujarat Technological University

SUBMITTED TO:-

PROF. BIJAL KHUMAN

130110107054-SHUKLA DEVANSHI

130110107055-SONAGARA CHIRAG

130110107056-SUTARIYA ASHISH

130110107057-SUTARIYA YASH

130110107058-SUTHAR HEMANT

130110107059-SHRINAND THAKKAR

130110107060-VIRALI THAKKAR

PRESENTED BY:-2

Contain:-

1. introduction

2. formulas

3. application

4. fourier cosine integral

5. fourier sine integral

6. reference

3

1:-INTRODUCTION

4

Fourier series are powerful tools for problems involving

functions that are periodic or are of interest on a finite

interval only.

many problems involve functions that are nonperiodic and

are of interest on the whole x-axis, we ask what can be done

to extend the method of Fourier series to such functions. This

idea will lead to “Fourier integrals.”

5

2:-FORMULAS

6

wvdvvfwBand

wvdvvfwA

where

dwwxwBwxwAxf

sin)(1

)(

cos)(1

)(

,

sin)(cos)()(0

This is called a representation of

f(x) by a Fourier integral.

integral from 0 to which represents f(x), namely,7

3:-APPLICATIONS

8

Applications of Fourier Integrals:-

The main application of Fourier integrals is in solving ODEs and

PDEs.

9

10

11

4:-FOURIER

COSINE

INTEGRAL

12

Fourier Cosine Integral

Just as Fourier series simplify if a function is even or

odd, so do Fourier integrals, and you can save

work. Indeed, if f has a Fourier integral

representation and is even. This holds because

the integrand of is odd. Then reduces to a Fourier

cosine integral

13

For even function f(x): B(w)=0,

0wvdvcos)v(f

2)w(A

0

dwwxcos)w(A)x(f

Therefor,

14

5:-FOURIER

SINE

INTEGRAL

15

Fourier Sine Integral

Note the change in : for even f the integrand

is even, hence the integral from to equals

twice the integral from 0 to infinite. Similarly, if

f has a Fourier integral representation and is

odd. This is true because the integrand of is

odd. Then a Fourier sine integral

16

For odd function f(x): A(w)=0

0wvdvsin)v(f

2)w(B

0

dwwxsin)w(B)x(f

Therefor,

17

18

19

20

21

6:-REFERENCE

22

1.Advanced Engineering Mathematics (10th Edition) by Erwin

Kreyszig

2.http://googleimages.co.in

3.http://wikiepedia.com

23

24