Topic: Fourier Series

TERM PAPER

SUMMITTED TO: SUBMITTED BY:

MR. GURPREET SINGH SIR. HAZRAT BELAL

ACKNOWLEDGEMENT

I HAVE GREAT SENSE OF HAPPINESS AND PRIDE IN WRITING THIS TERM PAPER. I HAVE WITNESSED THE UNTIRING EFFORTS MADE BY MY MANUFACTURING TEACHER MR. GURPREET SIR. I WOULD LIKE TO THANK My FATHER IN GIVING ME IDEAS FOR MAKING THIS TERM PAPER. I WOULD LIKE TO THANK THE AUTHOR OF THE BOOKS WHICH I USED FOR REFERENCE. I WOULD LIKE TO THANK THE HOST AND CREATOR OF THE WEB SITES FROM WHICH I GOT THE INFORMATION ABOUT THE TERM PAPER

TABLE OF CONTENTS:-

S.No Contents Page no

1) Abstract 4

2) About Fourier series 5

3) What is Fourier Series?6

4) Expalanation of the topic 7

5) Introduction to the problem 8

6) Example 9

7) Conclution 10

Abstract

Fourier series is used in approximation of functions in the form of their Sines and Cosines. Most of the single valued functions which occur in applied mathematics can be expressed in form of series and such a series is called Fourier Series.

In my research I have focused over whether a discontinuous function be represented as a Fourier Series by solving various examples and thus concluding with the necessary conditions that must be fulfilled to do so.

Introduction

Fourier Series-Expresses approximation of functions in the form of their Sines and Cosines. Most of the single valued functions which occur in applied mathematics can be expressed in form of series and such a series is called Fourier Series.

Eq(1)

The series in Equation 1 is called a trigonometric series or Fourier series.

expressing a function as a Fourier series is sometimes more advantageous than

expanding it as a power series. In particular, astronomical phenomena are usually periodic,as are heartbeats, tides, and vibrating strings, so it makes sense to express them in termsof periodic functions.

Discontinuous Function-A function that has a break, hole, or jump in the graphical representation.

Here |x| is a discontinuous function and the discontinuity is at 0.

Introduction to the problem

Can a discontinuous function be developed in the Fourier series? Comment.

Discussion on the problem-

As the computation for a discontinuous function in the form of fourier series must have some conditions to be fulfilled so, by solving a number of problems we can get the conclusion from each and put forward a set of conditions that must be fulfilled in order to develop a discontinuous function in fourier series.

Conditions of fourier expansion of any function

Any function f(x) can be developed as a fourier series

Where a0 ,an, bn are constants

F(x) is periodic,single valued and finite.
F(x) has a finite number of discontinuities in a period.

Within the limits (c,c+2π)

Where c may be any constant.

While calculating the Eulers Formulae it was assumed that function is discontinuous, Hence it was formulated as :

Functions having points of discontinuity

Let

F(x)=¥(x) a < x < c

=§(x) c < x < a + 2π

i.e. c is the point of discontinuity then Eulers formulae become,

Here c is the point of discontinuity. Both the limit on the left and the right exist and are different.

AT SUCH A POINT FOURIER GIVES THE VALUE OF f(x) AS ARITHMETIC MEAN OF THE TWO LIMITS.

i.e.

F(x)= ½ [ f(c - 0) + f(c + 0)]

Example 1-

Example-2

Conclusion

From the above two examples we must conclude the necessary conditions are as follows-

1. f must be periodic with period 2π

2. f must be piecewise continuous

at each position x = q where f is discontinuous, we must have

Hence, I conclude that a discontinuous function can be represented as a fourier expansion by taking arithmetic mean around the limit.

Submitted for –