NAME : ASHISH DAHIYA

COURSE : MTH 102

ROLL NO. : RK6003B45

REG. NO. : 11009100

SECTION : K6003

COURSE INST.: NITIN K. MISHRA

Annexure-I

1

TERM PAPER

ENGINEERING MATHEMATICS-II

TOPIC: EXPLAIN DIFFERENT TYPES OF WAVE FORM IN FOURIER SERIES?

DOA: 1/02/2011

DOS: 15/04/2011

Submitted to: Submitted by:

Mr. Nitin Kumar Mishra Mr. Ashish Dahiya

Deptt . Of Mathematics RK6003B45

11009100

K6003

2

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. NITIN K MISHRA 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

ASHISH DAHIYA

B.TECH (ECE+MBA)

RK6003B45

11009100

3

TABLE OF CONTENTS:-

S.No Contents Page no

1) Abstract 5

2) About Fourier series 6

3) What is Fourier series? 7

4) Waveforms in Fourier series 8

5) Explanation of the topic 9

6) Introduction to the problem 10

7) Let us look at some Examples 12

8) Conclusion 14

9) References 15

4

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.

In mathematics, a Fourier series decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate.

5

IntroductionFourier 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 terms of periodic functions.

Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent arbitrary functions. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.

6

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.

7

Conditions of fourier expansion of any function:Any function f(x) can be developed as a fourier series

Where a0 ,an, bn are constants

1. F(x) is periodic,single valued and finite.2. F(x) has a finite number of discontinuities in a period.

Within the limits (c,c+2π)Where c may be any constant.

WAVEFORMS IN FOURIER SERIES:The Fourier theorem is fairly general and also applies to periodic functions that have discontinuities and cannot be represented by a single analytic expression. For a periodic function f(x), provided that the following conditions are satisfied (Dirichlet conditions): 

(a)    f(x) is defined and single-valued except (perhaps) at a finite number in (-T, T),(b)    f(x) is periodic outside (-T, T) with period 2T,(c)    f(x) and f΄(x) are piecewise continuous in (-T, T), then f(x) can be expressed by the following series: 

 

Where,

8

 

Some simple examples of Fourier series are those of square, triangular and sawtooth waveforms: 

Square waveform

 

Triangular waveform

 

Sawtooth waveform

9

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.

10

i.e.

F(x)= ½ [ f(c - 0) + f(c + 0)]

LET US LOOK AT SOME EXAMPLES:

Example 1- A simple Fourier series

We now use the formula above to give a Fourier series expansion of a very simple function. Consider a sawtooth wave

In this case, the Fourier coefficients are given by

It can be proved that the Fourier series converges to ƒ(x) at every point x where ƒ is differentiable, and therefore:

11

Example-2 :Fourier's motivation

One notices that the Fourier series expansion of our function in example 1 looks much less simple than the formula ƒ(x) = x, and so it is not immediately apparent why one would need this Fourier series. While there are many applications, we cite Fourier's motivation of solving the heat equation. For example, consider a metal plate in the shape of a square whose side measures π meters, with coordinates (x, y) ∈ [0, π] × [0, π]. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by y = π, is maintained at the temperature gradient T(x, π) = x degrees Celsius, for x in (0, π), then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by

Here, sin h is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of Eq.1 by sinh(ny)/sinh(nπ). While our example function f(x) seems to have a needlessly complicated Fourier series, the heat distribution T(x, y) is nontrivial. The function T cannot be written as a closed-form expression. This method of solving the heat problem was made possible by Fourier's work.

We can also define the Fourier series for functions of two variables x and y in the square [−π, π]×[−π, π]:

12

Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression. In particular, the jpeg image compression standard uses the two-dimensional discrete cosine transform, which is a Fourier transform using the cosine basis functions.

Properties of Fourier series:

We say that ƒ belongs to    if ƒ is a 2π-periodic function on R which is k times differentiable, and its kth derivative is continuous.

If ƒ is a 2π-periodic odd function, then an = 0  for all n.

If ƒ is a 2π-periodic even function, then bn = 0  for all n.

If ƒ is integral, , and This result is known as the Riemann–Lévesque lemma.

A doubly infinite sequence {an} in is the sequence of Fourier coefficients of a

function in L1[0,2π] if and only if it is a convolution of two sequences in .

If , then the Fourier coefficients of the derivative f' can be expressed

in terms of the Fourier coefficients of the function f, via the formula

.

If , then . In particular, since tends to

zero, we have that tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n.

Parseval's theorem. If , then

.

13

Plancherel's theorem. If are coefficients and then

there is a unique function such that for every n.

The first convolution theorem states that if ƒ and g are in L1([−π, π]), then

, where ƒ ∗ g denotes the 2π-periodic convolution of ƒ and g. (The factor 2π is not necessary for 1-periodic functions.)

The second convolution theorem states that .

ConclusionFrom 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

3. 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.

14

REFERENCES O'Meara, T. (2000), Introduction to Engineering Mathematics , Berlin, New

York: Springer-Verlag, ISBN 978-3-540-66564-9 Conway, John Horton; Fung, Francis Y. C. (1997), The Sensual (FOURIER

SERIES) , Carus

Mathematical Monographs, The Mathematical Association of America

Bayer-Fluckinger, E.; Lewis, D.; and Ranicki, A. (Eds.). Quadratic Forms

Buell, D. A. Binary Forms: Classical Theory and Modern Computations. New York: Springer-Verlag, 1989.

Conway, J. H. and Fung, F. Y. The Sensual (Quadratic) Form. Washington, DC: Math. Assoc. Amer., 1997

15