Nilesh Math Term Paper

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LOVELY PROFESSIONAL

UNIVERSITY

TERM PAPER

ENGINEERING MATHEMATICS II

MTH-102

Topic: - Can a discontinuous function be developed in a

Fourier series

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D.O.S:- 13/05/2010

Submitted By: - Submitted To: -

NILESH TRIPATHI MRS.RAVINDER KAUR

Class- B.Tech (ECE)

Sec. D6905

Roll no. A11

Reg. No.:-10906520

ACKNOWLEDGEMENT

First and foremost I thank my teacher MRS.

RAVINDER KAUR who has given me this

Term Paper to bring out my creative

capabilities. I am also thankful to him for

their valuable suggestions on my term

paper.

I would like to acknowledge the assistance

provided to me by the library staff of L.P.U.

My heartfelt gratitude to my friends, for

helping me morally to complete my work in

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CONTENT

1) Introduction.............................................

.................................4

2) Periodic

Functions.......................................................

.............5

3) Definition of Fourier

series.......................................................6

4) Discontinuous

Function........................................................

....6

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5) Can a discontinuous function be Fourier

series? ......................8

6) Gibbs

Phenomenon.................................................

...................8

7) Formula for finding Fourier

series............................................9

8) Reference................................................

...................................10

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INTRODUCTION

In mathematics, a Fourier series decomposes a periodic function or periodic

signal into a sum 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 (17681830) for the purpose of solving the heat

equation in a metal plate. The heat equation is a partial differential equation. Prior to

Fourier's work, there was no known solution to the heat equation in a general situation,

although particular solutions were known if the heat source behaved in a simple way, in

particular, if the heat source was a sine or cosine wave. These simple solutions are now

sometimes called Eigen solutions. Fourier's idea was to model a complicated heat source as

a superposition (or linear combination) of simple sine and cosine waves, and to write the

solution as a superposition of the corresponding Eigen solutions. This superposition or

linear combination is called the Fourier series.

Fig: 1 The first four Fourier series approximations for a square wave.

Fourier series is named in honour of Joseph Fourier (1768-1830), who

made important contributions to the study of trigonometric series, after preliminary

investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.

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The Fourier series has many applications in electrical engineering,

vibration analysis, acoustics, optics, signal processing, image processing, quantum

mechanics, econometrics, etc.

PERIODIC FUNCTIONS

If the value of each ordinate f(t) repeats itself at equal intervals in the abscissa, then f(t) is

said to be a periodic function.

If f(t) = f(t+T) = f(t+2T) = ... then T is called the period of the function f(t).

DEFINATION OF FOURIER SERIES

Here we will express a non-sinusoidal periodic function into a fundamental and

its harmonics. A series of sines & cosines of an angle & its multiples of the form.

is called the Fourier series. where,

And

are called the Fourier coefficients of.

A periodic function can be expanded in a Fourier series. The series consists of the

following:

(i) A constant term a0 (called D.C. component in electrical work.)

(ii) A component at the fundamental frequency determined by the values of a1, b1.

(iii) Component of the harmonics determined by a2, a3...b2,b3....are known as Fourier

coefficients.

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The Fourier series does not always converge, and even when it does converge for a

specific value x0 of x, the sum of the series at x0 may differ from the value (x0) of the

function. It is one of the main questions in harmonic analysis to decide when Fourier series

converge, and when the sum is equal to the original function. If a function is square-

integrals on the interval [, ], then the Fourier series converges to the function at almostevery point. In engineering applications, the Fourier series is generally presumed to

converge everywhere except at discontinuities, since the functions encountered in

engineering are more well behaved than the ones that mathematicians can provide as

counter-examples to this presumption. In particular, the Fourier series converges absolutely

and uniformly to (x) whenever the derivative of (x) is square integral.

DISCONTINUOUS FUNCTION

1) This is probably the first discontinuous function we learned about. It's called a step

function, and its domain is still the entire set of Real numbers. (The open circles

mean that, for example, at x=2, the y-value is no longer 1, but 2).

Fig: 2 Step discontinuous function

There are clearly gaps when the function jumps to each new value. You can't run your

finger along the graph without lifting it to move to the next portion. This function isdiscontinuous.

2) The next example, at the right, is a Rational expression function where there is an

undefined value of x. The value of x can never equal zero, since division by zero is

not defined.

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Fig: 3

As a result, there is an asymptote at x=0; the graph has a break there. On either side

of this gap the graph approaches infinity. We can't run our finger along the graph

without lifting it to move to the next portion. This function is discontinuous.

3) The graph on the left is one you may have come across before. It is very mysterious

... the graph all by itself looks like the simple linear function y=x+2. If you examine

this function's actual equation, you will notice that it's a Rational expression. The x-

value of -3 is undefined. This means there must be a gap at -3, even though you

can't see it!

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Fig: 4

The values of x have corresponding points on the graph right up to -3 on eitherside, but there is no value for x=-3 itself. This one missing point can't be seen, so

although there is a gap, it isn't visible! This function is discontinuous.

4) There are many types of discontinuous functions, all of which exhibit one common

feature ... there is always a gap. At the right is a graph made from two different

equations:

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Fig: 5

Again notice that the domain is all Real numbers, but there is still a gap. This function is

also discontinuous.

CAN A DISCONTINUOUS FUNCTION BE DEVELOPED IN A

FOURIER SERIES

When comparing the square wave to its Fourier series representation

it is not clear that the two are equal. The fact that the square wave's Fourier series requires

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more terms for a given representation accuracy is not important. Can a discontinuousfunction, like the square wave, be expressed as a sum, even an infinite one, of continuous

ones? This issue brought Fourier much criticism from the French Academy of Science for

several years after its presentation on 1807. It was not resolved for also a century, and its

resolution is interesting and important to understand from a practical viewpoint. Theextraneous peaks in the square wave's Fourier series never disappear; they are termed

Gibb's phenomenon after the American physicist Josiah Willard Gibbs. They occur

whenever the signal is discontinuous, and will always be present whenever the signal has

jumps.

GIBBS PHENOMENON

In mathematics, the Gibbs phenomenon, named after the American physicist J.

Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise

continuously differentiable periodic function behaves at a jump discontinuity: the nth

partial sum of the Fourier series has large oscillations near the jump, which might increase

the maximum of the partial sum above that of the function itself. The overshoot does not

die out as the frequency increases, but approaches a finite limit.

For a discontinuous function f, converted to Fourier series, it should hold 3 conditions:

1) f must be periodic with period 2

2) f must be piecewise continuous

3) at each position x = q where f is discontinuous

f(q) =1/2 [lim xq- f(x) + lim xq+ f(x)]

Suppose we want to construct a Fourier series which converges to the function

s(x) = {1 if |x| < /2 {0 if |x| = /2

on x [2 , 2 ]. To do this, we define a new function S which agrees with s on ss domain,

and satisfies conditions (1), (2), and (3) above,

S(x) = {+1 if |x| < /2

{0 if |x| = /2

{-1 if /2 < |x| Extended periodically with period 2.

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Informally, it reflects the difficulty inherent in approximating a discontinuous function

by a

finite series of continuous sine and cosine waves. It is important to put emphasis on the

word finite because even though every partial sum of the Fourier series overshoots the

function it is approximating, the limit of the partial sums does not exhibit the same

overshoot.

It is impossible for a discontinuous function to have absolutely convergent Fourier

coefficients, since the function would thus be the uniform limit of continuous functions

and

therefore be continuous, a contradiction.

FORMULA FOR FINDING FOURIER SERIES

At a point of discontinuity, Fourier series gives the value of f(x) as the arithmetic mean

of

left and right limits.

At the point of discontinuity, x=c

At x=c,

f(x) = [ f(c-0) + f(c+0) ]

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REFERENCE

Advanced Engineering Mathematics by H.K.Dass

http://en.wikipedia.org/wiki/Fourier_series

http://www.worsleyschool.net/science/files/discontinuous/functions.html

http://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierS

eri

es/

http://cnx.org/content/m0041/latest/

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