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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
D.O.S:- 03/05/2010
Submitted By: - Submitted To: -
SACHIN GUPTA MISS SWATI
AGARWAL
Class- B.Tech (ECE)
Sec. E6902
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Roll no. B47
Reg. No.:-10905375
ACKNOWLEDGEMENT
First and foremost I thank my teacher MISS
SWATI AGARWAL who has given me thisTerm 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 assistanceprovided 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
time.
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CONTENT1) 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 Fourierseries............................................9
8) Reference................................................
...................................10
INTRODUCTION
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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 orlinear 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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