Termpaper-can a Discontinuous Function Be Developed in Furier Series-comment-math

7/30/2019 Termpaper-can a Discontinuous Function Be Developed in Furier Series-comment-math

1/13


2/13

INDEX

1. INTRODUCTION

2. HISTORY OF SERIES

3. A SIMPLE DESCRIPTION

4. DERIVATION

5. REFRENCES


3/13

ACKNOWLEDGEMENT

I hereby submit my term paper given to me by my teacherof the su bject MATH- 142 namely Mrs. Gurinderjit kaur. I

have prepared this term paper under the guidance of my

subject teacher. I would thank my class teacher, my subject

teacher and my friends who have helped me to complete the

term paper. I am also highly thankful to all the staff and

executives of the esteemed university namely LOVELY

PROFESSIONAL UNIVERSITY, PHAGWARA, JALANDHAR .

Omkar kumar jha

RH-4901-A12

10902923

B-tech ME(IV Sem)


4/13

INTRODUCTION

In mathematics, a Fourier series decomposes a periodicfunction or periodic signal into a sum of simple oscillatingfunctions, namely sines and cosines (or complexexponentials ). The study of Fourier series is a branch ofFourier analysis. Fourier series were introduced by JosephFourier (1768 1830) for the purpose of solving the heatequation in a metal plate.

The heat equation is a partial differential equation. Prior toFourier's work, there was no known solution to the heatequation in a general situation, although particularsolutions were known if the heat source behaved in a simpleway, in particular, if the heat source was a sine or cosinewave. These simple solutions are now sometimes calledeigensolutions. Fourier's idea was to model a complicatedheat source as a superposition (or linear combination) ofsimple sine and cosine waves, and to write the solution as asuperposition of the corresponding eigensolutions. Thissuperposition or linear combination is called the Fourier

series.Although the original motivation was to solve the heatequation, it later became obvious that the same techniquescould be applied to a wide array of mathematical andphysical problems.

The Fourier series has many applications in electricalengineering, vibration analysis, acoustics, optics, signalprocessing, image processing, quantum mechanics, econometrics, etc.

Fourier series is named in honour of Joseph Fourier (1768-1830), who made important contributions to the study oftrigonometric series, after preliminary investigations byLeonhard Euler, Jean le Rond d'Alembert, and DanielBernoulli. He applied this technique to find the solution of
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Periodic_functionhttp://en.wikipedia.org/wiki/Periodic_functionhttp://en.wikipedia.org/wiki/Sine_wavehttp://en.wikipedia.org/wiki/Complex_exponentialhttp://en.wikipedia.org/wiki/Complex_exponentialhttp://en.wikipedia.org/wiki/Fourier_analysishttp://en.wikipedia.org/wiki/Joseph_Fourierhttp://en.wikipedia.org/wiki/Joseph_Fourierhttp://en.wikipedia.org/wiki/Heat_equationhttp://en.wikipedia.org/wiki/Heat_equationhttp://en.wikipedia.org/wiki/Partial_differential_equationhttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspacehttp://en.wikipedia.org/wiki/Linear_combinationhttp://en.wikipedia.org/wiki/Electrical_engineeringhttp://en.wikipedia.org/wiki/Electrical_engineeringhttp://en.wikipedia.org/wiki/Oscillationhttp://en.wikipedia.org/wiki/Acousticshttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Image_processinghttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Econometricshttp://en.wikipedia.org/wiki/Trigonometric_serieshttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Jean_le_Rond_d%27Alemberthttp://en.wikipedia.org/wiki/Daniel_Bernoullihttp://en.wikipedia.org/wiki/Daniel_Bernoullihttp://en.wikipedia.org/wiki/Daniel_Bernoullihttp://en.wikipedia.org/wiki/Daniel_Bernoullihttp://en.wikipedia.org/wiki/Daniel_Bernoullihttp://en.wikipedia.org/wiki/Jean_le_Rond_d%27Alemberthttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Trigonometric_serieshttp://en.wikipedia.org/wiki/Econometricshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Image_processinghttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Acousticshttp://en.wikipedia.org/wiki/Oscillationhttp://en.wikipedia.org/wiki/Electrical_engineeringhttp://en.wikipedia.org/wiki/Electrical_engineeringhttp://en.wikipedia.org/wiki/Linear_combinationhttp://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspacehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Partial_differential_equationhttp://en.wikipedia.org/wiki/Heat_equationhttp://en.wikipedia.org/wiki/Heat_equationhttp://en.wikipedia.org/wiki/Heat_equationhttp://en.wikipedia.org/wiki/Joseph_Fourierhttp://en.wikipedia.org/wiki/Joseph_Fourierhttp://en.wikipedia.org/wiki/Joseph_Fourierhttp://en.wikipedia.org/wiki/Fourier_analysishttp://en.wikipedia.org/wiki/Complex_exponentialhttp://en.wikipedia.org/wiki/Complex_exponentialhttp://en.wikipedia.org/wiki/Complex_exponentialhttp://en.wikipedia.org/wiki/Sine_wavehttp://en.wikipedia.org/wiki/Periodic_functionhttp://en.wikipedia.org/wiki/Periodic_functionhttp://en.wikipedia.org/wiki/Periodic_functionhttp://en.wikipedia.org/wiki/Mathematics


5/13


6/13

HISTORY OF SERIES

Since Fourier's time, many different approaches to definingand understanding the concept of Fourier series have beendiscovered, all of which are consistent with one another, buteach of which emphasizes different aspects of the topic.Some of the more powerful and elegant approaches arebased on mathematical ideas and tools that were notavailable at the time Fourier completed his original work.Fourier originally defined the Fourier series for real-valuedfunctions of real arguments, and using the sine and cosinefunctions as the basis set for the decomposition.

Many other Fourier-related transforms have since beendefined, extending the initial idea to other applications.This general area of inquiry is now sometimes calledharmonic analysis. A Fourier series, however, can be usedonly for periodic functions.
http://en.wikipedia.org/wiki/Basis_%28linear_algebra%29http://en.wikipedia.org/wiki/List_of_Fourier-related_transformshttp://en.wikipedia.org/wiki/Harmonic_analysishttp://en.wikipedia.org/wiki/Harmonic_analysishttp://en.wikipedia.org/wiki/List_of_Fourier-related_transformshttp://en.wikipedia.org/wiki/Basis_%28linear_algebra%29


7/13

A SIMPLE DESCRIPTION

To recapitulate, Fourier series simplify the analysis ofperiodic, realvalued functions. Specifically, it can break up aperiodic function into an infinite series of sine and cosinewaves. This property makes Fourier series very useful inmany applications. We now give a few.

Consider the very common differential equation given by:

x00(t) + ax0(t) + b = f(t) (23)

This equation describes the motion of a damped harmonicoscillator that is driven by some function f(t). It can be usedto model an extensive variety of physical phenomena, suchas a driven mass on a spring, an analog circuit with acapacitor, resistor, and inductor, or a string vibrated atsome frequency. There are two parts to the solution ofequation (25). The first part is a transient that fades away(generally) fairly quickly. When the transient is gone, whatremains is the steadystate solution. This is what we willconcern ourselves with.

If f(t) is a sinusoid, then the solution is also a sinusoidwhich is not very difficult to find. The problem is that thedriver is generally not a simple sinusoid, but some otherperiodic function. In electronics, for example, a commondriving voltage function is the square wave s(t), a periodicfunction (whose period we shall say is 2 ) such that s(t) = 0for t < 0 and s(t) = 1 for 0 t < .

The physical property of oscillating systems that makesFourier Analysis useful is the property of superposition inother words, suppose the driving force


8/13

f1(t), along with some initial conditions, produces somesteady state solution x1(t), and that another driving force,f2(t) produces the steady state solution x2(t). Then thedriving force f3(t)= f1(t) + f2(t) produces the steadystate

response x3(t) = x1(t) + x2(t).Then, since we can represent any period driving functionas a Fourier series, and it is a simple matter to find thesteady state solution to a sinusoidallydriven oscillator, wecan find the response to the arbitrary driving function

f(x) = a0 +(ancos(nx) + bnsin(nx)).So suppose we had our square wave equation, where f(t) is

the square wave function. We could then decompose thesquare wave into sinusoidal components as follows:

and then just combine the cn and c n terms as before. Theresult would be an infinite sum of sin and cos terms of theform in equation (2). The steadystate response of the

system to the square wave would then just be the sums ofthe steadystate responses to the sinusoidal components ofthe square wave.

The basic equations of the Fourier series led to thedevelopment of the Fourier transform, which candecompose a nonperiodic function much like the Fourierseries decomposes a periodic function. Because this type of


9/13

analysis is very computationintensive, different FastFourierTransform algorithms have been devised, which lower theorder of growth of the number of operations from order(N 2)to order(n log(n)).

With these new techniques, Fourier series and Transformshave become an integral part of the toolboxes ofmathematicians and scientists. Today, it is used forapplications as diverse as file compression (such as theJPEG image format), signal processing in communicationsand astronomy, acoustics, optics, and cryptography.


10/13


11/13

presumed to converge everywhere except at discontinuities,since the functions encountered in engineering are morewell behaved than the ones that mathematicians canprovide as counter-examples to this presumption. In

particular, the Fourier series converges absolutely anduniformly to (x) whenever the derivative of (x) (which maynot exist everywhere) is square integrable Convergence ofFourier series. It is possible to define Fourier coefficients for more generalfunctions or distributions, in such cases convergence innorm or weak convergence is usually of interest .
http://en.wikipedia.org/wiki/Convergence_of_Fourier_serieshttp://en.wikipedia.org/wiki/Convergence_of_Fourier_serieshttp://en.wikipedia.org/wiki/Weak_convergence_%28Hilbert_space%29http://en.wikipedia.org/wiki/Weak_convergence_%28Hilbert_space%29http://en.wikipedia.org/wiki/Convergence_of_Fourier_serieshttp://en.wikipedia.org/wiki/Convergence_of_Fourier_serieshttp://en.wikipedia.org/wiki/Convergence_of_Fourier_series


12/13

Fourier Series of Discontinuous Functions Yes a discontinuous function can be developed into furrierseries..Recall that a generic Fourier series

can converge to a discontinuous function f, so long as threeconditions hold:1. f must be periodic with period2. f must be piecewise continuous

3. at each position x = q where f is discontinuous, we musthave

Suppose we want to construct a Fourier series whichconverges to the function

on2 ]. To do this, we define a new function S which agrees with s on s s domain, and satisfies conditions


13/13

REFRENCES1. http://wiki.answers.com/Q/Can_a_discontinuous_function_be_developed_i

n_a_Fourier_series

2. www.astro.indiana.edu/~jthorn/teach/soton/2007.../sep-of-vars.pdf

3. http://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierSeries/

4. www.toodoc.com/ can-a-discontinuous -function -developed-in-fourier -series -ebook.htmL

5. www-history.mcs.st-andrews.ac.uk/PrintHT/ Functions .html

6. A HAND BOOK OF HIGHER ENGINEERING METHEMATICS

7. JOURNOULS LIKE Weisstein, Eric W. , William E. Boyce and Richard C.DiPrima, Katznelson, Yitzhak, Felix Klein , Walter Rudin
http://wiki.answers.com/Q/Can_a_discontinuous_function_be_developed_in_a_Fourier_serieshttp://wiki.answers.com/Q/Can_a_discontinuous_function_be_developed_in_a_Fourier_serieshttp://wiki.answers.com/Q/Can_a_discontinuous_function_be_developed_in_a_Fourier_serieshttp://wiki.answers.com/Q/Can_a_discontinuous_function_be_developed_in_a_Fourier_serieshttp://www.astro.indiana.edu/~jthorn/teach/soton/2007.../sep-of-vars.pdfhttp://www.astro.indiana.edu/~jthorn/teach/soton/2007.../sep-of-vars.pdfhttp://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierSeries/http://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierSeries/http://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierSeries/http://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierSeries/http://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://en.wikipedia.org/wiki/Felix_Kleinhttp://en.wikipedia.org/wiki/Felix_Kleinhttp://en.wikipedia.org/wiki/Felix_Kleinhttp://en.wikipedia.org/wiki/Felix_Kleinhttp://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://www.toodoc.com/can-a-discontinuous-function-developed-in-fourier-series-ebook.htmLhttp://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierSeries/http://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierSeries/http://www.astro.indiana.edu/~jthorn/teach/soton/2007.../sep-of-vars.pdfhttp://wiki.answers.com/Q/Can_a_discontinuous_function_be_developed_in_a_Fourier_serieshttp://wiki.answers.com/Q/Can_a_discontinuous_function_be_developed_in_a_Fourier_series

Documents

Termpaper-can a Discontinuous Function Be Developed in Furier Series-comment-math