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# Class Note for MATH 322 with Professor Glickenstein at UA

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Date Created: 02/06/15

Chapter 11 Fourier Series Sections 1 5 Chapter 11 Fourier Series Definition Convergence 1 Fourier series 0 We saw before that the set of functions 17 cosX7 sinX cos2X7 sin2X7 7 cosmX7 sinmX7 where m is a nonnegative integer forms a complete orthogonal basis of the space of square integrable functions on 7r7 7r 0 This means that we can define the Fourier series of any square integrable function on 7r7 7r as fX 20 Z 2 cosnX bn sinnX7 n1 where 20 i fX dX and for n 2 1 27T 7 an 2 lW fX cosnX dX and bn lW fX sinnX dX 7T 7 7T 77139 Chapter 11 Fourier Series Fourier series Definition Cenerrli J Convergence Appli Convergence of Fourier series o If f is continuously differentiable on 7r7 7r except at possibly a finite number of points where it has a lefthand and a righthand derivative then the partial sum N fNX 20 Z 2 cosnX bn sinnX n1 with the a defined above converges to fX as N gt 00 if f is continuous at X At a point of discontinuity the Fourier series converges towards lfX fX l Mli I Chapter 11 Fourier Series Definition Convergence Convergence of Fourier series continued 0 Examples 0 Calculate the first three non zero Fourier coefficients of the rectangular wave function Z if 7T lt X g 0 fX ifo lt Xg W and fX27T fX c To what value does the above Fourier series converge if 0 X 0 o X 1 o X 7T 0 Experiment with the MIT applet called Fourier Coefficients o Gibbs phenomenon Near a point of discontinuity X0 the partial sums fNX exhibits oscillations which for small values of N are noticeable even far from X0 As N gt 00 the oscillations get quotcompressedquot near X0 but never disappear Chapter 11 Fourier Series Fourier eerles ZLperiodic functions Generalizations Even and oclcl functions Applications Complex form 2 Fourier series for 2L periodic functions 0 If instead of being 27r periodic the function f has period 2L we can obtain its Fourier series by re scaling the variable X 0 Indeed let gv f Then g is 27rperiodic and one 7r can write down its Fourier series as before Going back to the Xvariable one obtains fx 30 0 2 cos l bn sin 1 L where 20 fX dX and for n 2 1 7L L an 7L fXcos dX7 b fxsin dX 2L periodic functions Even and odd functions Complex form 3 Even and odd functions From the above formula it is easy to see that o If f is even then the bn39s are all zero and the Fourier series of f is a Fourier cosine series ie fX 20 2 2 cos lts nonzero coefficients are given by L L 20 0 fX dX7 an 2 0 fX cos dX 0 Similarly if f is odd then the an s are all zero and the Fourier series of f is a Fourier sine series fX f bnsin bn 0L fXsin dX Chapter 11 Fourier Series ZL periodic functions Even and oclcl functions ppicatIns Complex form Complex form of the Fourier series 0 The Fourier series of a function f fx 20 2 2 cos l bn sin 7 can be rewritten in complex form as fX f cn exp in7 7700 where the complex coefficients C are given by 1 L cn i 7L fXexp inW LX dX7 n 07 l17 l27 Chapter 11 Fourier Series Half range expansions Forced oscillations l5 Applications 5 Half range expansions 0 Sometimes if one only needs a Fourier series for a function defined on the interval 07 L it may be preferable to use a sine or cosine Fourier series instead of a regular Fourier series 0 This can be accomplished by extending the definition of the function in question to the interval L7 0 so that the extended function is either even if one wants a cosine series or odd if one wants a sine series 0 Such Fourier series are called halfrange expansions 0 Example Find the halfrange sine and cosine expansions of the function fX 1 on the interval 07 1 Chapter 11 Fourier Series Fourle Generali H alf range expa nsions Forced oscillations 6 Forced oscillations Applications Consider the forced and damped oscillator described by say by cy fX where b2 48C lt O b is positive and small and f is a periodic forcing function We know that the general solution to this equation is the sum of a particular solution and the general solution to the homogeneous equation ie yX yhX ypX Since the equation is linear the principle of superposition applies Using Fourier series we can think of f as a superposition of sines and cosines As a consequence if one of the terms in the forcing has a frequency close to the natural frequency of the oscillator one can expect the solution to be dominated by the corresponding mode 0 See the MIT applet called Harmonic Frequency Response Chapter 11 Fourier Series

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