ADVANCED ENGINEERING MATHEMATI
ADVANCED ENGINEERING MATHEMATI MATH 348
Colorado School of Mines
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This 3 page Class Notes was uploaded by Diana Prosacco on Monday October 5, 2015. The Class Notes belongs to MATH 348 at Colorado School of Mines taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/219615/math-348-colorado-school-of-mines in Mathematics (M) at Colorado School of Mines.
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Date Created: 10/05/15
E Kreyszig Advanced Engineering Mathematics 9 ed Section 117 pgs 506513 Lecture Fourier Integral Module 11 Suggested Problem Set 3 7 9 14 15 March 30 2009 E Kreyszig Advanced Engineering Mathematics 9 ed Section 118 pgs 513517 Lecture Fourier SineCosine Transform Module 11 Suggested Problem Set 1 5 6 March 30 2009 E Kreyszig Advanced Engineering Mathematics 9 ed Section 119 pgs 518528 Lecture Complex Fourier Transform Module 11 Suggested Problem Set 2 3 9 14a March 30 2009 Quote of Lecture 11 Eherybody s talking 7bout the stormy weather and what s a man do to but work out whether it s true Looking for a man with a focus and a temper who can open up a map and see between one and two Time to get it before you let it get to you Here he comes now stick to your guns and let him through Sonic Youth Teenage Riot 1988 1 OVERVIEW Now that we have ended the study of Fourier series which are used to represent piecewise continuous functions that have the additional feature of being periodic we as the question 0 Can these Fourier methods be applied to functions that do not have the extra structure of period icity Why would we like to do this Well we have seen already that Fourier series allow us to gather from a function its particular frequencies of oscillation and also their associated amplitudes With this description one can then gain insight into the overall energy of the function as well as how each of the oscillatory modes contributes to this energy Without the Fourier representation much of this information is inaccessible If we can port these methods over to functions without a periodic substructure then we may nd similar inter pretations and consequently Fourier methods would be invaluable to an extremely large class of physically relevant functions So how should we go about this Well we must rst notice that one of the fundamental assumption on functions with a Fourier series representation is that they can be de ned by using a nite portion of the realline This nite portion is considered to be the principleperiod and from this information the rest of the function is constructed by repeating the functions graph on this domain to the rest of R If we consider this principleperiod to be the 7LL which is a nite portion of R then we can destroy these concepts by taking the limit L gt oo Letting L become unbounded gives rise to a plausible heuristic derivation which results in the wellcelebrated complex Fourier transform or just Fourier transform for short Though the text chooses to consider rst the limit L gt 00 to arrive at the Fourier integral and symmetrically exploited to de ne the Fourier cosinesine transforms which upon reconsideration is used to de ne the more general complex Fourier transform we instead use the Fourier integral to go directly to the complex Fourier 1 MATH348 Advanced Eu 11151111 M quot 2 transform and show that the intermediary results as special cases1 In the following points I outline the key points of logic used in this derivation 1 Consider the Fourier series representation of a QLperiodic function in its full form 1 L 1 L 1 L iiL fvdv T 7L coswnvdv coswnc 7L s1nwnvdv s1nwnc w 2 Assume that the function f is absolutely integrable lfIldI lt oo and take the limit7 L gt oo 2 Once we have taken this limit we arrive at the Fourier integral a representation of a function that need not be periodic 140 coswx 3a sinwx dw 0 140 coswxdc 3a sinwxdx 7r x 7quot 700 3 Lastly consider the Fourier integral representation of the function f in its fullform fx Q fvcoswvdv coswI fvsinwvdv 31mm dw and again after some decent algebra we arrive at the complex Fourier transform pair 1 0 4W 7 0 Aweimx fw lmwe dam fx7m1wf dw Before we begin using this integral transform we make a note of it similarity to the complex Fourier series A 1 0 4W 7 0 Aweimx w fw wfxe dam fI727rWf aw ER L r empi Lfltxgte wdx M Z cwew up n7ltgtlt L l and highlight connections with previous logic stressing that the connection is found in the periodicity de stroying limit where L gt oo 4 0 Roll of coef cients In both cases the forward integral converts the function f into amplitude information f in the case of Fourier transform and 00 in the case of Fourier series These coef cients are then used represent f as a linear combination of oscillatory functions iiWC in the case of transform and em in the case of series The major difference between the two methods is that if the function f is periodic then frequency information is needed only for integer multiples of 7rL while in the case that f is not periodic increased frequency information is needed 0 Energy As with Fourier series the integralconversion of the function f to amplitude information makes accessible energy information associated with the function Emigs 0lt Elem Eumnsmm 0lt lfwl2dw 1The book does this to mimic the logic used for building the Fourier series This correspondence is nice but drags one and onehalf lectures of heavy symbolic manipulation into three I favor the shorter derivation since it is unlikely that you will need to ever repeat them 2Much is hidden here Formally we have to interchange two limiting processes This brings a good amount of fear to the table and mathematically we must dispel this fear by convergence tests It is unfortunate that the required uniform convergence cannot always be guaranteed and that because of this increased mathematical machinery must be constructed and applied The short story is that the methods work quite well and shouldn7t keep you up at night We have similar interpretations of this as we did with Fourier series Namely we represent the function f as a linear combination of trigonometric functions of varying frequency The weights of this linear combination are given by integrating the function f and are again thought of as amplitudes information for each oscillatory mode The major difference we see here is that since the function is more complicated in that it does not have the additional structure of periodicity we are required to use every possible frequency of oscillation for the trigonometric functions This results in the use of a continuous in nite sum as opposed to a discretely in nite sum in the case of Fourier series ne can recover the Fourier series as a special case of the Fourier transform by using the soicalled Dirac delta function7 but this is a cheat since it really isn7t a function at all and exists so that we can make integrals perform tricks we desire MATH348 Advanced Eu lllccllll M quot 0 Symmetry Arguments Lastly we have similar symmetry arguments With Fourier series we have5 ix Z 80006in 10 2 10 coswx ix Z Cwnkiw 2 MM sinwnx While for Fourier transforms we have the following statements fem we lt2 few u we V w mam Ew fltwgtcosltwxgtdw fem 7M lt2 few 71 we 00 mam E fltwgtsinltmgtw The takehome message is that if a function has evenodd symmetry then the complex Fourier series reduces to a Fourier cosinesine series while the complex Fourier transform reduces to a Fourier cosinesine transform 6 2 LECTURE GOALS Our goals with this material will be 0 Construct from the Fourier series an integral representation for functions which may not have a periodic structure 0 Understand the analogies between the Fourier series and Fourier integral representation of a func tion 3 LECTURE OBJECTIVES The objectives of these lessons will be 0 Consider the consequences of the limit L gt 00 on the Fourier series as a means to derive the Fourier integral 0 From the Fourier integral derive the Fourier transform 0 Compare and contrast the symmetry and spectral decomposition properties of Fourier transform to complex Fourier series 5There is a fair amount of algebra needed to show these statements However if we trust that complex Fourier series are equivalent to real Fourier series which we should then these equalities MUST be true 6Note that these statements de ne the inverse transforms taking amplitude data and representing f as a linear combination of oscillatory modes Symmetry arguments are needed to simplify the forward transforms
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