Introduction to Applied Mathematics
Introduction to Applied Mathematics MA 325
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This 6 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 325 at North Carolina State University taught by Robert White in Fall. Since its upload, it has received 8 views. For similar materials see /class/223679/ma-325-north-carolina-state-university in Mathematics (M) at North Carolina State University.
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Date Created: 10/15/15
Lecture 5 Complex Numbers and Fourier Transforms The imaginary number i sqrt1 is not a real number because 1392 1 is not positive The complex numbers are the set whose elements a bi where a and b are any real numbers called the real and imaginary parts of a bi One can view them as vectors in the complex plane where a is on the horizontal axis and b is on the vertical axis Then define addition of two complex numbers as vector addition the modulus of the complex number as the length of the vector and the conjugate of the complex number as a complex number corresponding to the re ection about the horizontal axis a ci bd cid modulus ofaib sq rtaquot2bquot2 The product and division of numbers are de ned by using 1392 1 aibcid E ac bd iadbc and aibcid E aibcid cidcid ac bd c2 612 iadbc c2 612 Using these de nitions one can compute more complicated functions of complex variables A very important function is the exponential function and we use its series representation e lz39x1l ix22ix33ix44 lix22ix44 z39x1l my3 cosx 139 sinx Euler39sFormula The modulus of e is equal to one because sin2x 0032x 1 Also x can be viewed as the angle from the real aXis to the vector representing the complex number e i2 7rn i2 7quot An important case is when x 27W so that w Ee and z E w e satisfy w lz wk w39kcmallww2wquot391 0 The last property is established by using wquot 1 and noting w is not equal to l wlww2 w 391 ww2 wquot ww2 wquot391 1 w ll w w2 w 030 thatforw lnot equal tozero lww2wquot3910 The compleX numbers wj with j 0n 1 are called the rim roots of unity because they solve x 1 For n 3 they the following vectors whose vector sum is zero ei21t32 Application of Complex Numbers to Laplace Transform Consider the Laplace transform if the complex exponential function em Here we make the natural extension to complex integrals By Euler s formula Lg Lc0sbt i Sinbt Lc0sbtiLSinbt39 Apply the definition of Laplace transform directly to em 81172 J39ers texbtdt 0 N limNgoo Ie39ste b39dt 0 N limmw j army 0 lim ersxl7N 1 ersxl70 1 NT sz39b sz39b 1 s ib 1 sib s ib sib s 1 32 b2 32 b2 By equating the real and imagery parts of Leibt we have derived the Laplace transforms for the sine and cosine function These rules could also have been derived by the direct application of the definition of the Laplace transform to the sine and cosine functions The Fourier Transform The Fourier transform of a function ft could be viewed as a variation on the Laplace transform where the real parameter s is replaced by an imaginary parameter iw and the domain of integration is extended to the entire real line Def39mtion The Fourier transform off is Ff j equot ftdt Basic Properties of Fourier Transform FCf CF Ff g Ff Fg Ff z39wFfand Ffg FfFgwhere fg j frgtrdr 59 H Application to Solution of uxx cu 139 Replace the variable t with x and compute the Fourier transform of both sides in the differential equation By using the third property the derivative property twice we obtain FumcuFf ia2Fu cFu F f W2 cFuFf Fu Ff 1 w2c Apply the fourth property the convolution property to conclude ufg where Fg a22c Discrete Fourier Transform The discrete Fourier transform is derived by truncating the integral to 0 1 replacing w by 27k and letting fjn The discrete Fourier transform of f fg H124 is complex vector whose k1h component is r171 r171 ea27rkjnf Iannle z j0 Another way to view this is as a matrix product from vectors in real n dimensional space to vectors in complex n dimensional space Ff where F has kj component 21 For example ifn 3 then 20 20 20 l l 1 F 0 1 2 1 21 22 0 22 4 1 22 1 Iff 1 72 n 3 and2 e42 12 132139 then 1 1 1 1 10 10 Ff 1 21 22 7 l72222 350 433i 2 l72222 350433i 122 21 In applications 71 is typically very large and an efficient method to do these computations is required In Matlab the command fft is an implementation of the fast Fourier transform This requires W2log2n operations which is much less than the 2112 operations for the matrixvector product The following are some simple examples gtgtfft172 n3 100 35000 43301i 35000 43301i Matlab Code fftt gm t 00011 n 1001 freq 111001 fftsin f 4sin2pi40t fftcos f 8cos2pi100t subp10t211 p10tfreqrealf sinfreqrea1f cos subp10t2 12 p10tfreqimagf sinfreqimagf cos The spikes in the Fourier transform plots identify the frequencies f and n f 4000 t real t cos 3000 0 2000 7 1000 1000 W 600 800 W 1000 frequency 1200 2000 t t 10007 J imaQUTKSinOD JI 0 V 1000 7 2000 W 600 800 W 1000 frequency 1200
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