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{TEXT 461 36 "Periodic Functions, Fourier Series, " }{TEXT 460 0 "" }} {PARA 227 "" 0 "" {TEXT 461 2 " " }{TEXT 461 34 " and Half-Ra nge Expansions" }{MPLTEXT 1 462 2 " " }{TEXT 460 0 "" }}{PARA 228 "" 0 "" {TEXT 463 0 "" }}{PARA 228 "" 0 "" {TEXT 463 220 "In this workshe et we introduce several Maple procedures for computing Fourier series \+ of periodic functions and plotting them. The programs are organized in the following sections (click on each + to view the contents). " } {TEXT 463 0 "" }}{PARA 228 "" 0 "" {TEXT 463 0 "" }}{SECT 1 {PARA 229 "" 0 "" {TEXT 464 1 " " }{TEXT 465 28 "0. Summary of the Procedures" } {TEXT 464 0 "" }}{PARA 230 "" 0 "" {TEXT 466 11 "Section 1. " }{TEXT 467 17 "ext ( f, a, b ) " }{TEXT 466 30 "plots the graph of a functio n " }{TEXT 467 2 "f " }{TEXT 466 12 " defined on " }{TEXT 467 5 "[a,b] " }{TEXT 466 45 ", when extended to the real line with period " } {TEXT 467 5 "b - a" }{TEXT 466 2 ". " }{TEXT 466 0 "" }}{PARA 231 "" 0 "" {TEXT 468 0 "" }}{PARA 230 "" 0 "" {TEXT 466 11 "Section 2. " } {TEXT 467 22 "Fourier ( f, a, b, N )" }{TEXT 466 32 " computes the su m of the first " }{TEXT 467 2 " N" }{TEXT 466 45 " terms of the Fouri er series of a function " }{TEXT 467 3 "f " }{TEXT 466 11 "defined o n " }{TEXT 467 5 "[a,b]" }{TEXT 466 47 " , when extended to the real l ine with period " }{TEXT 467 5 "b - a" }{TEXT 466 1 "." }{TEXT 466 0 "" }}{PARA 231 "" 0 "" {TEXT 468 0 "" }}{PARA 230 "" 0 "" {TEXT 466 11 "Section 3. " }{TEXT 467 16 "evenext ( f, L )" }{TEXT 466 5 " and " }{TEXT 467 15 "oddext ( f, L )" }{TEXT 466 73 " plot the graphs of th e even and odd half-range expansions of a function " }{TEXT 467 1 "f" }{TEXT 466 24 " defined originally on " }{TEXT 467 5 "[0,L]" }{TEXT 466 14 ", with period " }{TEXT 467 2 "2L" }{TEXT 466 1 "." }{TEXT 466 0 "" }}{PARA 231 "" 0 "" {TEXT 468 0 "" }}{PARA 230 "" 0 "" {TEXT 466 11 "Section 4. " }{TEXT 467 23 "Fourier_cos ( f, L, N )" }{TEXT 466 5 " and " }{TEXT 467 23 "Fourier_sin ( f, L, N )" }{TEXT 466 30 " comput e the sum of the first " }{TEXT 467 2 " N" }{TEXT 466 69 " terms of t he Fourier cosine and Fourier sine series of a function " }{TEXT 467 3 "f " }{TEXT 466 11 "defined on " }{TEXT 467 5 "[0,L]" }{TEXT 466 46 " , when extended to the real line with period " }{TEXT 467 2 "2L" }{TEXT 466 1 "." }{TEXT 466 0 "" }}}{SECT 1 {PARA 229 "" 0 "" {TEXT 464 1 " " }{TEXT 465 44 "1. Plotting Periodic Extensions of Functions" }{TEXT 464 0 "" }}{PARA 230 "" 0 "" {TEXT 466 23 "Recall that a funct ion " }{TEXT 467 1 "f" }{TEXT 466 21 " on the real line is " }{TEXT 467 8 "periodic" }{TEXT 466 13 " with period " }{TEXT 467 1 "p" } {TEXT 466 4 " if " }{TEXT 467 12 " f(x+p)=f(x)" }{TEXT 466 10 " for a ll " }{TEXT 467 2 "x " }{TEXT 466 7 "and if " }{TEXT 467 1 "p" }{TEXT 466 56 " is the smallest positive number with this property. If " } {TEXT 467 2 " f" }{TEXT 466 47 " is any given function defined on an \+ interval " }{TEXT 467 5 "[a,b]" }{TEXT 466 11 " of length " }{TEXT 467 9 "b - a = p" }{TEXT 466 33 ", there is a unique extension of " } {TEXT 467 2 " f" }{TEXT 466 35 " to a periodic function of period " } {TEXT 467 1 "p" }{TEXT 466 146 " on the entire real line. Our first pr ocedure allows us to sketch the graph of this periodic extension over \+ three consecutive intervals of length " }{TEXT 467 1 "p" }{TEXT 466 132 ". We will use this later when we want to compare the graph of thi s periodic extension with that of the truncated Fourier series of " } {TEXT 467 1 "f" }{TEXT 466 119 ". By a simple modification of this pro gram, you can plot the periodic extension over any number of intervals of length " }{TEXT 467 1 "p" }{TEXT 466 16 " that you wish. " }{TEXT 466 0 "" }}{PARA 230 "" 0 "" {TEXT 466 40 "The call sequence for this \+ procedure is " }{TEXT 467 15 "ext( f , a, b )" }{TEXT 466 9 " . Here \+ " }{TEXT 467 1 "f" }{TEXT 466 51 " is a function originally defined o n the interval " }{TEXT 467 5 "[a,b]" }{TEXT 466 59 ", and the output \+ is the graph of the periodic extension of " }{TEXT 467 3 "f " }{TEXT 466 10 "of period " }{TEXT 467 5 "b - a" }{TEXT 466 2 " :" }{TEXT 466 0 "" }}{PARA 232 "" 0 "" {TEXT 469 1 " " }{TEXT 469 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 8 "restart;" }{MPLTEXT 1 470 0 "" } }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 16 "ext:=proc(f,a,b)" } {MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 18 "local f1, f2,ff,p; " }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 7 "p:=b-a:" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 37 "f1:=x->Heaviside(x-a)-Heaviside(x-b):" }{MPLTEXT 1 470 0 "" }} {PARA 233 "> " 0 "" {MPLTEXT 1 470 18 "f2:=x->f(x)*f1(x):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 29 "ff:=x->f2(x-p)+f2( x)+f2(x+p):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 34 "plot(ff(x),x=a-p..b+p,color=blue);" }{MPLTEXT 1 470 0 "" }} {PARA 233 "> " 0 "" {MPLTEXT 1 470 4 "end:" }{MPLTEXT 1 470 0 "" }}} {PARA 228 "" 0 "" {TEXT 463 30 "Look at the following example:" } {TEXT 463 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 13 "f:=x->- x^2+1:" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 12 " ext(f,-2,2);" }{MPLTEXT 1 470 0 "" }}}{PARA 234 "" 0 "" {TEXT 471 24 " Here is another example:" }{TEXT 471 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 13 "f:=x->sin(x):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 17 "ext(f,Pi,3*Pi/2);" }{MPLTEXT 1 470 0 "" }}} {PARA 235 "" 0 "" {TEXT 472 49 "Try this procedure on a few examples o f your own!" }{TEXT 472 0 "" }}}{SECT 1 {PARA 229 "" 0 "" {TEXT 464 1 " " }{TEXT 465 49 "2. Fourier Series of Arbitrary Periodic Functions" }{TEXT 464 0 "" }}{PARA 230 "" 0 "" {TEXT 466 54 "In Fourier Analysis \+ we learn that a periodic function " }{TEXT 467 1 "f" }{TEXT 466 11 " o f period " }{TEXT 467 1 "p" }{TEXT 466 7 " has a " }{TEXT 467 14 "Four ier series" }{TEXT 466 13 " of the form " }{TEXT 466 0 "" }}{PARA 236 "" 0 "" {XPPEDIT 18 0 "Ao/2+Sum(An*cos(2*Pi*n*x/p)+Bn*sin(2*Pi*n*x/p), n = 1 .. infinity);" "6#,&*&%#AoG\"\"\"\"\"#!\"\"F&-%$SumG6$,&*&%#AnGF &-%$cosG6#*,F'F&%#PiGF&%\"nGF&%\"xGF&%\"pGF(F&F&*&%#BnGF&-%$sinGF1F&F& /F4;F&%)infinityGF&" }{TEXT 473 0 "" }}{PARA 230 "" 0 "" {TEXT 466 23 "where the coefficients " }{TEXT 467 2 "An" }{TEXT 466 5 " and " } {TEXT 467 2 "Bn" }{TEXT 466 27 " are given by the formulas " }{TEXT 466 0 "" }}{PARA 236 "" 0 "" {XPPEDIT 18 0 "An = 2*Int(f(x)*cos(2*Pi*n *x/p),x = 0 .. p)/p;" "6#/%#AnG*(\"\"#\"\"\"-%$IntG6$*&-%\"fG6#%\"xGF' -%$cosG6#*,F&F'%#PiGF'%\"nGF'F/F'%\"pG!\"\"F'/F/;\"\"!F6F'F6F7" } {TEXT 473 0 "" }}{PARA 236 "" 0 "" {XPPEDIT 18 0 "Bn = 2*Int(f(x)*sin( 2*Pi*n*x/p),x = 0 .. p)/p;" "6#/%#BnG*(\"\"#\"\"\"-%$IntG6$*&-%\"fG6#% \"xGF'-%$sinG6#*,F&F'%#PiGF'%\"nGF'F/F'%\"pG!\"\"F'/F/;\"\"!F6F'F6F7" }{TEXT 473 0 "" }}{PARA 231 "" 0 "" {TEXT 468 0 "" }}{PARA 230 "" 0 "" {TEXT 466 79 "Here is a procedure for general Fourier series computat ions. Its call sequence " }{TEXT 467 11 "Fourier ( f" }{TEXT 466 1 " " }{TEXT 467 12 ", a, b, N ) " }{TEXT 466 31 "gives you the sum of the \+ first " }{TEXT 467 2 " N" }{TEXT 466 45 " terms of the Fourier series of a function " }{TEXT 467 1 "f" }{TEXT 466 37 " originally defined on the interval " }{TEXT 467 5 "[a,b]" }{TEXT 466 60 ", extended peri odically to the entire real line with period " }{TEXT 467 10 "p = b - \+ a." }{TEXT 466 1 " " }{TEXT 466 0 "" }}{PARA 237 "" 0 "" {TEXT 474 1 " " }{TEXT 474 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 24 "Fou rier := proc(f,a,b,N)" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 41 "local n, p; global Series, An, Bn, A0, x;" } {MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 7 "p:=b-a:" } {MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 20 "for n fro m 1 to N do" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 52 " An(n):= (2/p)*int( f(x)*cos(2*Pi*n*x/p), x=a..b );" } {MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 52 " Bn(n):= (2/p)*int( f(x)*sin(2*Pi*n*x/p), x=a..b );" }{MPLTEXT 1 470 0 "" }} {PARA 233 "> " 0 "" {MPLTEXT 1 470 4 "od: " }{MPLTEXT 1 470 0 "" }} {PARA 233 "> " 0 "" {MPLTEXT 1 470 32 "A0:= (1/p)*int( f(x) , x=a..b ) ;" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 89 "Serie s := A0 + convert( [seq(An(n)*cos(n*2*Pi*x/p)+Bn(n)*sin(n*2*Pi*x/p), n =1..N)], `+`);" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 4 "end:" }{MPLTEXT 1 470 0 "" }}}{PARA 226 "" 0 "" {TEXT 458 1 " " }{TEXT 475 54 "Let us first test this procedure in two trivial cases: " }{TEXT 458 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 15 "f:=x ->cos(2*x):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 18 "Fourier(f,0,Pi,4);" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 " > " 0 "" {MPLTEXT 1 470 8 "f:=x->1:" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 19 "Fourier(f,-1,1,10);" }{MPLTEXT 1 470 0 "" }}}{PARA 238 "" 0 "" {TEXT 476 78 "Both answers are what we should ha ve expected. Here is a less trivial example:" }{TEXT 476 0 "" }} {EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 14 "f:=x->exp(-x):" } {MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 18 "Fourier(f ,-1,1,4);" }{MPLTEXT 1 470 0 "" }}}{PARA 239 "" 0 "" {TEXT 477 85 "You can also see the coefficients in floating-point form, if that is what you prefer:" }{TEXT 477 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 14 "evalf(Series);" }{MPLTEXT 1 470 0 "" }}}{PARA 230 "" 0 "" {TEXT 466 63 "Let us plot this partial sum as well as the periodic ext ension " }{TEXT 466 4 "of " }{TEXT 467 12 "f = exp(-x) " }{TEXT 466 2 "on" }{TEXT 467 7 " [-1,1]" }{TEXT 466 91 " on the same coordinate a xes (we use the procedure \"ext\" we wrote in the previous section):" }{TEXT 466 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 12 "with(p lots):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 25 " g1:=plot(Series,x=-3..3):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 16 "g2:=ext(f,-1,1):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 17 "display(\{g1,g2\});" }{MPLTEXT 1 470 0 "" }}}{PARA 228 "" 0 "" {TEXT 463 276 "To make things a little more exci ting, let us try to see the convergence of the partial sums of the Fou rier series toward the function by animating it (the first line below \+ takes some time to execute; do not forget to play the animation after \+ you execute the display command):" }{TEXT 463 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 73 "for N from 0 to 30 do Fourier(f,-1,1,N): \+ G(N):=plot(Series, x=-3..3): od:" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 47 "g1:=display(seq(G(N),N=0..30),i nsequence=true):" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 17 "display(\{g1,g2\});" }{MPLTEXT 1 470 0 "" }}}{PARA 230 "" 0 "" {TEXT 466 67 "As can be seen, the Fourier series (in red) \+ converges pointwise to " }{TEXT 467 1 "f" }{TEXT 466 257 " (in blue) \+ except at the points of discontinuity where it converges to the averag e of the right and left limits. There you can also observe an \"oversh oot\" in the Fourier series that does not go away by adding more terms to the series. This is known as the " }{TEXT 467 17 "Gibbs phenomenon " }{TEXT 466 61 "and will be discussed in the last section of this wo rksheet. " }{TEXT 466 0 "" }}}{SECT 1 {PARA 229 "" 0 "" {TEXT 464 1 " \+ " }{TEXT 465 33 "3. Plotting Half-Range Expansions" }{TEXT 464 0 "" }} {PARA 230 "" 0 "" {TEXT 466 5 "Let " }{TEXT 467 1 "f" }{TEXT 466 29 " be defined on the interval " }{TEXT 467 5 "[0,L]" }{TEXT 466 16 ". W e can extend " }{TEXT 467 1 "f" }{TEXT 466 55 " either as an even or \+ an odd function to the interval " }{TEXT 467 6 "[-L,L]" }{TEXT 466 67 " and then to the entire real line as a periodic function of period " }{TEXT 467 2 "2L" }{TEXT 466 19 ". These are called " }{TEXT 467 21 "h alf-range expansions" }{TEXT 466 3 " of" }{TEXT 467 3 " f " }{TEXT 466 44 ". The resulting Fourier series contain only " }{TEXT 467 4 "co s " }{TEXT 466 4 "and " }{TEXT 467 3 "sin" }{TEXT 466 70 " terms, resp ectively. Here we write two procedures with call sequences" }{TEXT 467 17 " evenext( f , L )" }{TEXT 466 5 " and " }{TEXT 467 15 "oddext( f , L )" }{TEXT 466 56 " to sketch these even or odd extensions on th e interval " }{TEXT 467 8 "[-3L,3L]" }{TEXT 466 41 ". They are very si milar to the procedure " }{TEXT 467 15 "ext( f , a, b )" }{TEXT 466 15 " in section 1. " }{TEXT 466 0 "" }}{PARA 231 "" 0 "" {TEXT 468 0 " " }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 18 "evenext:=proc(f,L)" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 16 "local f1 ,f2,fe; " }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 35 "f1:=x->Heaviside(x)-Heaviside(x-L):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 18 "f2:=x->f(x)*f1(x):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 62 "fe:=x->f2(x)+f2(x-2*L)+f 2(x+2*L)+f2(-x)+f2(-x-2*L)+f2(-x+2*L):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 35 "plot(fe(x),x=-3*L..3*L,color=blue);" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 4 "end:" } {MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 17 "o ddext:=proc(f,L)" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 16 "local f1,f2,fo; " }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 35 "f1:=x->Heaviside(x)-Heaviside(x-L):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 18 "f2:=x->f(x)*f1(x):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 62 "fo:=x->f 2(x)+f2(x-2*L)+f2(x+2*L)-f2(-x)-f2(-x-2*L)-f2(-x+2*L):" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 35 "plot(fo(x),x=-3*L..3 *L,color=blue);" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 4 "end:" }{MPLTEXT 1 470 0 "" }}}{PARA 230 "" 0 "" {TEXT 466 64 "Here is the first example, the two different ways of extending " } {TEXT 467 11 "f(x)=exp(x)" }{TEXT 466 17 " on the interval " }{TEXT 467 5 "[0,1]" }{TEXT 466 1 "." }{TEXT 466 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 13 "f:=x->exp(x):" }{MPLTEXT 1 470 0 "" }}} {EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 13 "evenext(f,1);" } {MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 12 "o ddext(f,1);" }{MPLTEXT 1 470 0 "" }}}{PARA 230 "" 0 "" {TEXT 466 32 "A s the second example, consider " }{TEXT 467 14 "f(x)=1/(x^2+1)" } {TEXT 466 17 " on the interval " }{TEXT 467 5 "[0,2]" }{TEXT 466 1 ":" }{TEXT 466 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 16 "f:=x- >1/(x^2+1):" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 13 "evenext(f,2);" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 12 "oddext(f,2);" }{MPLTEXT 1 470 0 "" }}}{PARA 230 "" 0 "" {TEXT 466 38 "Let us also use \"ext\" on the interval " }{TEXT 467 6 "[-2,2]" }{TEXT 466 2 " :" }{TEXT 466 0 "" }} {EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 12 "ext(f,-2,2);" }{MPLTEXT 1 470 0 "" }}}{PARA 228 "" 0 "" {TEXT 463 46 "Why are the first and th e last plots the same?" }{TEXT 463 0 "" }}}{SECT 1 {PARA 229 "" 0 "" {TEXT 464 1 " " }{TEXT 465 33 "4. Fourier Cosine and Sine Series" } {TEXT 464 0 "" }}{PARA 230 "" 0 "" {TEXT 466 116 "Let us now write pro cedures to compute Fourier series of half-range expansions defined in \+ the previous section. Let " }{TEXT 467 2 "f " }{TEXT 466 26 " be a fun ction defined on " }{TEXT 467 7 "[0,L]. " }{TEXT 466 34 "The even half -range expansion of " }{TEXT 467 1 "f" }{TEXT 466 42 " has a Fourier cosine series of the form " }{TEXT 466 0 "" }}{PARA 240 "" 0 "" {XPPEDIT 18 0 "Ao/2+Sum(An*cos(Pi*n*x/L),n = 1 .. infinity);" "6#,&*&% #AoG\"\"\"\"\"#!\"\"F&-%$SumG6$*&%#AnGF&-%$cosG6#**%#PiGF&%\"nGF&%\"xG F&%\"LGF(F&/F3;F&%)infinityGF&" }{TEXT 475 1 " " }{TEXT 458 0 "" }} {PARA 230 "" 0 "" {TEXT 466 23 "where the coefficients " }{TEXT 467 2 "An" }{TEXT 466 13 " are given by" }{TEXT 466 0 "" }}{PARA 236 "" 0 "" {XPPEDIT 18 0 "An = 2*Int(f(x)*cos(Pi*n*x/L),x = 0 .. L)/L;" "6#/%#An G*(\"\"#\"\"\"-%$IntG6$*&-%\"fG6#%\"xGF'-%$cosG6#**%#PiGF'%\"nGF'F/F'% \"LG!\"\"F'/F/;\"\"!F6F'F6F7" }{TEXT 473 0 "" }}{PARA 231 "" 0 "" {TEXT 468 0 "" }}{PARA 226 "" 0 "" {TEXT 475 43 "Similarly, the odd ha lf-range expansion of " }{TEXT 478 2 " f" }{TEXT 475 39 " has a Fouri er sine series of the form" }{TEXT 458 1 " " }{TEXT 458 0 "" }}{PARA 240 "" 0 "" {XPPEDIT 18 0 "Sum(Bn*sin(Pi*n*x/L),n = 1 .. infinity);" " 6#-%$SumG6$*&%#BnG\"\"\"-%$sinG6#**%#PiGF(%\"nGF(%\"xGF(%\"LG!\"\"F(/F .;F(%)infinityG" }{TEXT 458 1 " " }{TEXT 458 0 "" }}{PARA 226 "" 0 "" {TEXT 475 23 "where the coefficients " }{TEXT 478 2 "Bn" }{TEXT 475 13 " are given by" }{TEXT 458 1 " " }{TEXT 458 0 "" }}{PARA 240 "" 0 " " {XPPEDIT 18 0 "Bn = 2*Int(f(x)*sin(Pi*n*x/L),x = 0 .. L)/L;" "6#/%#B nG*(\"\"#\"\"\"-%$IntG6$*&-%\"fG6#%\"xGF'-%$sinG6#**%#PiGF'%\"nGF'F/F' %\"LG!\"\"F'/F/;\"\"!F6F'F6F7" }{TEXT 458 1 " " }{TEXT 458 0 "" }} {PARA 231 "" 0 "" {TEXT 468 0 "" }}{PARA 230 "" 0 "" {TEXT 466 52 "The following are two procedures with call sequences" }{TEXT 467 24 " Fou rier_cos( f , L, N )" }{TEXT 466 5 " and " }{TEXT 467 23 "Fourier_sin( f , L, N )" }{TEXT 466 61 " to compute these series. The output is th e sum of the first " }{TEXT 467 1 "N" }{TEXT 466 72 " terms of the co rresponding series. Here is the Fourier cosine series: " }{TEXT 479 1 " " }{TEXT 480 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 26 "Fo urier_cos := proc(f,L,N)" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 38 "local n; global Series_cos, An, A0, x;" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 20 "for n from 1 to N \+ do" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 50 " An (n):= (2/L)*int( f(x)*cos(n*Pi*x/L), x=0..L );" }{MPLTEXT 1 470 0 "" } }{PARA 233 "> " 0 "" {MPLTEXT 1 470 4 "od: " }{MPLTEXT 1 470 0 "" }} {PARA 233 "> " 0 "" {MPLTEXT 1 470 32 "A0:= (1/L)*int( f(x) , x=0..L ) ;" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 69 "Serie s_cos := A0 + convert( [seq(An(n)*cos(n*Pi*x/L), n=1..N)], `+`);" } {MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 4 "end:" } {MPLTEXT 1 470 0 "" }}}{PARA 241 "" 0 "" {TEXT 481 36 "and here is the Fourier sine series:" }{TEXT 481 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 26 "Fourier_sin := proc(f,L,N)" }{MPLTEXT 1 470 0 "" }} {PARA 233 "> " 0 "" {MPLTEXT 1 470 34 "local n; global Series_sin, Bn, x;" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 20 "for n from 1 to N do" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 50 " Bn(n):= (2/L)*int( f(x)*sin(n*Pi*x/L), x=0..L );" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 4 "od: " } {MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 64 "Series_si n := convert( [seq(Bn(n)*sin(n*Pi*x/L), n=1..N)], `+`);" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 4 "end:" }{MPLTEXT 1 470 0 "" }}}{PARA 242 "" 0 "" {TEXT 482 54 "Let us test these procedur es on a few simple examples:" }{TEXT 482 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 8 "f:=x->5:" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 20 "Fourier_cos(f,1,10);" }{MPLTEXT 1 470 0 "" }} }{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 17 "f:=x->(cos(x))^2:" } {MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 23 "Fourier_c os(f,Pi/2,10);" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 15 "f:=x->sin(3*x):" }{MPLTEXT 1 470 0 "" }}{PARA 233 " > " 0 "" {MPLTEXT 1 470 21 "Fourier_sin(f,Pi,10);" }{MPLTEXT 1 470 0 " " }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 20 "f:=x->sin(x)*cos(x) :" }{MPLTEXT 1 470 0 "" }}{PARA 233 "> " 0 "" {MPLTEXT 1 470 23 "Fouri er_sin(f,Pi/2,10);" }{MPLTEXT 1 470 0 "" }}}{PARA 228 "" 0 "" {TEXT 463 166 "Let us now test them on a less trivial example, a rectangular pulse. We sketch both the Fourier cosine series and the graph of the \+ even extension using the procedure " }{TEXT 483 16 "evenext( f , L )" }{TEXT 463 25 " in the previous section:" }{TEXT 463 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 12 "with(plots):" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 25 "f:=x->2*Heavisid e(x-1)-1:" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 20 "Fourier_cos(f,2,10);" }{MPLTEXT 1 470 0 "" }}} {EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 30 "g1:=plot(Series_cos, x=- 6..6):" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 17 "g2:=evenext(f,2):" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 17 "display(\{g1,g2\});" }{MPLTEXT 1 470 0 "" } }}{PARA 243 "" 0 "" {TEXT 484 93 "Finally, here is an example for the \+ Fourier sine series and plotting the odd extension using " }{TEXT 485 16 "oddext( f , L ) " }{TEXT 484 1 ":" }{TEXT 484 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 14 "f:= x -> -x+1:" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 20 "Fourier_sin(f,1,10);" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 29 "g1:=plot(Series_sin,x=-3..3):" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 16 "g2:=oddext(f,1):" }{MPLTEXT 1 470 0 " " }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 17 "display(\{g1,g2\}); " }{MPLTEXT 1 470 0 "" }}}{PARA 244 "" 0 "" {TEXT 486 168 "Try to come up with your own examples; choose some of the problems you have done \+ by hand before, find the Fourier series using these procedures and com pare the answers." }{TEXT 486 0 "" }}}{SECT 1 {PARA 229 "" 0 "" {TEXT 464 1 " " }{TEXT 465 35 "5. An Example: The Gibbs Phenomenon" }{TEXT 464 0 "" }}{PARA 226 "" 0 "" {TEXT 475 12 "Consider the" }{TEXT 458 1 " " }{TEXT 475 20 "Heaviside function " }{TEXT 478 3 "f " }{TEXT 475 21 "which is identically " }{TEXT 478 1 "0" }{TEXT 475 24 " for ne gative values of " }{TEXT 478 1 "x" }{TEXT 475 17 " and identically " }{TEXT 478 1 "1" }{TEXT 475 38 " otherwise (this is also known as the \+ " }{TEXT 478 18 "unit step function" }{TEXT 475 49 "). Let us restrict this function to the interval " }{TEXT 478 6 "[-1,1]" }{TEXT 475 69 " and then extend it periodically to the entire real line with period " }{TEXT 478 1 "2" }{TEXT 475 31 "; here is the resulting graph: " } {TEXT 458 2 " " }{TEXT 458 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 22 "f:= x -> Heaviside(x):" }{MPLTEXT 1 470 0 "" }}} {EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 12 "ext(f,-1,1);" }{MPLTEXT 1 470 0 "" }}}{PARA 230 "" 0 "" {TEXT 466 71 "Let us compute the 5th a nd 15th partial sums of the Fourier series of " }{TEXT 467 1 "f" } {TEXT 466 2 " :" }{TEXT 466 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 18 "Fourier(f,-1,1,5);" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 19 "Fourier(f,-1,1,15);" }{MPLTEXT 1 470 0 "" }}}{PARA 230 "" 0 "" {TEXT 466 82 "Looking at the above pat tern, it is not hard to guess that the Fourier series of " }{TEXT 467 1 "f" }{TEXT 466 14 " has the form" }{TEXT 466 0 "" }}{PARA 240 " " 0 "" {XPPEDIT 18 0 "1/2+2*Sum(sin((2*n-1)*Pi*x)/(2*n-1),n = 1 .. inf inity)/Pi;" "6#,&*&\"\"\"F%\"\"#!\"\"F%*(F&F%-%$SumG6$*&-%$sinG6#*(,&* &F&F%%\"nGF%F%F%F'F%%#PiGF%%\"xGF%F%F1F'/F3;F%%)infinityGF%F4F'F%" } {TEXT 458 1 " " }{TEXT 458 0 "" }}{PARA 230 "" 0 "" {TEXT 466 87 "You \+ can easily check this mathematically. As we know, this Fourier series \+ converges to " }{TEXT 467 1 "f" }{TEXT 466 63 " everywhere except at t he integer points where it converges to " }{TEXT 467 3 "0.5" }{TEXT 466 89 ". There is an \"overshoot\" near the points of discontinuity, \+ which is usually referred to " }{TEXT 466 7 "as the " }{TEXT 467 18 "G ibbs phenomenon. " }{TEXT 466 52 "To show the mechanism of this phenom enon near, say, " }{TEXT 467 3 "x=0" }{TEXT 466 315 ", we can of cours e proceed as in the last example of section 2. However, knowing the ex plicit form of the series allows us to animate as many plots as we wan t much more quickly. The following 200 plots would have taken signific antly longer to animate using the old method. Play the animation after you get the plot." }{MPLTEXT 1 487 1 " " }{TEXT 466 0 "" }}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 80 "S:= N -> 0.5 + (2/Pi)* convert( [seq(1/(2*n-1)*sin((2*n-1)*Pi*x), n=1..N)], `+`);" }{MPLTEXT 1 470 0 " " }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 63 "g1:=display(seq(plo t(S(N),x=0..0.1),N=1..200),insequence=true):" }{MPLTEXT 1 470 0 "" }}} {EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 32 "g2:=plot(1,x=0..0.1,colo r=blue):" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 12 "with(plots):" }{MPLTEXT 1 470 0 "" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 470 17 "display(\{g1,g2\});" }{MPLTEXT 1 470 0 "" }}} {PARA 230 "" 0 "" {TEXT 466 146 "As you can see, the overshoot never d ies but it gets pushed towards the point of discontinuity. The amount \+ of the overshoot is seen to be around " }{TEXT 467 4 "0.08" }{TEXT 466 45 ". The actual limiting amount, which is about " }{TEXT 467 7 "0 .08948" }{TEXT 466 53 ", can be computed using Maple. Can you find out how? " }{TEXT 466 0 "" }}}{PARA 245 "" 0 "" {TEXT 488 0 "" }}{PARA 246 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }