{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 28 "Module 6: Fourier Expansions" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "Here is the sine series on the interval [ 0 ,L]:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " The sine Fourier approximation: " }{XPPEDIT 18 0 "sum(b[n]*sin(n*p i*x/L),n = 1 .. N);" "6#-%$sumG6$*&&%\"bG6#%\"nG\"\"\"-%$sinG6#**F*F+% #piGF+%\"xGF+%\"LG!\"\"F+/F*;F+%\"NG" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 11 " where " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "int(f(x)*sin(n*Pi*x/L),x = 0 .. L)/in t(sin(n*Pi*x/L)^2,x = 0 .. L);" "6#*&-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%$ sinG6#**%\"nGF,%#PiGF,F+F,%\"LG!\"\"F,/F+;\"\"!F3F,-F%6$*$-F.6#**F1F,F 2F,F+F,F3F4\"\"#/F+;F7F3F4" }{TEXT -1 8 " , for " }{XPPEDIT 18 0 "1 < = n;" "6#1\"\"\"%\"nG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Here \+ is the cosine series on the interval [0 ,L]:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The cosine Fourier approximatio n: " }{XPPEDIT 18 0 "sum(a[n]*cos(n*pi*x/L),n = 0 .. N);" "6#-%$sumG6$ *&&%\"aG6#%\"nG\"\"\"-%$cosG6#**F*F+%#piGF+%\"xGF+%\"LG!\"\"F+/F*;\"\" !%\"NG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 " where " } {XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(f(x)*cos(n*Pi*x/L),x = 0 .. L)/int(cos(n*Pi*x/L)^2,x = 0 .. \+ L);" "6#*&-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"nGF,%#PiGF,F+F,% \"LG!\"\"F,/F+;\"\"!F3F,-F%6$*$-F.6#**F1F,F2F,F+F,F3F4\"\"#/F+;F7F3F4 " }{TEXT -1 8 " , for " }{XPPEDIT 18 0 "1 <= n;" "6#1\"\"\"%\"nG" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(f(x),x = 0 .. L)/int(1^2,x = 0 .. L);" "6#*&-%$intG6$-%\"fG6#%\"xG/F*;\"\"!%\"LG\"\"\"-F%6$*$F/\"\"#/F*;F-F.! \"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We illustrate these expansions with several functions. " }}{PARA 0 "" 0 "" {TEXT -1 13 "For f(x) = x:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "L:=1;\nN:=3;\nf:=x->x;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "b:=n->evalf(int(sin(n*Pi*x/L )*f(x),x=0..L)/\n int(sin(n*Pi*x/L)^2,x=0..L));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "approxs:=x->sum('b(n)*sin(n* Pi*x/L)',n=1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "a:=n->if n = 0 then evalf(i nt(f(x),x=0..L)/int(1^2,x=0..L))\nelse evalf(int(cos(n*Pi*x/L)*f(x),x= 0..L)/int(cos(n*Pi*x/L)^2,x=0..L))\nfi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "approxc:=x->sum('a(n)*cos(n*Pi*x/L)','n'=0..N);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "plot([[x,f(x),x=0..L],[x,app roxs(x),x=-L..L],[x,approxc(x),x=-L..L]],\n color=[BLACK,RED,GREEN]) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "For f(x) = Heaviside(x-1/ 2):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "L:=1;\nN:=3;\nf:=x->piecewise(0<=x and x<=L/2,0,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "b:=n->evalf(int(sin(n*Pi* x/L)*f(x),x=0..L)/int(sin(n*Pi*x/L)^2,x=0..L));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "approxs:=x->sum('b(n)*sin(n*Pi*x/L)',n=1..N); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 122 "a:=n->if n = 0 then evalf(1/L*int(f(x),x=0..L ))\nelse evalf(int(cos(n*Pi*x/L)*f(x),x=0..L)/int(cos(n*Pi*x/L)^2,x=0. .L))\nfi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "approxc:=x->su m('a(n)*cos(n*Pi*x/L)','n'=0..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "plot([[x,f(x),x=0..L],[x,approxs(x),x=-L..L],[x,appro xc(x),x=-L..L]],\n color=[BLACK,RED,GREEN]);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "convert(Heaviside(x-1/2),piecewise);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Here is the Fourier trigonometr ic series on [-L, L]:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " with " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "int(f(x)*cos(n*Pi*x/L),x = -L .. L)/i nt(cos(n*Pi*x/L)^2,x = -L .. L);" "6#*&-%$intG6$*&-%\"fG6#%\"xG\"\"\"- %$cosG6#**%\"nGF,%#PiGF,F+F,%\"LG!\"\"F,/F+;,$F3F4F3F,-F%6$*$-F.6#**F1 F,F2F,F+F,F3F4\"\"#/F+;,$F3F4F3F4" }{TEXT -1 8 " , for " }{XPPEDIT 18 0 "1 <= n;" "6#1\"\"\"%\"nG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "a [0];" "6#&%\"aG6#\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(f(x),x = -L .. L)/int(1^2,x = -L .. L);" "6#*&-%$intG6$-%\"fG6#%\"xG/F*;,$%\"L G!\"\"F.\"\"\"-F%6$*$F0\"\"#/F*;,$F.F/F.F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " and " } {XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(f(x)*sin(n*Pi*x/L),x = -L .. L)/int(sin(n*Pi*x/L)^2,x = -L . . L);" "6#*&-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#**%\"nGF,%#PiGF,F+F ,%\"LG!\"\"F,/F+;,$F3F4F3F,-F%6$*$-F.6#**F1F,F2F,F+F,F3F4\"\"#/F+;,$F3 F4F3F4" }{TEXT -1 8 " , for " }{XPPEDIT 18 0 "1 <= n;" "6#1\"\"\"%\"n G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "We i llustrate this expansion with several functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "For f(x) = |x|." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "L: =1;\nN:=10;\nf:=x->abs(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "a:=n->if n = 0 then evalf(int(f(x),x=-L..L)/(2*L))\nelse evalf(in t(cos(n*Pi*x/L)*f(x),x=-L..L)/\n int(cos(n*Pi*x /L)^2,x=-L..L))\nfi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "b:= n->evalf(int(sin(n*Pi*x/L)*f(x),x=-L..L)/\n int(sin( n*Pi*x/L)^2,x=-L..L));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "a pprox:=x->sum('a(n)*cos(n*Pi*x/L)','n'=0..N) + sum('b(n)*sin(n*Pi*x/L) ','n'=1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot([[x,f( x),x=-L..L],[x,approx(x),x=-3*L/2..3*L/2]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For f(x) = | sin( " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT -1 0 "" }{TEXT -1 6 " x) |:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "L:=1;\nN:=10;\nf:=x->abs(sin(Pi*x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "a:=n->if n = 0 then evalf(int(f(x) ,x=-L..L)/int(1^2,x=-L..L))\nelse evalf(int(cos(n*Pi*x/L)*f(x),x=-L..L )/\n int(cos(n*Pi*x/L)^2,x=-L..L))\nfi;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 86 "b:=n->evalf(int(sin(n*Pi*x/L)*f(x),x=-L..L)/\n int(sin(n*Pi*x/L)^2,x=-L..L));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 83 "approx:=x->sum('a(n)*cos(n*Pi*x/L)','n'=0..N) + sum ('b(n)*sin(n*Pi*x/L)','n'=1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot([[x,f(x),x=-L..L],[x,approx(x),x=-3*L/2..3*L/2]] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 28 "For f(x) = 2*Heaviside(x)-1:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "L:=1;\nN:=3;\nf:=x->piecewise(x<=0,-1,1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "a:=n->if n = 0 then evalf(i nt(f(x),x=-L..L)/int(1^2,x=-L..L))\nelse evalf(int(cos(n*Pi*x/L)*f(x), x=-L..L)/\n int(cos(n*Pi*x/L)^2,x=-L..L))\nfi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "b:=n->evalf(int(sin(n*Pi*x/L)*f(x), x=-L..L)/\n int(sin(n*Pi*x/L)^2,x=-L..L));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "approx:=x->sum('a(n)*cos(n*Pi*x/L)','n'=0 ..N) + sum('b(n)*sin(n*Pi*x/L)','n'=1..N);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "plot([[x,f(x),x=-L..L],[x,approx(x),x=-3*L/2..3*L/2 ]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "F or f(x) as indicated below." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "L:=1;\nN:=3;\nf:=x->piecewise(x<=-1/2,2*(x+1),-1/2 " 0 "" {MPLTEXT 1 0 148 "a:=n->if n = \+ 0 then evalf(int(f(x),x=-L..L)/int(1^2,x=-L..L))\nelse evalf(int(cos(n *Pi*x/L)*f(x),x=-L..L)/\n int(cos(n*Pi*x/L)^2,x=-L..L))\nfi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "b:=n->evalf(int(sin(n*Pi* x/L)*f(x),x=-L..L)/\n int(sin(n*Pi*x/L)^2,x=-L..L));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "approx:=x->sum('a(n)*cos(n*P i*x/L)','n'=0..N) + sum('b(n)*sin(n*Pi*x/L)','n'=1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot([[x,f(x),x=-L..L],[x,approx(x) ,x=-3*L/2..3*L/2]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "For f(x) = 4 Heaviside(x) x (1-x):" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "L:=1;\nN:=3;\nf:=x->piecewis e(x<=0,0,4*x*(1-x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "a: =n->if n = 0 then evalf(int(f(x),x=-L..L)/int(1^2,x=-L..L))\nelse eval f(int(cos(n*Pi*x/L)*f(x),x=-L..L)/\n int(cos(n*Pi*x/L)^2,x=-L. .L))\nfi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "b:=n->evalf(in t(sin(n*Pi*x/L)*f(x),x=-L..L)/\n int(sin(n*Pi*x/L)^2,x=-L..L ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "approx:=x->sum('a(n) *cos(n*Pi*x/L)','n'=0..N) + sum('b(n)*sin(n*Pi*x/L)','n'=1..N);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot([[x,f(x),x=-L..L],[x,ap prox(x),x=-3*L/2..3*L/2]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 17 " Remark in Closing" }}{PARA 0 "" 0 "" {TEXT -1 122 "I have now presente d three methods for using Maple to compute the coefficients. In Works heet 2, I typed in each line for " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#% \"nG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" } {TEXT -1 174 ". This would be tiresome. In this worksheet, I made each of a and b a function so that the approximation used a(n) and b(n). I n Worksheet 3, I defined the a's and b's with a " }{TEXT 256 7 "do loo p" }{TEXT -1 97 " construction. So that you can compare, I repeat one \+ approximation that I did above, but use the " }{TEXT 257 7 "do loop" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "For f(x) = 2*Heaviside(x)-1:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "L:=1;\nN:=3;\nf:=x->piecewise(x<=0,-1,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "for n from 0 to N do\n a[n]:=int (f(x)*cos(n*Pi*x/L),x=-L..L)/int(cos(n*Pi*x/L)^2,x=-L..L):\nod;\nn:='n ':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "for n from 1 to N do \n b[n]:=int(f(x)*sin(n*Pi*x/L),x=-L..L)/int(sin(n*Pi*x/L)^2,x=-L..L ):\nod;\nn:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "approx: =x->sum(a[n]*cos(n*Pi*x/L),n=0..N) + sum(b[n]*sin(n*Pi*x/L),n=1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot([[x,f(x),x=-L..L],[x ,approx(x),x=-3*L/2..3*L/2]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 28 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }