{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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 "Times " 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 266 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "Grey Emphasis" -1 270 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 45 "Fourier series of piecewise linea r functions " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, \+ B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "load Fourier series and Fourier transfo rm procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 270 9 "fourier.m" }{TEXT -1 37 " contains the code for the proce dure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 25 " used in this workshe et. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple sess ion by a command similar to the one that follows, where the file path \+ gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \+ \"K:\\\\Maple/procdrs/fourier.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 79 "A procedure for constructing a pi ecewise linear function through given points: " }{TEXT 0 22 "convert(. .,pwlinear,x)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "convert/pwlinear: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 18 "Call ing Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 263 2 " " }{TEXT -1 29 " c onvert(pts, pwlinear, x) " }{TEXT 264 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 11 " pts - " }{TEXT -1 60 " a l ist of points with non-decreasing first coordinates." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 10 " x - \+ " }{TEXT 265 75 "a variable to be used in the description of each pi ecewise linear function " }{XPPEDIT 18 0 "m*x+c" "6#,&*&%\"mG\"\"\"%\" xGF&F&%\"cGF&" }{TEXT 266 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 16 "convert/pwlinear" }{TEXT -1 27 " converts a list of points " }{XPPEDIT 18 0 "[[x[1],y[1]],[x[2],y[ 2]], `. . . `,[[x[n],y[n]]]" "6#7&7$&%\"xG6#\"\"\"&%\"yG6#F(7$&F&6#\" \"#&F*6#F/%'.~.~.~G7#7$&F&6#%\"nG&F*6#F7" }{TEXT -1 96 " to a piecewis e linear function whose graph consists of line segments joining the gi ven points. " }}{PARA 0 "" 0 "" {TEXT -1 89 "If two consecutive points have the same first coordinate, a \"jump\" discontinuity results." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 261 16 "How to activate:" }{TEXT 256 1 "\n" } {TEXT -1 154 "To make the procedure active open the subsection, place \+ the cursor anywhere after the prompt [ > and press [Enter].\nYou can \+ then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 " convert/pwlinear: implementation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 238 "A user can make his own con versions known to the convert function by defining a Maple procedure i n the following way. If the procedure `convert/f` is defined, then the function call convert(a,f,x,y,...) will invoke `convert/f`(a,x,y,...) ;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 797 "`convert/pwlinear` := proc(pts::listlist,x::name)\n \+ local n,i,p,q,m,c,L; \n n := nops(pts);\n L := NULL;\n for i to n-1 do\n p := pts[i];\n if not type(p,list(realcons)) or nops(p)<>2 then\n error \"1st argument must be a list of poin ts\"\n end if;\n q := pts[i+1];\n if not type(q,list(re alcons)) or nops(q)<>2 then\n error \"1st argument must be a l ist of points\"\n end if;\n if signum(p[1]-q[1])=1 then\n \+ error \"the points must have increasing 1st coordinates\"\n \+ end if;\n if p[1]<>q[1] then\n m := (q[2]-p[2])/(q[1]-p[ 1]);\n c := p[2]-m*p[1];\n if i " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 97 " be the function of period 4 obtained by extending the piecewise l inear function on the interval " }{XPPEDIT 18 0 "[-2,2]" "6#7$,$\"\"#! \"\"F%" }{TEXT -1 64 " whose graph consists of the line segments which join the points" }{XPPEDIT 18 0 "``(-2,0),``(-1,1),``(1,1)" "6%-%!G6$ ,$\"\"#!\"\"\"\"!-F$6$,$\"\"\"F(F--F$6$F-F-" }{TEXT -1 6 ", and " } {XPPEDIT 18 0 "``(2,0)" "6#-%!G6$\"\"#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 63 "We can construct a Fourier series for this func tion as follows." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 60 "Note that we can extend a function f defined on an interv al " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 39 " to a perio dic function g with period " }{XPPEDIT 18 0 "b-a" "6#,&%\"bG\"\"\"%\" aG!\"\"" }{TEXT -1 22 " by the the formula " }{XPPEDIT 18 0 "g(x)=f( x-(b-a)*floor((x-a)/(b-a))" "6#/-%\"gG6#%\"xG-%\"fG6#,&F'\"\"\"*&,&%\" bGF,%\"aG!\"\"F,-%&floorG6#*&,&F'F,F0F1F,,&F/F,F0F1F1F,F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "convert([[-2,0],[-1,1],[1,1],[2,0]],pwlinear,x);\nf \+ := unapply(%,x):\nf_ := x -> f(x-4*floor((x+2)/4)):\nFourierSeries(f(x ),x=-2..2,numterms=6):\ng := unapply(%,x);\nplot([f_(x),g(x)],x=-2..6, color=[blue,red]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6%7 $,&%\"xG\"\"\"\"\"#F)2F(!\"\"7$F)2F(F)7$,&F(F,F*F)1F)F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,.#\"\" $\"\"%\"\"\"*(F/F0%#PiG!\"#-%$cosG6#,$*&F2F09$F0#F0\"\"#F0F0*(F;F0F2F3 -F56#F8F0!\"\"*(#F/\"\"*F0F2F3-F56#,$F8#F.F;F0F0*(#F/\"#DF0F2F3-F56#,$ F8#\"\"&F;F0F0*&#F;FBF0*&F2F3-F56#,$F8F.F0F0F?F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 493 178 178 {PLOTDATA 2 "6&-%'CURVESG6$7ip7$$!\"#\"\"!$F* F*7$$!3MLLL$Q6G\">!#<$\"3Hmmmm;')=()!#>7$$!3ommmmFiD=F/$\"3ELLLLBxV5 E$F87$$!35++]AHe)e\"F/$\"3&*)****\\xqT6%F87$$!3nmmmwxE.:F/$\"3MLLLLAKn \\F87$$!3ymmm1rQ<9F/$\"3:KLLL*Gh#eF87$$!3ommmOk]J8F/$\"3=LLLLc$\\o'F87 $$!3+++]UR.Y7F/$\"34+++v0mRvF87$$!3_LLL[9cg6F/$\"3wkmm;bQ%R)F87$$!37++ +0vJ\"3\"F/$\"3w)*****\\\\#o=*F87$$!3smmmhN2-5F/$\"3wKLL$Qk#z**F87$$!3 CMLL$[5-?*F8$\"\"\"F*7$$!3E,++]`oz$)F8Fho7$$!3'zmmm\")3Do'F8Fho7$$!3%> +++:v2*\\F8Fho7$$!3IKLLL\">1D$F8Fho7$$!3Ymmmm())yr\"F8Fho7$$\"3fg%)*** ***f-w!#@Fho7$$\"3[*********y,u\"F8Fho7$$\"3!)*******RP)4MF8Fho7$$\"3. 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[2aSxf!*)F27$F[z$\"3(pPdsl#=5')F27$$\"3kLLLo " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 97 " be the function of period 6 obtained by extend ing the piecewise linear function on the interval " }{XPPEDIT 18 0 "[- 3,3]" "6#7$,$\"\"$!\"\"F%" }{TEXT -1 65 " whose graph consists of the \+ line segments which join the points " }{XPPEDIT 18 0 "``(-3,0),``(-2,0 ),``(-2,1),``(-1,1),``(-1,2),``(1,2),``(1,1),``(2,1),``(2,0)" "6+-%!G6 $,$\"\"$!\"\"\"\"!-F$6$,$\"\"#F(F)-F$6$,$F-F(\"\"\"-F$6$,$F1F(F1-F$6$, $F1F(F--F$6$F1F--F$6$F1F1-F$6$F-F1-F$6$F-F)" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "``(3,0)" "6#-%!G6$\"\"$\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 64 "We can construct a Fourier series for this func tion as follows. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 275 "convert([[-3,0],[-2,0],[-2,1],[-1,1],[-1,2] ,[1,2],[1,1],\n [2,1],[2,0],[3,0]],pwlinear,x);\nf := u napply(%,x):\nf_ := x -> f(x-6*floor((x+3)/6)):\nFourierSeries(f(x),x= -3..3,numterms=16):\ng := unapply(%,x);\nplot([f_(x),g(x)],x=-3..9,col or=[blue,red],numpoints=80);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIE CEWISEG6'7$\"\"!2%\"xG!\"#7$\"\"\"2F)!\"\"7$\"\"#2F)F,7$F,2F)F07$F'1F0 F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG% &arrowGF(,.\"\"\"F-**\"\"#F-\"\"$#F-F/%#PiG!\"\"-%$cosG6#,$*&F2F-9$F-# F-F0F-F-*&#F/\"\"&F-*(F0F1F2F3-F56#,$F8#F=F0F-F-F3**#F/\"\"(F-F0F1F2F3 -F56#,$F8#FEF0F-F-*&#F/\"#6F-*(F0F1F2F3-F56#,$F8#FLF0F-F-F3**#F/\"#8F- F0F1F2F3-F56#,$F8#FTF0F-F-F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 551 208 208 {PLOTDATA 2 "6&-%'CURVESG6$7du7$$!\"$\"\"!$F*F*7$$!3%30CR\"R2T G!#F;#F/F+7$$!33R6HBu\\!4#F/F+7$$!3!='pb/k5`?F/ 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x)F/$\"3b*QibDl;J#Fjs7$$\"3?wg)3$RI#z)F/$\"3Ktfrb'yfa#Fjs7$$\"3Ln$\\f$ Gx7))F/$\"3>*z[!Ra68DFjs7$$\"3YeE,T " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 51 "Construct a sequence of points along the graph of " } {XPPEDIT 18 0 "y = x-x^3/16;" "6#/%\"yG,&%\"xG\"\"\"*&F&\"\"$\"#;!\"\" F+" }{TEXT -1 6 " with " }{TEXT 267 1 "x" }{TEXT -1 13 " coordinates \+ " }{XPPEDIT 18 0 "x=-4,-3,-2,-1,0,1,2,3,4" "6+/%\"xG,$\"\"%!\"\",$\"\" $F',$\"\"#F',$\"\"\"F'\"\"!F-F+F)F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 49 "Form a piecewise linear function on the interval " } {XPPEDIT 18 0 "[-4,4]" "6#7$,$\"\"%!\"\"F%" }{TEXT -1 103 " which pass es through these points, and then extend this function to a periodic f unction with period 8." }}{PARA 0 "" 0 "" {TEXT -1 64 "We can construc t a Fourier series for this function as follows. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "pts := [se q([i,i-i^3/16],i=-4..4)];\nconvert(%,pwlinear,x);\nf := unapply(%,x): \nf_ := x -> f(x-8*floor((x+4)/8)):\nFourierSeries(f(x),x=-4..4,numter ms=4):\ng := unapply(%,x);\nplot([f_(x),g(x)],x=-4..12,color=[blue,red ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7+7$!\"%\"\"!7$!\"$#!#@ \"#;7$!\"##F*\"\"#7$!\"\"#!#:F-7$F(F(7$\"\"\"#\"#:F-7$F1#\"\"$F17$F=# \"#@F-7$\"\"%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6)7$,& %\"xG#!#@\"#;#\"#@\"\"%!\"\"2F(!\"$7$,&F(#F1F+#\"#:\"\")F/2F(!\"#7$,&F (#\"\"*F+#\"\"$F7F/2F(F/7$,$F(#F6F+2F(\"\"\"7$,&F(F<#F?F7FE2F(\"\"#7$, &F(F4#F6F7FE2F(F?7$,&F(F)#F-F.FE1F?F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*(,&\"\"\"F/*$-%%sqrtG6# \"\"#F/F/F/%#PiG!\"#-%$sinG6#,$*&F5F/9$F/#F/\"\"%F/\"\"'*&#\"\"$F4F/*& F5F6-F86#,$F;#F/F4F/F/!\"\"**#F4FBF/,&F0F/F/FHF/F5F6-F86#,$F;#FBF>F/F/ F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 445 310 310 {PLOTDATA 2 "6&-%'CUR VESG6$7cp7$$!\"%\"\"!$F*F*7$$!3ommmmFiDQ!#<$!3S(***\\(=,()G#!#=7$$!3OL LLLbC^OF/$!3o.++vBSxXF27$$!3A+++N3_*\\$F/$!3!Q*\\iS!*yolF27$$!3kmmmOhz ZLF/$!3o,+D1d0_\"!#?$\"3P6(**** \\([D9Fdr7$$\"3%z******f\"yZoF2$\"31*****\\7XMR'F27$$\"3#GLLLe%)eL)F2$\"3!)**\\(oz\"*[ \"yF27$$\"31mmmmV4_)*F2$\"3U***\\(o%QjB*F27$$\"37LLL`MzX8F/$\"3#)**** \\<)3?8\"F/7$$\"3#GLLLTb7l\"F/$\"3g******p6$QI\"F/7$$\"3@mmm^y\"*G=F/$ \"3x*\\i:HmPS\"F/7$$\"3g*******G!e1?F/$\"3-+]i&>m()\\\"F/7$$\"33mmm^;% Q;#F/$\"31+DJ!pz#p9F/7$$\"3eKLL8I5@BF/$\"33+++&=$zR9F/7$$\"3M+++!H%=mE F/$\"3%***\\iX/4v8F/7$$\"3B+++!y$[IGF/$\"3$****\\7H%GW8F/7$$\"35+++qKy %*HF/$\"3$***\\(o8yMJ\"F/7$$\"3#omm;b7i;$F/$\"3'**\\Pf-YV4\"F/7$$\"3`L LLL=kPLF/$\"3'z***\\P4X$p)F27$$\"3SLLLGu1&\\$F/$\"3<,]7.+CFmF27$$\"3EL LLBI\\_OF/$\"3O/+vo!H5c%F27$$\"3*)*****\\z(HAQF/$\"3$4+D1WTBL#F27$$\"3 _mmmmD5#*RF/$\"3#Q1+]7Ql.\"!#>7$$\"3%fmm;+#[oTF/$!3`\"*\\(=FE8@#F27$$ \"3NlmmO9'[M%F/$!3d!)*\\iN1j_%F27$$\"3EKLL`-S)\\%F/$!3b')***\\K.:a'F27 $$\"3=******p!R>l%F/$!3_#**\\PH+nb)F27$$\"3]KLL)>lx\"[F/$!3=*\\7G#oJt5 F/7$$\"3#emmmK\"f$)\\F/$!35***\\ihj4H\"F/7$$\"3kKLLVf!\\:&F/$!3')**\\P k[aT8F/7$$\"3W******f0AE`F/$!3!******\\NmOP\"F/7$$\"3M)*****>kThcF/$!3 p***\\(yb^O9F/7$$\"3a)****\\.wN#eF/$!3u*\\il]?pY\"F/7$$\"3u)*****\\ct& )fF/$!3w**\\PMaK(\\\"F/7$$\"3m)****\\D'ylhF/$!3k+DcJ_u19F/7$$\"3e)**** *fo$eM'F/$!3!3+]iwmaI\"F/7$$\"3?KLL8QSpmF/$!3j+++b.YB6F/7$$\"3p******* f!)[,(F/$!3\"=++]P%\\N#*F27$$\"3!GLL$esSrrF/$!3K/]7`p0oxF27$$\"3%fmmm \"R$zK(F/$!3%o+]7`>1I'F27$$\"3%GLL$)*)e!*\\(F/$!3c.]7GGK'p%F27$$\"3s** ****zQ=qwF/$!3I+++Dh-#4$F27$$\"3plmm^\"*>JyF/$!3q4]7`z]#e\"F27$$\"3mJL LBW@#*zF/$!3C4>+D\"y*)H(Fdr7$$\"3Ammmw<_g\")F/$\"3m$**\\il\"*[]\"F27$$ \"3.******H\"H)G$)F/$\"3m))*\\P4tF3$F27$$\"3%emm;L!Q$\\)F/$\"3\"*))\\P 41WDYF27$$\"3mKLLL:$zl)F/$\"3=*)***\\73\"ohF27$$\"3'fmm;R,-$))F/$\"3a& *\\(o/QJy(F27$$\"3E******\\7Z-!*F/$\"3#e**\\7y+*)Q*F27$$\"3GKLL)f(Qo\" *F/$\"3&)*\\iS-=A.\"F/7$$\"32nmmYRIM$*F/$\"37+++qfaD6F/7$$\"3?mmm13lt' *F/$\"33++vy&GkJ\"F/7$$\"3`+++b#>>%)*F/$\"3')*\\P4ez5T\"F/7$$\"3JLLLq( =5+\"!#;$\"3/+]i0)*3)\\\"F/7$$\"3jmm;$*3[;5F_cl$\"31+DJD$)4p9F/7$$\"3' ******f,V>.\"F_cl$\"33+++Xo5S9F/7$$\"3KLLL\"p&Qn5F_cl$\"3#****\\(G=lt8 F/7$$\"3)******p'eB$3\"F_cl$\"3&***\\P\\F$RM\"F/7$$\"3gmmmUg3*4\"F_cl$ \"3++++qO@98F/7$$\"3CLL$ej&)f6\"F_cl$\"3`,voH&)o-6F/7$$\"3/+++H_)G8\"F _cl$\"3*Q+]Pp8)3))F27$$\"3/+++'Hg!\\6F_cl$\"3)H*****\\h$eo'F27$$\"3/++ +j`Bl6F_cl$\"3%)***\\igeGc%F27$$\"3-++]\"o$!3#[] o$[%3sU64i9F/7$$!3^LL$3R&f1CF/$!3 1]aKY[(zY\"F/7$$!31++vV\"fQO#F/$!3_Z^%))pV?Z\"F/7$FW$!3?#p`5)RaE!=Z\"F/7$Ffn$!39%>Qn`k\"o9F/7$$!3immmm5R$3# F/$!3u_kri'HpX\"F/7$F[o$!3]&3C#y.nS9F/7$F`o$!3=@W>]qH#R\"F/7$Feo$!3n$Q \")fl1hK\"F/7$Fjo$!3^^\\3%)*399\"F/7$F_p$!3-`I#eXY(G5F/7$Fdp$!3Fr6`eX! 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Then extend this function to a periodic f unction with period 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 238 "pts := [seq(op([[i,i-i^3/16],[i+1, i-i^3/16]]),i=-4..4)];\nconvert(%,pwlinear,x);\nf := unapply(%,x):\nf_ := x -> f(x-8*floor((x+4)/8)):\nFourierSeries(f(x),x=-4..4,numterms=3 2):\ng := unapply(%,x):\nplot([f_(x),g(x)],x=-4..12,color=[blue,red]); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ptsG747$!\"%\"\"!7$!\"$F(7$F*#! #@\"#;7$!\"#F,7$F0#F*\"\"#7$!\"\"F27$F5#!#:F.7$F(F77$F(F(7$\"\"\"F(7$F <#\"#:F.7$F3F>7$F3#\"\"$F37$FCFB7$FC#\"#@F.7$\"\"%FF7$FIF(7$\"\"&F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6+7$\"\"!2%\"xG!\"$7$#!# @\"#;2F)!\"#7$#F*\"\"#2F)!\"\"7$#!#:F.2F)F'7$F'2F)\"\"\"7$#\"#:F.2F)F3 7$#\"\"$F32F)FC7$#\"#@F.2F)\"\"%7$F'1FIF)" }}{PARA 13 "" 1 "" {GLPLOT2D 500 244 244 {PLOTDATA 2 "6&-%'CURVESG6$7gx7$$!\"%\"\"!$F*F*7 $$!3OLLLLbC^O!#F/$!3++++++++:F/7$$!3;+++&*Rj$)>F/Fbo7$$!3iLL$3VxL(>F/Fb 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