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" }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple s ession by a command similar to the one that follows, where the file pa th gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "re ad \"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 45 "Fourier series of general periodi c functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Given a function " }{XPPEDIT 18 0 "f(x);" "6#-% \"fG6#%\"xG" }{TEXT -1 13 " defined for " }{XPPEDIT 18 0 "alpha <= x; " "6#1%&alphaG%\"xG" }{XPPEDIT 18 0 "`` < beta;" "6#2%!G%%betaG" } {TEXT -1 16 ", we can extend " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG " }{TEXT -1 24 " to a periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-% #f_G6#%\"xG" }{TEXT -1 13 " with period " }{XPPEDIT 18 0 "T = beta-alp ha;" "6#/%\"TG,&%%betaG\"\"\"%&alphaG!\"\"" }{TEXT -1 55 " defined for all real numbers as follows. For a number " }{TEXT 277 1 "x" }{TEXT -1 14 " greater than " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 19 ", keep subtracting " }{TEXT 278 1 "T" }{TEXT -1 7 " from " }{TEXT 280 1 "x" }{TEXT -1 28 " until the resulting number " }{XPPEDIT 18 0 " u = x-k*T;" "6#/%\"uG,&%\"xG\"\"\"*&%\"kGF'%\"TGF'!\"\"" }{TEXT -1 6 " , for " }{TEXT 279 1 "k" }{TEXT -1 43 " a positive integer, lies in th e interval " }{XPPEDIT 18 0 "alpha <= x;" "6#1%&alphaG%\"xG" } {XPPEDIT 18 0 "`` < beta;" "6#2%!G%%betaG" }{TEXT -1 26 ". Similarly, \+ for a number " }{TEXT 282 1 "x" }{TEXT -1 11 " less than " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 34 ", add a suitable integer multi ple " }{XPPEDIT 18 0 "k*`.`*T;" "6#*(%\"kG\"\"\"%\".GF%%\"TGF%" } {TEXT -1 4 " of " }{TEXT 281 1 "T" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "u = x+k*T;" "6#/%\"uG,&%\"xG\"\"\"*&%\"kGF'%\"TGF'F'" }{TEXT -1 23 " lies in the interval " }{XPPEDIT 18 0 "alpha <= x;" "6#1%&alphaG %\"xG" }{XPPEDIT 18 0 "`` < beta;" "6#2%!G%%betaG" }{TEXT -1 36 ". In \+ each of these two cases define " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 27 " for the extended function " }{XPPEDIT 18 0 "f_(x)" " 6#-%#f_G6#%\"xG" }{TEXT -1 7 " to be " }{XPPEDIT 18 0 "f(u)" "6#-%\"fG 6#%\"uG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "More succinctl y, given any real number " }{TEXT 283 1 "x" }{TEXT -1 51 ", there is a unique (positive or negative) integer " }{TEXT 284 1 "k" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "u=x+k*T" "6#/%\"uG,&%\"xG\"\"\"*&%\"kGF'% \"TGF'F'" }{TEXT -1 23 " lies in the interval " }{XPPEDIT 18 0 "alpha <= x;" "6#1%&alphaG%\"xG" }{XPPEDIT 18 0 "`` < beta;" "6#2%!G%%betaG " }{TEXT -1 16 ", and we define " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 16 " to be equal to " }{XPPEDIT 18 0 "f(u)" "6#-%\"fG6#% \"uG" }{TEXT -1 2 ". 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Fb\\o-F$6%7$7$Fd[lF(Fc[lFj[oFb\\o-F$6%7$7$F]cmF(F\\cmFj[oFb\\o-F$6%7$7 $Fd_nF(Fc_nFj[oFb\\o-F$6%7$Fi[oFb[oFj[oFb\\o-F$6%7$7$$\"3a************ **pF,F(7$F\\^o$\"3_*****H7nId&F,-Fi[l6&F[\\lF(F(F\\\\l-Fc\\o6#\"\"$-F$ 6%7$7$$\"3;+++++++ZFcsF(7$Fj^oF_^oFa^oFc^o-F$6&7&F_\\o7$Fd[lF`\\o7$F]c mF`\\o7$Fd_nF`\\o-%'SYMBOLG6#%'CIRCLEGF^gl-%&STYLEG6#%&POINTG-F$6&F__o -Fd_o6#%(DIAMONDGF^glFg_o-F$6&F__o-Fd_o6#%&CROSSGF^glFg_o-F$6&7&7$F($ \"3))******e/,EvF,7$F]cmFi`o7$Fd_nFi`o7$Fc[oFi`oF^glFg_oFc_o-F$6&7#7$F (F`\\oFc_oFh[lFg_o-F$6&F`aoF]`oFh[lFg_o-F$6&F`aoFb`oFh[lFg_o-F$6&7#7$F d[lFi`oFh[lFg_oFc_o-%)POLYGONSG6&7&7$$\"+++++r!#5$!\"%!\"\"7$$\"+++++p FaboFbbo7$Ffbo$!++++]AFabo7$F_boFibo7%7$$\"++++]iFaboFibo7$$\"\"(Fdbo$ !#:Fd\\l7$$\"++++]xFaboFibo-Fh_o6#%,PATCHNOGRIDG-%&COLORG6&F[\\l$Fe_nF dboF)$\"\"*Fdbo-F[bo6&7&7$$\"++++5Z!\"*Fbbo7$$\"++++!p%FgdoFbbo7$FidoF ibo7$FedoFibo7%7$$\"++++DYFgdoFibo7$$\"#ZFdboFcco7$$\"++++vZFgdoFiboFh coF[do-F$6$7$7$F\\^o$!3A+++++++SF,7$Fj^oF[foF[do-%%TEXTG6&7$F($!\"(Fd \\lQ\"a6\"-F\\do6&F[\\lF)F)$\"\"\"Fd\\l-%%FONTG6$Fd_oF`do-F_fo6&7$Fd[l FbfoQ\"bFefoFffoFjfo-F_fo6&7$FacoFbfoQ'u=x-2TFefoFa^o-F[go6$%*HELVETIC AGF`do-F_fo6&7$FbeoFbfoQ\"xFefoFa^oFego-%*AXESSTYLEG6#%%NONEG-%+AXESLA BELSG6%F[hoQ!Fefo-F[go6#%(DEFAULTG-%%VIEWG6$;Fc\\lFc[oFfho" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "C urve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22 " "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "C urve 29" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "Now suppose t hat we have a periodic function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 13 " with period " }{TEXT 285 1 "T" }{TEXT -1 16 ". The function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 87 " can be described completely by specifying how its values are obtained on \+ any interval " }{XPPEDIT 18 0 "alpha<=x" "6#1%&alphaG%\"xG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 26 "Fourier series .. summary " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 39 " be a function de fined on the interval " }{XPPEDIT 18 0 "[alpha, beta];" "6#7$%&alphaG% %betaG" }{TEXT -1 10 " and let " }{XPPEDIT 18 0 "T = beta-alpha;" "6# /%\"TG,&%%betaG\"\"\"%&alphaG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 14 "We can extend " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 36 " to a periodic function with period " }{XPPEDIT 18 0 " T;" "6#%\"TG" }{TEXT -1 21 " by specifying that " }{XPPEDIT 18 0 "f(x +k*T) = f(x);" "6#/-%\"fG6#,&%\"xG\"\"\"*&%\"kGF)%\"TGF)F)-F%6#F(" } {TEXT -1 19 " for every integer " }{TEXT 262 1 "k" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 261 14 "Fourier series" }{TEXT 258 1 " " }{TEXT -1 16 "of the fu nction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = c+``;" "6#/-% \"FG6#%\"xG,&%\"cG\"\"\"%!GF*" }{XPPEDIT 18 0 "Sum(a[k]*cos(2*k*Pi*x/T )+b[k]*sin(2*k*Pi*x/T),k = 1 .. infinity);" "6#-%$SumG6$,&*&&%\"aG6#% \"kG\"\"\"-%$cosG6#*,\"\"#F,F+F,%#PiGF,%\"xGF,%\"TG!\"\"F,F,*&&%\"bG6# F+F,-%$sinG6#*,F1F,F+F,F2F,F3F,F4F5F,F,/F+;F,%)infinityG" }{TEXT -1 12 " ------- (i)" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "c+a[1]*cos(2*Pi*x/T)+b[1]*sin(2*Pi*x/T)+a[2]*cos(4*Pi*x/T)+b[2]*sin(4 *Pi*x/T)+a[3]*cos(6*Pi*x/T)+b[3]*sin(6*Pi*x/T)+` . . . `;" "6#,2%\"cG \"\"\"*&&%\"aG6#F%F%-%$cosG6#**\"\"#F%%#PiGF%%\"xGF%%\"TG!\"\"F%F%*&&% \"bG6#F%F%-%$sinG6#**F.F%F/F%F0F%F1F2F%F%*&&F(6#F.F%-F+6#**\"\"%F%F/F% F0F%F1F2F%F%*&&F56#F.F%-F86#**FAF%F/F%F0F%F1F2F%F%*&&F(6#\"\"$F%-F+6#* *\"\"'F%F/F%F0F%F1F2F%F%*&&F56#FKF%-F86#**FOF%F/F%F0F%F1F2F%F%%(~.~.~. ~GF%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 10 " where " }{XPPEDIT 18 0 "c = 1/T;" "6#/%\"cG*&\" \"\"F&%\"TG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = alpha . . beta)" "6#-%$IntG6$-%\"fG6#%\"xG/F);%&alphaG%%betaG" }{TEXT -1 2 ", \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = 2/T;" "6#/ &%\"aG6#%\"kG*&\"\"#\"\"\"%\"TG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Int(f(x)*cos(2*k*Pi*x/T),x = alpha .. beta);" "6#-%$IntG6$*&-%\"fG6#% \"xG\"\"\"-%$cosG6#*,\"\"#F+%\"kGF+%#PiGF+F*F+%\"TG!\"\"F+/F*;%&alphaG %%betaG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 6 "and " } {XPPEDIT 18 0 "b[k] = 2/T;" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\"%\"TG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin(2*k*Pi*x/T),x = alpha .. \+ beta);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#*,\"\"#F+%\"kGF+%#Pi GF+F*F+%\"TG!\"\"F+/F*;%&alphaG%%betaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 261 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 95 "We are assuming here that the integrals involved in the definition of the Fourier coefficients " }{XPPEDIT 18 0 "c, a[k]" "6$%\"cG&%\"aG 6#%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" } {TEXT -1 47 " all exist, and give finite real number values." }}{PARA 15 "" 0 "" {TEXT -1 58 "Under suitable conditions the Fourier series c onverges to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 53 ", \+ in which case the Fourier series (i) is called the " }{TEXT 263 24 "Fo urier series expansion" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 14 ".\nIn fact, if " }{XPPEDIT 18 0 "f(x)" "6#- %\"fG6#%\"xG" }{TEXT -1 24 " and the derivative f '(" }{TEXT 265 1 "x " }{TEXT -1 6 ") are " }{TEXT 263 20 "piecewise continuous" }{TEXT -1 14 ", that is, if " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 9 " and f '(" }{TEXT 264 1 "x" }{TEXT -1 165 ") are continuous except \+ at only a finite number points in any period, and do not have infinite discontinuities at these points, then the Fourier series converges to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 22 " at every poi nt where " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 46 " is c ontinuous.\nAt any point of discontinuity " }{XPPEDIT 18 0 "x = d;" "6 #/%\"xG%\"dG" }{TEXT -1 75 ", the Fourier series converges to the aver age of the left and right limits:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "([Limit(f(x),x = d,le ft)]+[Limit(f(x),x = d,right)])/2 = (f(d-``)+f(d+``))/2;" "6#/*&,&7#-% &LimitG6%-%\"fG6#%\"xG/F-%\"dG%%leftG\"\"\"7#-F(6%-F+6#F-/F-F/%&rightG F1F1\"\"#!\"\"*&,&-F+6#,&F/F1%!GF:F1-F+6#,&F/F1F@F1F1F1F9F:" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "and consider the corres ponding (dilated) function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" } {TEXT -1 12 " defined by " }{XPPEDIT 18 0 "g(x)=f(x/Pi)" "6#/-%\"gG6#% \"xG-%\"fG6#*&F'\"\"\"%#PiG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " g(x)=(Pi-x)/2" "6#/-%\"gG6#%\"xG*&,&%#PiG\"\"\"F'!\"\"F+\"\"#F," } {TEXT -1 4 ", " }{XPPEDIT 18 0 "0<=x " "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " is periodic with period 2 . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The following picture shows the graphs of " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 372 "f := x -> Pi*(1-x)/2:\n'f(x)'=f(x);\nf_ := x -> f(x-2*floor(x/2)):\ng := x -> (Pi-x)/2:\n'g(x)'=g(x);\ng_ := x -> \+ g(x-2*Pi*floor(x/(2*Pi))):\npi := evalf(Pi):\nplot([f_(x),g_(x)],x=-2. .10,y,color=[COLOR(RGB,.4,0,.8),red],thickness=[1,2],\n xtickmarks=[0 =`0`,2=`2`,pi=`p`,4=`4`,6=`6`,2*pi=`2p`,8=`8`,3*pi=`3p`,10=`10`],\n \+ ytickmarks=3,font=[SYMBOL,10],labelfont=[HELVETICA,10]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*(\"\"#!\"\"%#PiG\"\"\",&F-F-F' F+F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*&\"\"#!\"\" %#PiG\"\"\"F-*&F*F+F'F-F+" }}{PARA 13 "" 1 "" {GLPLOT2D 543 179 179 {PLOTDATA 2 "6(-%'CURVESG6%7]t7$$!\"#\"\"!$\"3c'*[zEjzq:!#<7$$!3!***** *\\2<#p=F-$\"3!RS(f))HOl8F-7$$!3,+++]TVQF=7$$!37******\\lfs**F=$!3\\**3:gQ_/V!#?7$$ !3')****\\P\"40p)F=$!37\"['yOM%p0#F=7$$!3i+++DiF=$!3[(f-;1$)z$fF=7$$!3&3++]U.6.&F=$!3;TPB)[C^!yF=7$$!3\"3 ++]s:.!QF=$!3\\!3ekGT%Q(*F=7$$!3y+++D!G&pDF=$!3)>Co%3erO\"F-7$$!39s*****\\AjP#FW$!3i;193O1n:F-7$$\"3'[-+D1z Ob&FW$\"3qBL7GE2i:F-7$$\"3?-+]i!o$[8!#>$\"3#4Z(z5ih\\:F-7$$\"3*=+](=#o 89#Fbq$\"38=;Z$zfr`\"F-7$$\"3f,++v$oV$HFbq$\"3Mld9wLqC:F-7$$\"3)4++voo ._%Fbq$\"3+gS\\T0z*\\\"F-7$$\"3Q++++!pj5'Fbq$\"3?aB%oqx[Z\"F-7$$\"3=** ***\\ip$y#*Fbq$\"3FV*Qv._]U\"F-7$$\"3!)*****\\-P]C\"F=$\"3!>`N#ojAv8F- 7$$\"3c******\\rVz=F=$\"3%)4(G'H]dv7F-7$$\"3K*****\\FPQ^#F=$\"3a()=-\" pBf<\"F-7$$\"3V++](Ga*=QF=$\"3uVj%4!R;4(*F=7$$\"3b,+++82C^F=$\"3=8Qn\" *34fwF=7$$\"3%4++]2>OF'F=$\"3'pcG#paQ`eF=7$$\"3K+++]o;BuF=$\"3QALyY+oZ SF=7$$\"3B,++v`G<()F=$\"3K\"4w7U$)[,#F=7$$\"3@+++!RS6+\"F-$!3%p'R6B/K \"z\"FW7$$\"32+++?O3J6F-$!3[\"y*)zoc!f?F=7$$\"3#*******\\o-h7F-$!3AB%[ <<+-5%F=7$$\"31+++I7D'Q\"F-$!3w3,/L,AngF=7$$\"3>+++5cZ6:F-$!3w$zJV4SU. )F=7$$\"32++vV8>D;F-$!3ybY1jE[?)*F=7$$\"3'*****\\xq!*QvzJ_s1; \"F-7$$\"3()**\\i!e;l!=F-$!3!)H\\M#GtoE\"F-7$$\"3+++v$3ET(=F-$!393,^TS 2t8F-7$$\"3'**\\7`$3$z!>F-$!3@(p#4@WF-$!3]'Gv1![F z9F-7$$\"3C]ilizje>F-$!3.\"em/*\\#e]\"F-7$$\"35+vVQ.av>F-$!3MvyD!=vB` \"F-7$$\"39D\"Gj_\"*R)>F-$!3#G_`^F]ca\"F-7$$\"3'*\\(=UrUC*>F-$!3kp\"\\ +PD*e:F-7$$\"3Bv$4@!R*3+#F-$\"3wv\\k)=#Rp:F-7$$\"31+++!4X$4?F-$\"3sG$ \\P4+KD'>\"F-7$$\"3)****\\()Q)orBF-$\"3%)H3M5d\\p)*F=7$$\"3#*****\\<_$\\] #F-$\"3yW^w=%fkx(F=7$$\"3z***\\(Q7)Gi#F-$\"38>%p$\\wwBfF=7$$\"3m****** fs#3u#F-$\"3#Hpt*ze2rSF=7$$\"3'****\\(QFBqGF-$\"3\"zL(4t$z$Q?F=7$$\"3E ++]<#Q'**HF-$\"3_yE)4AmGo&!#@7$$\"3G+++NG'G7$F-$!3'*yg<**[#*H>F=7$$\"3 I++]_u3YKF-$!3#39tXmKb'QF=7$$\"3M++v8%fYP$F-$!3q)Hw\"4j8&)eF=7$$\"3P++ +v8B.NF-$!3:d%zP&*RZ!zF=7$$\"3Q++Dr0I@OF-$!3_:2b^lOf(*F=7$$\"3R++]n(p$ RPF-$!3G(>K\\J*Rh6F-7$$\"39++DYLsmQF-$!3W\")oj[eWh8F-7$$\"3))*****\\#p 2%*RF-$!3il:M#Q#\\h:F-7$$\"3&\\i:5rWB+%F-$\"3MdTMjK6n:F-7$$\"3+]7.(\\7 1,%F-$\"3?(3SaDETb\"F-7$$\"31vo/$G!))=SF-$\"3GjRA:G:F-7$$\"3A]P4TOoVSF-$\"312Q#QAy@]\"F-7$$\"3M+]78#>-1 %F-$\"3+nc,3U?w9F-7$$\"3c+v=d.H$4%F-$\"3%oQ*RwhDC9F-7$$\"3!****\\7]hj7 %F-$\"3-3JyW\"3BP\"F-7$$\"3W**\\P*y.D>%F-$\"33\\0b\"37%o7F-7$$\"3!**** *\\xgkeUF-$\"3c))zJ=g^k6F-7$$\"3')*******=+QP%F-$\"3-Bm(QhBj$)*F=7$$\" 3%)****\\-V&*)[%F-$\"3[hLdWq[F!)F=7$$\"3/++v)*QK8YF-$\"339(G]*R*Q2'F=7 $$\"3E+++&\\$pPZF-$\"3ynS[X4I?TF=7$$\"3Y****\\d%zh'[F-$\"3i6q3m\"[?5#F =7$$\"3e******>am%*\\F-$\"3%==%**o'Q&z$)FW7$$\"31++]nQO?^F-$!35x#y%47n !*=F=7$$\"3k*****\\JigC&F-$!3!>[Yc!y8lQF=7$$\"3!)***\\i-#on`F-$!3KW+ii d`vdF=7$$\"3%*****\\PP$fo(F=7$$\"3L++D\"pRVi&F-$!35.w(o $\\52)*F=7$$\"3#)*****\\kx$fdF-$!3aehT:w#G>\"F-7$$\"3K++v[k/?eF-$!3)** p(\\\\f7)G\"F-7$$\"3%*****\\__r!)eF-$!3iR#zNGCMQ\"F-7$$\"3u**\\Pa'\\5 \"fF-$!3d4+i]M2J9F-7$$\"3a***\\i0%QTfF-$!3]z2mAoj>]:F-7$$\"3U\\7`f#pW*fF -$!3AF@)*f'3@c\"F-7$$\"3;******fG0-gF-$\"3utu%=qrvc\"F-7$$\"3p*****\\l 1;8'F-$\"3!R8Rl2pSO\"F-7$$\"3A+++]/;hiF-$\"31%zI7Xm0;\"F-7$$\"3#)***\\ PWb&yjF-$\"37Z=ki#G;w*F=7$$\"3Y****\\P/&f\\'F-$\"3)[vvH,#fF=7$$\"3=****\\<3;%*pF-$\"3SCj[Ak;s\"*FW7$$ \"3n****\\K8R?rF-$!3ID%*oGE5\"*=F=7$$\"3;++]Z=iYsF-$!3m8Pg@p#R(QF=7$$ \"3#)***\\([_.qtF-$!3\"fN6_4+D\"eF=7$$\"3[******\\'[M\\(F-$!3g(**=)oK2 ^xF=7$$\"3Z***\\P/^Ei(F-$!3!e2vRsz0y*F=7$$\"3W****\\PM&=v(F-$!3S:J\"zh 35=\"F-7$$\"3A+]7o?29yF-$!3AvqN99uy7F-7$$\"35++v)p!HwyF-$!3[L5!3@ukP\" F-7$$\"3f*\\iS,+u!zF-$!3&=,B!41MD9F-7$$\"3)***\\PH$4&QzF-$!3w\"*\\C2q? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Hence the Fourier series of " }{XPPEDIT 18 0 "f(x);" "6#- %\"fG6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(sin*k*Pi*x/k,k = 1 .. infinity) = sin*Pi*x+sin*2*Pi *x/2+sin*3*Pi*x/3+sin*4*Pi*x/4+sin*5*Pi*x/5+sin*6*Pi*x/6+sin*7*Pi*x/7+ sin*8*Pi*x/8+` . . . `;" "6#/-%$SumG6$*,%$sinG\"\"\"%\"kGF)%#PiGF)%\"x GF)F*!\"\"/F*;F)%)infinityG,4*(F(F)F+F)F,F)F)*,F(F)\"\"#F)F+F)F,F)F4F- F)*,F(F)\"\"$F)F+F)F,F)F6F-F)*,F(F)\"\"%F)F+F)F,F)F8F-F)*,F(F)\"\"&F)F +F)F,F)F:F-F)*,F(F)\"\"'F)F+F)F,F)FF-F)*,F (F)\"\")F)F+F)F,F)F@F-F)%(~.~.~.~GF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The following picture \+ shows the graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 58 " together with the graph of the truncated Fourier series: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=sin*Pi*x+sin* 2*Pi*x/2+sin*3*Pi*x/3+sin*4*Pi*x/4" "6#/%\"yG,**(%$sinG\"\"\"%#PiGF(% \"xGF(F(*,F'F(\"\"#F(F)F(F*F(F,!\"\"F(*,F'F(\"\"$F(F)F(F*F(F/F-F(*,F'F (\"\"%F(F)F(F*F(F1F-F(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "f := x -> Pi*(1-x)/2 :\nf_ := x -> f(x-2*floor(x/2)):\nplot([f_(x),sin(Pi*x)+sin(2*Pi*x)/2+ sin(3*Pi*x)/3+sin(4*Pi*x)/4],x=-2..5,y,\n color=[COLOR(RGB,.4,0,. 8),magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 650 211 211 {PLOTDATA 2 "6 &-%'CURVESG6$7^p7$$!\"#\"\"!$\"3c'*[zEjzq:!#<7$$!3GLL$3#*>u%=F-$\"3NAy *))4C6L\"F-7$$!3om;z43m9hC(\\()>lF=7$$!3]m;Hn7\\l7F-$\"3 )zryX2F.<%F=7$$!3WL$ekO9o7\"F-$\"3!4sv,T&*>*>F=7$$!3:***\\7oCA$)*F=$!3 fUM>W&3aj#!#>7$$!3qJLe9_>Z$)F=$!3%)H:%eo>if#F=7$$!3-++Dc#Gp'oF=$!3]\"Q nme<9#\\F=7$$!3lkmmT%*F=7$$!3!p****\\sZL\\#F=$!3#fl4?=U\"z6F-7$$!38)*****\\bMN%=?`\"F-7$$!3!)))**\\7Qe b:FR$!3%3$zSt7OY:F-7$$!3h#))***\\P*[U'!#?$!3U\\BHFTqg:F-7$$!3+!))*\\i: Uf=Fbq$!3geXBab(yc\"F-7$$\"3hA,+D101FFbq$\"3MDITscam:F-7$$\"3AD,]7G_rs Fbq$\"3;;3ZXUPf:F-7$$\"3y7+++&*p$=\"FR$\"3w1'G&=G?_:F-7$$\"3$Q,+]P)))4 IFR$\"3\")p(f26Rj!R\"F-7$$\"3^nmT&G\"H5=F=$\"3Gl(QmVOkG\"F-7$$\"3 9NLLej%yQ$F=$\"3e\\LDlYjQ5F-7$$\"3gOLLLCCCZF=$\"3cw.y119(G)F=7$$\"3-,+ vo()yyiF=$\"3_7aFL\\EXeF=7$$\"3!GLLLoD[l(F=$\"3eKPJ-7z$o$F=7$$\"3u.+vo ibk\"*F=$\"3)pJ\"HW>J78F=7$$\"3a+]i!o<-1\"F-$!3]2fQ`6(*e%*FR7$$\"3qLL3 -$=-@\"F-$!3+0cPn85-LF=7$$\"3kL$3xplzM\"F-$!3mQ(3\")RKeY&F=7$$\"3gmm\" H([a'\\\"F-$!3?'QjYi3(*z(F=7$$\"3wm;ayo(3l\"F-$!3Ey*3+,&RA5F-7$$\"3?+] 7VLA&y\"F-$!30fV4VfUL7F-7$$\"3eL3FuAxd=F-$!3!*H6lPbQZ8F-7$$\"3'pm;a?@. $>F-$!3'4!z?K^Mh9F-7$$\"3[$e9;ue!\\>F-$!3au2C&)zx!\\\"F-7$$\"3A+D\"yF' zn>F-$!3c[OFQ3@?:F-7$$\"3[e9\"f/lr(>F-$!3M&3!zks#\\`\"F-7$$\"3'pT5S\"Q `')>F-$!3eAlI\"pV'\\:F-7$$\"3)f*)f!)>=7*>F-$!3uSZc/>+d:F-7$$\"3Av$4@e- f*>F-$!3OfH#y6gVc\"F-7$$\"3Xa)eh'pe+?F-$\"39:'3DKu)p:F-7$$\"3ZL$3-Nr_+ #F-$\"3u'R]#4h^i:F-7$$\"3%p;/EUYF/#F-$\"3r[Y=./l.:F-7$$\"3)******\\\\@ -3#F-$\"3M,*=rp%yW9F-7$$\"3=+]P*eXN:#F-$\"3#))34!zugH8F-7$$\"3Q++v$opo A#F-$\"33w#**3EIW@\"F-7$$\"3c+](oMf(oBF-$\"311O_#*\\]:**F=7$$\"3#)*** \\ii.j_#F-$\"3TFc-R_!3W(F=7$$\"3%GLL$oT'ym#F-$\"3&*)o*yVu<<_F=7$$\"3'3 ++DE5!>GF-$\"3o6d_[,)H%GF=7$$\"3Mm;a)3rf&HF-$\"3xBG:R_/;pFR7$$\"3*4++v W0d5$F-$!3GB@b'GeKd:F-7$$\"3]5!p!*\\ Dg*RF-$!3Qb;L&>`Xc\"F-7$$\"3*\\7`%*eE1+%F-$\"3K>NQ#37)p:F-7$$\"3)RN@-x G)4SF-$\"3s#GM1Ld`b\"F-7$$\"3&Qe*)4&4.>SF-$\"3cW]))yD!4a\"F-7$$\"3rUg_ 7`VPSF-$\"3!)plQvI*>^\"F-7$$\"3o+D1u'Re0%F-$\"39'4))=d$3$[\"F-7$$\"3]< a8(R[E4%F-$\"3%z9\"*[ck_U\"F-7$$\"3LM$3-7d%HTF-$\"3_*>%*ybXuO\"F-7$$\" 31oTg2R5(>%F-$\"3u[LtKi=h7F-7$$\"3#4++]p]ZE%F-$\"31*\\sv!p#\\:\"F-7$$ \"3$QL$e*R7)>WF-$\"3jWW'e^lN6*F=7$$\"3'pmmmV,&eXF-$\"3#fB#p:K/NpF=7$$ \"3<+](o(GP1ZF-$\"3dD5_kSG7YF=7$$\"3g+]78Z!z%[F-$\"35^1Yzr5*Q#F=7$$\" \"&F*$F*F*-%&COLORG6&%$RGBG$\"\"%!\"\"F*$\"\")Feel-F$6$7iz7$F($\"3u>)y JRurz*!#L7$$!3qmT5!\\F4)>F-$\"3q#=.RA*)fQ#F=7$$!3QL$3-)\\&='>F-$\"3qY, !3pB\"3ZF=7$$!31+DJqCyU>F-$\"3G1/?bxB0pF=7$$!3vmmTg*4P#>F-$\"3+F57z8T@ *)F=7$$!3WL3_]uj/>F-$\"3uxtRSo#32\"F-7$$!37+]iS\\c&)=F-$\"3\\T'>?Y!pA7 F-7$$!3gm\"H2V#\\m=F-$\"3aNp[;`$\\M\"F-7$F/$\"3zz'*y4>&fV\"F-7$$!3'\\7 `>.D3$=F-$\"3MhA1drS*[\"F-7$$!3k;H2V,B9=F-$\"3yPoW%[)R>:F-7$$!3I3F>a_j (z\"F-$\"3(etK/Gnq_\"F-7$$!3)**\\7`OS5y\"F-$\"3Ai.?Px99:F-7$$!3M$3_ve] yu\"F-$\"3'RO)H&\\ZfV\"F-7$F4$\"3<3f78<528F-7$$!3O$e*[LaLx;F-$\"3%ek5? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 25 " can be drawn as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "f := x -> piecewi se(x<1,0,2):\n'f(x)'=f(x);\nf_ := x -> f(x-3*floor(x/3)):\n'f_(x)'='f( x-3*floor(x/3))';\n``=value(rhs(%));\nplot(f_(x),x=-1..8,y,color=COLOR (RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6 #%\"xG-%*PIECEWISEG6$7$\"\"!2F'\"\"\"7$\"\"#%*otherwiseG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*&\"\"$F,-%&floo rG6#,$*&F.!\"\"F'F,F,F,F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PI ECEWISEG6$7$\"\"!2,&%\"xG\"\"\"*&\"\"$F--%&floorG6#,$*&F/!\"\"F,F-F-F- F5F-7$\"\"#%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 563 169 169 {PLOTDATA 2 "6'-%'CURVESG6#7_r7$$!\"\"\"\"!$\"\"#F*7$$!3k+++DhDQ!)!#=F +7$$!3g***\\(oKNJjF0F+7$$!3W****\\([iF+7$$!314+DJqUMLF@F+7$$!3)y++]7\"p06F @F+7$$!3'e2vV[r][&!#?F+7$$\"3;3F*\\7Gon)!#A$F*F*7$$\"3/I\\(o/3'ecFJFO7 $$\"3I$**\\7yWI7\"F@FO7$$\"3)Q*\\PMFTPAF@FO7$$\"3[%***\\(o!y^LF@FO7$$ \"3#o******\\_#4yF@FO7$$\"3#****\\7VsmA\"F0FO7$$\"3S***\\i?j(\\@F0FO7$ $\"3'))**\\7)R&G2$F0FO7$$\"3A++D\"ex@)\\F0FO7$$\"3Q)**\\i&zP&)oF0FO7$$ \"3K+]i:d@kyF0FO7$$\"3F-++vM0V))F0FO7$$\"3[,]i!*Q8u#*F0FO7$$\"3q++D1V@ 0(*F0FO7$$\"3M^i:5W)H\")*F0FO7$$\"3'3]iS^a2#**F0FO7$$\"3=Ec,m&RY(**F0F O7$$\"3/vozhC&G+\"!#`%*o]g3+$F`qFO7$$\"3 WD\"yvJ()o+$F`qFO7$$\"3B8`%*Q4%*=IF`qFO7$$\"3f+DJgX*4.$F`qFO7$$\"3W]7y X!4#zIF`qFO7$$\"3s++DJNUFJF`qFO7$$\"3R+vVGa(f@$F`qFO7$$\"32+]iDt_/LF`q FO7$$\"3[***\\Ppdb\\$F`qFO7$$\"3#***\\7eX)Rp$F`qFO7$$\"37++]U,N!y$F`qF O7$$\"3K+](os:n'QF`qFO7$$\"3<](oaKaL\"RF`qFO7$$\"3-+D1CH**fRF`qFO7$$\" 3;P4rtDlrRF`qFO7$$\"3uu$fLA7L)RF`qFO7$$\"3I$f$=[?9*)RF`qFO7$$\"3K7y+t= (\\*RF`qFO7$$\"3KJ?$yp,3+%F`qF+7$$\"3W\\ilA:j1SF`qF+7$$\"3gCJ&>#3&*HSF `qF+7$$\"3u***\\77qK0%F`qF+7$$\"3g**\\7$fM'\\TF`qF+7$$\"3Y*****\\1**fC %F`qF+7$$\"3t***\\itYXV%F`qF+7$$\"3'***\\7.j(ph%F`qF+7$$\"3l***\\PBL&> [F`qF+7$$\"3P*****\\kR:+&F`qF+7$$\"3e++]P.(e>&F`qF+7$$\"3e**\\7GG'>P&F `qF+7$$\"3_++]K%yWc&F`qF+7$$\"3S**\\781iXdF`qF+7$$\"33**\\P*\\$HSeF`qF +7$$\"3n**\\i&Qm\\$fF`qF+7$$\"3su$f$)R1\"efF`qF+7$$\"3y\\P46kC\")fF`qF +7$$\"35VtF99.()fF`qF+7$$\"3IP4YIk&f,'F`qFO7$$\"3()*\\ilVEv-'F`qFO7$$\"3'*\\ 7.ik!Q2'F`qFO7$$\"31++](['3?hF`qFO7$$\"3Q*\\7GG))p@'F`qFO7$$\"3e**\\7y +*QJ'F`qFO7$$\"3,,++qfa+lF`qFO7$$\"31++vy&G9p'F`qFO7$$\"3')*\\P4ezgy'F `qFO7$$\"3_+]7$eI2)oF`qFO7$$\"3F]Pf.y@CpF`qFO7$$\"3/+D1C]qnpF`qFO7$$\" 3-(oz\"HodypF`qFO7$$\"3#\\(oHM'[%*)pF`qFO7$$\"3Ipa&o`%)[*pF`qFO7$$\"3z iSTR/K+qF`qF+7$$\"3HcE(>Mcd+(F`qF+7$$\"3y\\7`WA>6qF`qF+7$$\"3mCcwae$H. (F`qF+7$$\"3a*****\\YzY0(F`qF+7$$\"3Y**\\P*)>OarF`qF+7$$\"3Q***\\P^WSD (F`qF+7$$\"3*3+++**eBV(F`qF+7$$\"3L**\\78%zCi(F`qF+7$$\"3v**\\(o\"*[W! yF`qF+7$$\"\")F*F+-%*THICKNESSG6#F,-%+AXESLABELSG6$Q\"x6\"Q\"yFjal-%&C OLORG6&%$RGBG$\"\"%F)FO$\"\"\"F*-%%VIEWG6$;F(Faal%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The constant coefficient in the Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 " is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c = 1/3;" "6#/% \"cG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 3);" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"!\"\"$" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/3" "6#/%!G*&\" \"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2,x=1..3)" "6#-% $IntG6$\"\"#/%\"xG;\"\"\"\"\"$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 4/3" "6#/%!G*&\"\"%\"\"\"\"\"$!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "f := x -> piecewise(x<1,0,2):\nc=1/3*Int('f(x )',x=0..3);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" cG,$*&#\"\"\"\"\"$F(-%$IntG6$-%\"fG6#%\"xG/F0;\"\"!F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"%\"\"$" }}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "The coefficients of the cosine terms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 51 "T he coefficients of the cosine terms are given by: " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = 2/3;" "6#/&%\"aG6#%\"kG*&\"\" #\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos(2*k*Pi *x/3),x = 1 .. 3);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#*,\"\"#F +%\"kGF+%#PiGF+F*F+\"\"$!\"\"F+/F*;F+F3" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/3;" "6#/%!G*&\"\"#\"\"\" \"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*cos(2*k*Pi*x/3),x = \+ 1 .. 3);" "6#-%$IntG6$*&\"\"#\"\"\"-%$cosG6#*,F'F(%\"kGF(%#PiGF(%\"xGF (\"\"$!\"\"F(/F/;F(F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 4/3;" "6#/%!G*&\"\"%\"\"\"\"\"$!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos(2*k*Pi*x/3),x = 1 .. 3);" "6#-% $IntG6$-%$cosG6#*,\"\"#\"\"\"%\"kGF+%#PiGF+%\"xGF+\"\"$!\"\"/F.;F+F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 4/3" "6#/%!G*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 287 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "3/(2*k*Pi)" "6#*&\"\"$\"\"\"*(\"\"#F%% \"kGF%%#PiGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(2*k*Pi*x/3);" "6#-%$sinG6#*,\"\"#\"\"\"%\"kGF(%#PiGF(%\"xGF(\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, ``],[``, ``],[1, ``]);" "6#-%*PIECEW ISEG6%7$\"\"$%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/(k*Pi);" "6#/%!G*&\"\"#\"\"\"*&%\"kGF'%#PiGF'!\"\"" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "sin(2*k*Pi*x/3);" "6#-%$sinG6#*,\"\"#\"\"\"%\"kGF(%# PiGF(%\"xGF(\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, \+ ``],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"$%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=- 2/(k*Pi)" "6#/%!G,$*&\"\"#\"\"\"*& %\"kGF(%#PiGF(!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(2*k*Pi/3); " "6#-%$sinG6#**\"\"#\"\"\"%\"kGF(%#PiGF(\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "k = 1,2,3,4,` . . . `;" "6'/%\"kG\"\"\"\"\"#\"\"$\"\"%% (~.~.~.~G" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "sin(2*k*Pi/3);" "6#-%$sin G6#**\"\"#\"\"\"%\"kGF(%#PiGF(\"\"$!\"\"" }{TEXT -1 17 " has the value s " }{XPPEDIT 18 0 "sqrt(3)/2,-sqrt(3)/2,0,sqrt(3)/2,-sqrt(3)/2,0,sqr t(3)/2,` . . . `;" "6**&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\",$*&-F%6#F'F(F) F*F*\"\"!*&-F%6#F'F(F)F*,$*&-F%6#F'F(F)F*F*F/*&-F%6#F'F(F)F*%(~.~.~.~G " }{TEXT -1 22 ", so the coefficients " }{XPPEDIT 18 0 "a[k]" "6#&%\"a G6#%\"kG" }{TEXT -1 6 " are: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1] = -sqrt(3)/Pi,a[ 2] = sqrt(3)/(2*Pi),a[3] = 0,a[4] = -sqrt(3)/(4*Pi),a[5] = sqrt(3)/(5* Pi),a[6] = 0,a[7] = -sqrt(3)/(7*Pi),` . . . `;" "6*/&%\"aG6#\"\"\",$*& -%%sqrtG6#\"\"$F'%#PiG!\"\"F//&F%6#\"\"#*&-F+6#F-F'*&F3F'F.F'F//&F%6#F -\"\"!/&F%6#\"\"%,$*&-F+6#F-F'*&F?F'F.F'F/F//&F%6#\"\"&*&-F+6#F-F'*&FH F'F.F'F//&F%6#\"\"'F;/&F%6#\"\"(,$*&-F+6#F-F'*&FTF'F.F'F/F/%(~.~.~.~G " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matri x([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [a[k], `|`, -sqrt(3)/Pi, s qrt(3)/(2*Pi), 0, -sqrt(3)/(4*Pi), sqrt(3)/(5*Pi), 0, -sqrt(3)/(7*Pi), sqrt(3)/(8*Pi), 0, -sqrt(3)/(10*Pi)]])" "6#-%'matrixG6#7$7.%\"kG%\"|g rG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"aG6#F(F),$*& -%%sqrtG6#F,F*%#PiG!\"\"F>*&-F;6#F,F**&F+F*F=F*F>\"\"!,$*&-F;6#F,F**&F -F*F=F*F>F>*&-F;6#F,F**&F.F*F=F*F>FC,$*&-F;6#F,F**&F0F*F=F*F>F>*&-F;6# F,F**&F1F*F=F*F>FC,$*&-F;6#F,F**&F3F*F=F*F>F>" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "f := x -> piecewise(x<1,0,2 ):\ninterface(showassumed=0): k := 'k': assume(k,integer):\na[k]=2/3*I nt('f(x)'*cos(2*k*Pi*x/3),x=1..3);\n``=value(rhs(%));\naa := unapply(r hs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..10)],[a[k],`|`,seq(aa(k ),k=1..10)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%#k|irG,$*& #\"\"#\"\"$\"\"\"-%$IntG6$*&-%\"fG6#%\"xGF--%$cosG6#,$*,F+F-F,!\"\"F'F -%#PiGF-F5F-F-F-/F5;F-F,F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$ **\"\"#\"\"\"-%$sinG6#,$**F'F(\"\"$!\"\"%#k|irGF(%#PiGF(F(F(F0F/F1F/F/ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7.%\"kG%\"|grG\"\"\" \"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"aG6#F(F),$*&F,#F*F+% #PiG!\"\"F<,$*(F+F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "The coeffi cients of the sine terms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 49 "The coefficients of the sine terms \+ are given by: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b[ k] = 2/3;" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x)*sin(2*k*Pi*x/3),x = 1 .. 3);" "6#-%$IntG6$*&-% \"fG6#%\"xG\"\"\"-%$sinG6#*,\"\"#F+%\"kGF+%#PiGF+F*F+\"\"$!\"\"F+/F*;F +F3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/3;" "6#/%!G*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(2*sin(2*k*Pi*x/3),x = 1 .. 3);" "6#-%$IntG6$*&\"\"# \"\"\"-%$sinG6#*,F'F(%\"kGF(%#PiGF(%\"xGF(\"\"$!\"\"F(/F/;F(F0" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` \+ = 4/3;" "6#/%!G*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin(2*k*Pi*x/3),x = 1 .. 3);" "6#-%$IntG6$-%$sinG6#*,\"\"#\"\" \"%\"kGF+%#PiGF+%\"xGF+\"\"$!\"\"/F.;F+F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -4/3;" "6#/%!G,$*&\"\" %\"\"\"\"\"$!\"\"F*" }{TEXT -1 1 " " }{TEXT 288 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "3/(2*k*Pi)" "6#*&\"\"$\"\"\"*(\"\"#F%%\"kGF%%#PiGF%!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(2*k*Pi*x/3);" "6#-%$cosG6#*,\" \"#\"\"\"%\"kGF(%#PiGF(%\"xGF(\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, ``],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"$% !G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -2/(k*Pi);" "6# /%!G,$*&\"\"#\"\"\"*&%\"kGF(%#PiGF(!\"\"F," }{TEXT -1 2 " " } {XPPEDIT 18 0 "cos(2*k*Pi*x/3);" "6#-%$cosG6#*,\"\"#\"\"\"%\"kGF(%#PiG F(%\"xGF(\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, ``] ,[``, ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"$%!G7$F(F(7$\"\"\"F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=- 2/(k*Pi)" "6#/%!G,$*&\"\"#\"\"\"*& %\"kGF(%#PiGF(!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1-cos(2*k*Pi /3));" "6#-%!G6#,&\"\"\"F'-%$cosG6#**\"\"#F'%\"kGF'%#PiGF'\"\"$!\"\"F0 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/(k*Pi);" "6#/%!G*&\"\"#\"\"\"*&% \"kGF'%#PiGF'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(cos(2*k*Pi/3)-1 );" "6#-%!G6#,&-%$cosG6#**\"\"#\"\"\"%\"kGF,%#PiGF,\"\"$!\"\"F,F,F0" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "k = \+ 1, 2, 3, 4, ` . . . `" "6'/%\"kG\"\"\"\"\"#\"\"$\"\"%%(~.~.~.~G" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "cos(2*k*Pi/3)-1;" "6#,&-%$cosG6#**\"\" #\"\"\"%\"kGF)%#PiGF)\"\"$!\"\"F)F)F-" }{TEXT -1 16 " has the values \+ " }{XPPEDIT 18 0 "-3/2,-3/2,0,-3/2,-3/2,0,-3/2,-3/2,0,-3/2,` . . . `; " "6-,$*&\"\"$\"\"\"\"\"#!\"\"F(,$*&F%F&F'F(F(\"\"!,$*&F%F&F'F(F(,$*&F %F&F'F(F(F+,$*&F%F&F'F(F(,$*&F%F&F'F(F(F+,$*&F%F&F'F(F(%(~.~.~.~G" } {TEXT -1 22 ", so the coefficients " }{XPPEDIT 18 0 "b[k];" "6#&%\"bG6 #%\"kG" }{TEXT -1 6 " are: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[1] = -3/Pi,b[2] = -3/(2*P i),b[3] = 0,b[4] = -3/(4*Pi),b[5] = -3/(5*Pi),b[6] = 0,b[7] = -3/(7*Pi ),` . . . `;" "6*/&%\"bG6#\"\"\",$*&\"\"$F'%#PiG!\"\"F,/&F%6#\"\"#,$*& F*F'*&F0F'F+F'F,F,/&F%6#F*\"\"!/&F%6#\"\"%,$*&F*F'*&F;F'F+F'F,F,/&F%6# \"\"&,$*&F*F'*&FBF'F+F'F,F,/&F%6#\"\"'F7/&F%6#\"\"(,$*&F*F'*&FMF'F+F'F ,F,%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2 , 3, 4, 5, 6, 7, 8, 9, 10], [b[k], `|`, -3/Pi, -3/(2*Pi), 0, -3/(4*Pi) , -3/(5*Pi), 0, -3/(7*Pi), -3/(8*Pi), 0, -3/(10*Pi)]])" "6#-%'matrixG6 #7$7.%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.& %\"bG6#F(F),$*&F,F*%#PiG!\"\"F;,$*&F,F**&F+F*F:F*F;F;\"\"!,$*&F,F**&F- F*F:F*F;F;,$*&F,F**&F.F*F:F*F;F;F?,$*&F,F**&F0F*F:F*F;F;,$*&F,F**&F1F* F:F*F;F;F?,$*&F,F**&F3F*F:F*F;F;" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "f := x -> p iecewise(x<1,0,2):\ninterface(showassumed=0): k := 'k': assume(k,integ er):\nb[k]=2/3*Int('f(x)'*sin(2*k*Pi*x/3),x=1..3);\n``=value(rhs(%)); \nbb := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..10)],[ b[k],`|`,seq(bb(k),k=1..10)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#%#k|irG,$*&#\"\"#\"\"$\"\"\"-%$IntG6$*&-%\"fG6#%\"xGF--%$sinG6#, $*,F+F-F,!\"\"F'F-%#PiGF-F5F-F-F-/F5;F-F,F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\",&-%$cosG6#,$**F'F(\"\"$!\"\"%#k|irG F(%#PiGF(F(F(F(F0F(F1F0F2F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7$7.%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"* \"#57.&%\"bG6#F(F),$*&F,F*%#PiG!\"\"F;,$*(F,F*F+F;F:F;F;\"\"!,$*(F,F*F -F;F:F;F;,$*(F,F*F.F;F:F;F;F>,$*(F,F*F0F;F:F;F;,$*(F,F*F1F;F:F;F;F>,$* (F,F*F3F;F:F;F;Q)pprint326\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "It follows that the Fouri er series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " i s: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "F(x) = 4/3-sqrt(3)/Pi;" "6#/-%\"FG6#%\"xG,&*&\"\"%\" \"\"\"\"$!\"\"F+*&-%%sqrtG6#F,F+%#PiGF-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(2/3*Pi*x)-3/Pi;" "6#,&-%$cosG6#**\"\"#\"\"\"\"\"$!\"\"%#PiGF )%\"xGF)F)*&F*F)F,F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(2*Pi*x/3)+ sqrt(3)/(2*Pi);" "6#,&-%$sinG6#**\"\"#\"\"\"%#PiGF)%\"xGF)\"\"$!\"\"F) *&-%%sqrtG6#F,F)*&F(F)F*F)F-F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(4/ 3*Pi*x)-3/(2*Pi);" "6#,&-%$cosG6#**\"\"%\"\"\"\"\"$!\"\"%#PiGF)%\"xGF) F)*&F*F)*&\"\"#F)F,F)F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(4/3*Pi* x)-sqrt(3)/(4*Pi);" "6#,&-%$sinG6#**\"\"%\"\"\"\"\"$!\"\"%#PiGF)%\"xGF )F)*&-%%sqrtG6#F*F)*&F(F)F,F)F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos (8/3*Pi*x)-3/(4*Pi);" "6#,&-%$cosG6#**\"\")\"\"\"\"\"$!\"\"%#PiGF)%\"x GF)F)*&F*F)*&\"\"%F)F,F)F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(8*Pi *x/3);" "6#-%$sinG6#**\"\")\"\"\"%#PiGF(%\"xGF(\"\"$!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``+sqrt(3)/(5*Pi);" "6#,&%!G\"\"\"*&-%%sqrtG6#\"\" $F%*&\"\"&F%%#PiGF%!\"\"F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(10*Pi* x/3)-3/(5*Pi);" "6#,&-%$cosG6#**\"#5\"\"\"%#PiGF)%\"xGF)\"\"$!\"\"F)*& F,F)*&\"\"&F)F*F)F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(10*Pi*x/3)- sqrt(3)/(7*Pi);" "6#,&-%$sinG6#**\"#5\"\"\"%#PiGF)%\"xGF)\"\"$!\"\"F)* &-%%sqrtG6#F,F)*&\"\"(F)F*F)F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos( 14*Pi*x/3)-3/(7*Pi);" "6#,&-%$cosG6#**\"#9\"\"\"%#PiGF)%\"xGF)\"\"$!\" \"F)*&F,F)*&\"\"(F)F*F)F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(14*Pi *x/3)+sqrt(3)/(8*Pi);" "6#,&-%$sinG6#**\"#9\"\"\"%#PiGF)%\"xGF)\"\"$! \"\"F)*&-%%sqrtG6#F,F)*&\"\")F)F*F)F-F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(16*Pi*x/3)-3/(8*Pi);" "6#,&-%$cosG6#**\"#;\"\"\"%#PiGF)%\"xGF) \"\"$!\"\"F)*&F,F)*&\"\")F)F*F)F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "s in(16/3*Pi*x)+`. . . `;" "6#,&-%$sinG6#**\"#;\"\"\"\"\"$!\"\"%#PiGF)% \"xGF)F)%'.~.~.~GF)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``=4/3+`` " "6# /%!G,&*&\"\"%\"\"\"\"\"$!\"\"F(F$F(" }{XPPEDIT 18 0 "Sum(``(-2/(k*Pi)) *sin(2/3*k*Pi)*cos(2*k*Pi*x/3)+``(2/(k*Pi))*(cos(2*k*Pi/3)-1)*sin(2*k* Pi*x/3),k = 1 .. infinity);" "6#-%$SumG6$,&*(-%!G6#,$*&\"\"#\"\"\"*&% \"kGF.%#PiGF.!\"\"F2F.-%$sinG6#**F-F.\"\"$F2F0F.F1F.F.-%$cosG6#*,F-F.F 0F.F1F.%\"xGF.F7F2F.F.*(-F)6#*&F-F.*&F0F.F1F.F2F.,&-F96#**F-F.F0F.F1F. F7F2F.F.F2F.-F46#*,F-F.F0F.F1F.F " 0 "" {MPLTEXT 1 0 124 "FS := (x,n) -> 4/3+sum((-2 /(k*Pi))*sin(2/3*k*Pi)*cos(2*k*Pi*x/3)+\n (2/(k*Pi))*(cos(2*k*Pi/3) -1)*sin(2*k*Pi*x/3),k=1..n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FS Gf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&#\"\"%\"\"$\"\"\"-%$sumG6$, &*,\"\"#F1%\"kG!\"\"%#PiGF9-%$sinG6#,$**F7F1F0F9F8F1F:F1F1F1-%$cosG6#, $*&#F7F0F1*(F8F1F:F19$F1F1F1F1F9*,F7F1F8F9F:F9,&-FAF=F1F1F9F1-F " 0 "" {MPLTEXT 1 0 8 "FS(x,7);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8#\"\"%\" \"$\"\"\"*(%#PiG!\"\"F&#F'\"\"#-%$cosG6#,$**F,F'F&F*F)F'%\"xGF'F'F'F** (F&F'F)F*-%$sinGF/F'F**&F+F'*(F)F*F&F+-F.6#,$**F%F'F&F*F)F'F2F'F'F'F'F '*&#F&F,F'*&F)F*-F5F9F'F'F**&#F'F%F'*(F)F*F&F+-F.6#,$**\"\")F'F&F*F)F' F2F'F'F'F'F**&#F&F%F'*&F)F*-F5FDF'F'F**&#F'\"\"&F'*(F)F*F&F+-F.6#,$** \"#5F'F&F*F)F'F2F'F'F'F'F'*&#F&FNF'*&F)F*-F5FQF'F'F**&#F'\"\"(F'*(F)F* F&F+-F.6#,$**\"#9F'F&F*F)F'F2F'F'F'F'F**&#F&FenF'*&F)F*-F5FhnF'F'F*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The fo llowing picture compares the graphs of some truncated Fourier series w ith the graph of the function f(" }{TEXT 286 1 "x" }{TEXT -1 2 ")." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "f := x -> piecewise(x<1,0,2):\nf_ := x -> f(x-3*floor(x/3)):\nFS \+ := (x,n) -> 4/3+sum((-2/(k*Pi))*sin(2/3*k*Pi)*cos(2*k*Pi*x/3)+\n (2 /(k*Pi))*(cos(2*k*Pi/3)-1)*sin(2*k*Pi*x/3),k=1..n):\nplot([f_(x),seq(F S(x,i),i=1..7)],x=-1..5,y,\n color=[black,red,blue,green,magenta,c oral,brown,navy],linestyle=[3,1$5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 649 275 275 {PLOTDATA 2 "6,-%'CURVESG6%7ip7$$!\"\"\"\"!$\"\"#F*7$$!3/+ ++]2<#p)!#=F+7$$!3[++]7bBavF0F+7$$!3++++D$3XF'F0F+7$$!3c*****\\F)H')\\ F0F+7$$!3J++]i3@/PF0F+7$$!3V++]7F+7$$!33')****\\7;)=\"!#?F+7$$\"3W7+DJ&Rox#FJ$F*F*7$$\"3$4,+D JS=u'FJFN7$$\"3%4+v$4Toq5FFFN7$$\"3!3++v=%=n9FFFN7$$\"3]++vVV=gAFFFN7$ $\"3?++++X=`IFFFN7$$\"3f****\\7[=RYFFFN7$$\"3**)****\\7&=DiFFFN7$$\"3y (*****\\d=(R*FFFN7$$\"3m****\\P'=pD\"F0FN7$$\"3A++vVrZ4>F0FN7$$\"3y+++ ]c.iDF0FN7$$\"3;+++DMe6PF0FN7$$\"32,++]>q0]F0FN7$$\"3h******\\U80jF0FN 7$$\"3'4+++0ytb(F0FN7$$\"3O++v=n&f7)F0FN7$$\"3w****\\(QNXp)F0FN7$$\"3M **\\7.HeK!*F0FN7$$\"3-++v=/jq$*F0FN7$$\"3\")*\\il**F0 FN7$$\"3%)\\P4rN@i**F0FN7$$\"3i(oa5&pW+5!#1\"FgrF+7$$\"3.+++!y?#>6FgrF+7$$\"3'****\\(3wY_7FgrF+7$$\"3#) ******HOTq8FgrF+7$$\"37++v3\">)*\\\"FgrF+7$$\"3:++DEP/B;FgrF+7$$\"3=++ ](o:;v\"FgrF+7$$\"3=++v$)[op=FgrF+7$$\"3%*****\\i%Qq*>FgrF+7$$\"3&**** \\(QIKH@FgrF+7$$\"3#****\\7:xWC#FgrF+7$$\"37++]Zn%)oBFgrF+7$$\"3y***** *4FL(\\#FgrF+7$$\"3#)****\\d6.BEFgrF+7$$\"3(****\\(o3lWFFgrF+7$$\"3<+] iX)p@\"GFgrF+7$$\"3!*****\\A))ozGFgrF+7$$\"3;+]PCK-5HFgrF+7$$\"3(**** \\iid.%HFgrF+7$$\"3')*\\(=F[_bHFgrF+7$$\"3y**\\7G?pqHFgrF+7$$\"3]\\PfG cFyHFgrF+7$$\"3o*\\i!H#fe)HFgrF+7$$\"3auoHH5l*)HFgrF+7$$\"3%)\\7`HGW$* HFgrF+7$$\"3sCcwHYB(*HFgrF+7$$\"3e******Hk-,IFgrFN7$$\"3%)****\\FL!e1$ FgrFN7$$\"36+++D-eIJFgrFN7$$\"3u***\\(=_(zC$FgrFN7$$\"3M+++b*=jP$FgrFN 7$$\"3g***\\(3/3(\\$FgrFN7$$\"33++vB4JBOFgrFN7$$\"3u*****\\KCnu$FgrFN7 $$\"3t**\\(=_D8\"QFgrFN7$$\"3s***\\(=n#f(QFgrFN7$$\"36+D1Mg.2RFgrFN7$$ \"30+]P\\`9QRFgrFN7$$\"3!)\\7.2+q`RFgrFN7$$\"3++vokYDpRFgrFN7$$\"35Dc^ $*>.xRFgrFN7$$\"3=]PMA$4[)RFgrFN7$$\"3_7yv')zp))RFgrFN7$$\"3Gv=<^me#*R FgrFN7$$\"3/Qfe:`Z'*RFgrFN7$$\"3P+++!)RO+SFgrF+7$$\"3m++D;:*R1%FgrF+7$ $\"30++]_!>w7%FgrF+7$$\"3O++v)Q?QD%FgrF+7$$\"3G+++5jypVFgrF+7$$\"3<++] Ujp-XFgrF+7$$\"3++++gEd@YFgrF+7$$\"39++v3'>$[ZFgrF+7$$\"37++D6Ejp[FgrF +7$$\"\"&F*F+-%'COLOURG6&%$RGBGF*F*F*-%*LINESTYLEG6#\"\"$-F$6%7]s7$F($ \"3@GR)4mcU#Fgr7$$!3@****\\i!G\">!*F0$\"3eyV/&=0GT#Fgr7$F.$ \"3==6A$Q%)[R#Fgr7$$!3#3+]78.K7)F0$\"3a%ROPU(*=N#Fgr7$F2$\"35>=)*GQY%H #Fgr7$$!3o***\\(=>P9pF0$\"3[=(ot'fj8AFgr7$F5$\"3.Bt2Q?-<@Fgr7$$!3y**** ***H./j&F0$\"3Rr\\u7%yb+#Fgr7$F8$\"3y\"f*R<'>>)=Fgr7$$!3%****\\(oXDXVF 0$\"3UR[!QaZ*[$\"3IYA9T[#pL\"Fgr7$$!3T++]iy:+>F0$\"3M!pxi$)p^>\"Fgr7$FA$ \"3WJ%yj$oqb5Fgr7$FD$\"3=w6NWlXj\"*F07$FH$\"3\"\\lz,lCQ%yF07$F\\o$\"3v T!eYUD`i'F07$Fbo$\"35i-*GUv]_&F07$Feo$\"3Pis$=()=r`%F07$Fho$\"3.yL*\\; 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You c an advance one frame at a time by using the ->| button in the context \+ bar. 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"" }}{PARA 0 "" 0 "" {TEXT -1 61 "Truncated Fourier series can be obtained using the procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". 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D)Q;#Fbr7$$\"3!=CRhfD%yrFbr$\"3U(3-Hve(f@Fbr7$$\"33?:Al9#H>(Fbr$\"31!> z$3&HQ9#Fbr7$Fiel$\"3ZCe['\\*)*=@Fbr7$$\"3s\"[yAV:2E(Fbr$\"3q\"=&\\Y* \\%)*>Fbr7$F\\fl$\"3x:f'*3XP>>Fbr7$$\"3z6H)=/<#GtFbr$\"3iYy:CFbr7$ $\"3aeE^`0UUtFbr$\"3ek*e\"[8Z9>Fbr7$$\"3I0C9lSictFbr$\"3#)=I7B:S@>Fbr7 $$\"31_@xwv#3P(Fbr$\"3@=O(f\"3VL>Fbr7$$\"3cX;.+YB*R(Fbr$\"3iG:]IRzn>Fb r7$F_fl$\"3SLk'p%o^1?Fbr7$$\"3!y.V/TKeX(Fbr$\"3I<*RW'psQ?Fbr7$$\"3]O\\ f(>BS[(Fbr$\"3_9_t.V4d?Fbr7$$\"3I')3<\"f=\")\\(Fbr$\"3s&*[^]u')f?Fbr7$ $\"3ANou%)R@7vFbr$\"3E1xfOIJe?Fbr7$$\"38%yA$y$4j_(Fbr$\"3;\"GUqS\\F0#F br7$Fbfl$\"3M[6())RWP/#Fbr7$$\"3lvg)3`:2g(Fbr$\"31E1xLzs')>Fbr7$Fefl$ \"3q_?f'=:D&>Fbr7$$\"3C#R6zpJVn(Fbr$\"3k)4rXKUF&>Fbr7$$\"35n$\\f5Pwo(F br$\"3Sgc$*y'=f&>Fbr7$$\"34Tt)R^U4q(Fbr$\"3yAwFO8xh>Fbr7$$\"3&fJD?#zC9 xFbr$\"3-y(=1vt)p>Fbr7$$\"3zk75Q(e3u(Fbr$\"3cJW^=O`!*>Fbr7$Fhfl$\"3st* ff.eE,#Fbr7$$\"3K$ouHB@Gy(Fbr$\"3/k2\"36eR-#Fbr7$$\"3A^@x6H<)z(Fbr$\"3 &)\\yGwG/L?Fbr7$$\"37>'p0fCN\"yFbr$\"3)=hm9\\7#R?Fbr7$$\"3#z3n$pi()GyF br$\"3gfbP6$*)H?Fbr7$F[gl$\"3:w XE***o--#Fbr7$$\"3m![yA\\T^%zFbr$\"3Gw`'*\\Z[z>Fbr7$F^gl$\"3sR4*fV&[f> Fbr-Fagl6&Fcgl$\"*++++\"!\")FenFen-Fegl6#\"\"\"-%+AXESLABELSG6$Q\"x6\" Q!Fdes-%%VIEWG6$;F(F^gl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 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 18 "Define a function \+ " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " by: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([-1, -2 <= x and x < 0],[0, 0 <= x and x < 1],[1, 1 <= x and x < 2]);" "6# /-%\"fG6#%\"xG-%*PIECEWISEG6%7$,$\"\"\"!\"\"31,$\"\"#F.F'2F'\"\"!7$F43 1F4F'2F'F-7$F-31F-F'2F'F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" " 6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "4;" "6#\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "In order to draw the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 ", we start by def ining " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the \+ interval " }{XPPEDIT 18 0 "-2 <= x;" "6#1,$\"\"#!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 2;" "6#2%!G\"\"#" }{TEXT -1 36 ", and then extend the defin ition of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 146 " to \+ all real numbers by periodicity. For a technical reason, a different s ymbol is needed for the extended periodic function, so it is denoted b y " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "f := x -> piecewise(x<0,-1,x<1,0,1):\n'f(x)'=f(x);\nf_ := x -> f( x-4*floor((x+2)/4)):\n'f_(x)'='f(x-4*floor((x+2)/4))';\nplot(f_(x),x=- 4..8,color=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$!\"\"2F'\"\"!7$F.2F'\"\" \"7$F1%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-% \"fG6#,&F'\"\"\"*&\"\"%F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,\"\"#F,F,F4" } }{PARA 13 "" 1 "" {GLPLOT2D 658 172 172 {PLOTDATA 2 "6'-%'CURVESG6#7_t 7$$!\"%\"\"!$F*F*7$$!3z******\\TVQP!#$F/F+7$$!3'*******f 13EJF/F+7$$!3y***\\(3a(Q4$F/F+7$$!3/++]d,nhIF/F+7$$!3=+](=`nb/$F/F+7$$ !3()***\\i!\\YHIF/F+7$$!3$**\\PMf89-$F/F+7$$!3c**\\i!GiL,$F/F+7$$!3i* \\7y'4J0IF/F+7$$!3o*****\\lfs*HF/$\"\"\"F*7$$!3w***\\P\"40pGF/FW7$$!3% )****\\s@%3u#F/FW7$$!3J++]U.6.DF/FW7$$!3')****\\-G&pD#F/FW7$$!3#***** \\iXmH@F/FW7$$!3(*****\\AjP-?F/FW7$$!33+v$4KYW*>F/$!\"\"F*7$$!3'***\\P >j^')>F/Fho7$$!3%)*\\7yJ'ey>F/Fho7$$!3%****\\iJc1(>F/Fho7$$!3!***\\78j za>F/Fho7$$!3))******4j$*Q>F/Fho7$$!3%)***\\PI;s!>F/Fho7$$!3-++](H'\\v =F/Fho7$$!3;+++&Gc?\"=F/Fho7$$!33++]sih[M[$>FarFho7$$!3'*****\\i\"R(e7FarF ho7$$!3D)**\\(ok\"p?*!#>Fho7$$!3(p***\\78WEeFgsFho7$$!3M'*\\PMP?OTFgsF ho7$$!3q&**\\i:mfW#FgsFho7$$!3P&\\(=nt%3g\"FgsFho7$$!3a]*\\7y&Gdv!#?Fh o7$$\"3dE0v$4@!R*)!#@F+7$$\"3'e0+++!4X$*FgtF+7$$\"3a+++]K$*Q7FarF+7$$ \"3_++++cT%Q#FarF+7$$\"3A*****\\<_$\\]FarF+7$$\"3a'******fs#3uFarF+7$$ \"3d****\\(QFBq)FarF+7$$\"3e-++v@Q'***FarF+7$$\"3E]P4c(Rt+\"F/FW7$$\"3 E+vo%HT],\"F/FW7$$\"3E]7GLGuA5F/FW7$$\"3E+](=PW/.\"F/FW7$$\"3E+D1\\u%e /\"F/FW7$$\"3G++DE0Dh5F/FW7$$\"3E+]i!oc?4\"F/FW7$$\"3G+++NG'G7\"F/FW7$ $\"3G++vV^Z%=\"F/FW7$$\"3I++]_u3Y7F/FW7$$\"3M++v8%fYP\"F/FW7$$\"3P+++v 8B.:F/FW7$$\"3R++]n(p$RF/ FW7$$\"3&\\i:5rWB+#F/Fho7$$\"3+]7.(\\71,#F/Fho7$$\"31vo/$G!))=?F/Fho7$ $\"36+D1p![r-#F/Fho7$$\"3A]P4TOoV?F/Fho7$$\"3!***\\78#>-1#F/Fho7$$\"3m *\\(=d.H$4#F/Fho7$$\"3!****\\7]hj7#F/Fho7$$\"3))**\\P*y.D>#F/Fho7$$\"3 !*****\\xgkeAF/Fho7$$\"3')*******=+QP#F/Fho7$$\"3%)****\\-V&*)[#F/Fho7 $$\"3E+++&\\$pPFF/Fho7$$\"3e******>am%*HF/Fho7$$\"3k*****\\JigC$F/Fho7 $$\"3%*****\\P \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 35 "The Fourier series coefficients of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 30 " can be determined as follows ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "c = 1/4;" "6#/%\"cG*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = -2 .. 2);" "6#-%$IntG6$-%\"fG6#%\"x G/F);,$\"\"#!\"\"F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = 1/4;" "6#/%!G*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-1);" "6#-%!G6#,$\"\"\"!\"\"" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = -1/4;" "6#/%!G,$*&\"\"\"F'\"\" %!\"\"F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "f := x -> piecewise(x<0,-1,x<1,0,1) :\nc=1/4*Int('f(x)',x=-2..2);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*&#\"\"\"\"\"%F(-%$IntG6$-%\"fG6#%\"xG/F0;!\"# \"\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#!\"\"\"\"%" }}} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "The coefficients of th e cosine terms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 51 "The coefficients of the cosine terms are \+ given by: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = 1/2;" "6#/&%\"aG6#%\"kG*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x)*cos(k*Pi*x/2),x = -2 .. 2);" "6#-%$IntG6$*&-% \"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+\"\"#!\"\"F+/F*;,$F2F3F2 " }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = 1/2;" "6#/% !G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((-1)*`.`*c os(k*Pi*x/2),x = -2 .. 0)+1/Pi;" "6#,&-%$IntG6$*(,$\"\"\"!\"\"F)%\".GF )-%$cosG6#**%\"kGF)%#PiGF)%\"xGF)\"\"#F*F)/F2;,$F3F*\"\"!F)*&F)F)F1F*F )" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1*`.`*cos(k*Pi*x/2),x = 1 .. 2) ;" "6#-%$IntG6$*(\"\"\"F'%\".GF'-%$cosG6#**%\"kGF'%#PiGF'%\"xGF'\"\"#! \"\"F'/F/;F'F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*& \"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(-2/(k*Pi))*sin(k*Pi*x/2);" "6#* &-%!G6#,$*&\"\"#\"\"\"*&%\"kGF*%#PiGF*!\"\"F.F*-%$sinG6#**F,F*F-F*%\"x GF*F)F.F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([0, ``],[``, ``], [-2, ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$\"\"#!\"\"F(" } {XPPEDIT 18 0 "``+1/2;" "6#,&%!G\"\"\"*&F%F%\"\"#!\"\"F%" }{XPPEDIT 18 0 "``(2/(k*Pi))*sin(k*Pi*x/2);" "6#*&-%!G6#*&\"\"#\"\"\"*&%\"kGF)%# PiGF)!\"\"F)-%$sinG6#**F+F)F,F)%\"xGF)F(F-F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([2, ``],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6 %7$\"\"#%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0+1/(k *Pi);" "6#/%!G,&\"\"!\"\"\"*&F'F'*&%\"kGF'%#PiGF'!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(0-sin(k*Pi/2));" "6#-%!G6#,&\"\"!\"\"\"-%$sinG6 #*(%\"kGF(%#PiGF(\"\"#!\"\"F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ -1/(k*Pi);" "6#/%!G,$*&\"\"\"F'*&%\"kGF'%#PiGF'!\"\"F+" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "sin(k*Pi/2)" "6#-%$sinG6#*(%\"kG\"\"\"%#PiGF(\"\"#! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "k = 1,2,3,4,` . . . `;" "6'/%\"kG\"\"\"\"\"#\"\"$\"\"%%(~.~.~.~G " }{TEXT -1 3 " , " }{XPPEDIT 18 0 "sin(k*Pi/2)" "6#-%$sinG6#*(%\"kG\" \"\"%#PiGF(\"\"#!\"\"" }{TEXT -1 16 " has the values " }{XPPEDIT 18 0 "1,0,-1,0,1,0,-1,` . . . `;" "6*\"\"\"\"\"!,$F#!\"\"F$F#F$,$F#F&%(~.~. ~.~G" }{TEXT -1 22 ", so the coefficients " }{XPPEDIT 18 0 "a[k]" "6#& %\"aG6#%\"kG" }{TEXT -1 6 " are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1] = -1/Pi,a[2] = 0,a[3] = 1/(3*Pi),a[4] = 0,a[5] = \+ -1/(5*Pi),a[6] = 0,a[7] = 1/(7*Pi),` . . . `;" "6*/&%\"aG6#\"\"\",$*&F 'F'%#PiG!\"\"F+/&F%6#\"\"#\"\"!/&F%6#\"\"$*&F'F'*&F4F'F*F'F+/&F%6#\"\" %F0/&F%6#\"\"&,$*&F'F'*&F>F'F*F'F+F+/&F%6#\"\"'F0/&F%6#\"\"(*&F'F'*&FI F'F*F'F+%(~.~.~.~G" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] , [a[k], `|`, -1/Pi, 0, 1/(3*Pi), 0, -1/(5*Pi), 0, 1/(7*Pi), 0, -1/(9* Pi), 0, 1/(11*Pi), 0]]);" "6#-%'matrixG6#7$70%\"kG%\"|grG\"\"\"\"\"#\" \"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#770&%\"aG6#F(F),$*&F*F*%#P iG!\"\"F=\"\"!*&F*F**&F,F*F,$*&F*F**&F.F*F*&F*F**&F0F* F,$*&F*F**&F2F*F*&F*F**&F4F*F" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "f := x -> piecewise(x<0,-1,x<1,0,1):\ninterface(showassumed=0 ): k := 'k': assume(k,integer):\na[k]=1/2*Int('f(x)'*cos(k*Pi*x/2),x=- 2..2);\n``=value(rhs(%));\naa := unapply(rhs(%),k): k := 'k':\nmatrix( [[k,`|`,seq(k,k=1..12)],[a[k],`|`,seq(aa(k),k=1..12)]]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6#%#k|irG,$*&#\"\"\"\"\"#F+-%$IntG6$*&-% \"fG6#%\"xGF+-%$cosG6#,$**F,!\"\"F'F+%#PiGF+F4F+F+F+/F4;!\"#F,F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(-%$sinG6#,$*(\"\"#!\"\"%#k|irG \"\"\"%#PiGF/F/F/F.F-F0F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7$70%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"# 5\"#6\"#770&%\"aG6#F(F),$*&F*F*%#PiG!\"\"F=\"\"!,$*&F*F**&F,F*F,$*&F*F**&F.F*F,$*&F*F**&F0F*F,$*&F*F**&F2F*F,$*&F*F**&F4F*FQ)pprint336\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "The coefficients of the sine terms " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 49 "The coefficients of the sine terms are given by: " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b[k] = 1/2;" "6#/&% \"bG6#%\"kG*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f (x)*sin(k*Pi*x/2),x = -2 .. 2);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$s inG6#**%\"kGF+%#PiGF+F*F+\"\"#!\"\"F+/F*;,$F2F3F2" }{TEXT -1 2 " " }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((-1)*`.`*sin(k*Pi*x/2),x = -2 \+ .. 0)+1/2;" "6#,&-%$IntG6$*(,$\"\"\"!\"\"F)%\".GF)-%$sinG6#**%\"kGF)%# PiGF)%\"xGF)\"\"#F*F)/F2;,$F3F*\"\"!F)*&F)F)F3F*F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(1*`.`*sin(k*Pi*x/2),x = 1 .. 2);" "6#-%$IntG6$*(\" \"\"F'%\".GF'-%$sinG6#**%\"kGF'%#PiGF'%\"xGF'\"\"#!\"\"F'/F/;F'F0" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\" " }{XPPEDIT 18 0 "``(2/(k*Pi))*cos(k*Pi*x/2);" "6#*&-%!G6#*&\"\"#\"\" \"*&%\"kGF)%#PiGF)!\"\"F)-%$cosG6#**F+F)F,F)%\"xGF)F(F-F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([0, ``],[``, ``],[-2, ``]);" "6#-%*PIE CEWISEG6%7$\"\"!%!G7$F(F(7$,$\"\"#!\"\"F(" }{XPPEDIT 18 0 "``+1/2;" "6 #,&%!G\"\"\"*&F%F%\"\"#!\"\"F%" }{XPPEDIT 18 0 "``(-2/(k*Pi))*cos(k*Pi *x/2);" "6#*&-%!G6#,$*&\"\"#\"\"\"*&%\"kGF*%#PiGF*!\"\"F.F*-%$cosG6#** F,F*F-F*%\"xGF*F)F.F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2, ` `],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"#%!G7$F(F(7$\"\"\"F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/(k*Pi)" "6#/%!G*&\"\"\"F&*&%\"kGF &%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(1-cos*k*Pi)+1/(k*Pi)" "6#,&-%!G6#,& \"\"\"F(*(%$cosGF(%\"kGF(%#PiGF(!\"\"F(*&F(F(*&F+F(F,F(F-F(" } {XPPEDIT 18 0 "``(-cos*k*Pi+cos(k*Pi/2))" "6#-%!G6#,&*(%$cosG\"\"\"%\" kGF)%#PiGF)!\"\"-F(6#*(F*F)F+F)\"\"#F,F)" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(k*Pi);" "6#/%!G*&\"\"\"F&*&%\"kGF&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(1-2*cos*k*Pi+cos(k*Pi/2));" "6#-%!G6#,(\"\"\"F'**\"\"#F'%$cos GF'%\"kGF'%#PiGF'!\"\"-F*6#*(F+F'F,F'F)F-F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(k*Pi);" "6#/%!G*&\"\"\"F&*&%\"kGF&%#PiGF&!\"\"" } {XPPEDIT 18 0 "``(1-2*(-1)^k+cos(k*Pi/2));" "6#-%!G6#,(\"\"\"F'*&\"\"# F'),$F'!\"\"%\"kGF'F,-%$cosG6#*(F-F'%#PiGF'F)F,F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "k = 1, 2, 3, 4, ` . . . `" "6'/%\"kG\"\"\"\"\"#\"\"$\" \"%%(~.~.~.~G" }{TEXT -1 17 ", the expression " }{XPPEDIT 18 0 "1-2*(- 1)^k+cos(k*Pi/2)" "6#,(\"\"\"F$*&\"\"#F$),$F$!\"\"%\"kGF$F)-%$cosG6#*( F*F$%#PiGF$F&F)F$" }{TEXT -1 16 " has the values " }{XPPEDIT 18 0 "3,- 2,3,0,3,-2,3,0,3,-2,3,0,3,-2,3,` . . . `;" "62\"\"$,$\"\"#!\"\"F#\"\"! F#,$F%F&F#F'F#,$F%F&F#F'F#,$F%F&F#%(~.~.~.~G" }{TEXT -1 22 ", so the c oefficients " }{XPPEDIT 18 0 "b[k];" "6#&%\"bG6#%\"kG" }{TEXT -1 6 " a re: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[1] = 3/Pi,b [2] = -1/Pi,b[3] = 1/Pi,b[4] = 0,b[5] = 3/(5*Pi),b[6] = -1/(3*Pi),b[7] = 3/(7*Pi),` . . . `;" "6*/&%\"bG6#\"\"\"*&\"\"$F'%#PiG!\"\"/&F%6#\" \"#,$*&F'F'F*F+F+/&F%6#F)*&F'F'F*F+/&F%6#\"\"%\"\"!/&F%6#\"\"&*&F)F'*& F>F'F*F'F+/&F%6#\"\"',$*&F'F'*&F)F'F*F'F+F+/&F%6#\"\"(*&F)F'*&FKF'F*F' F+%(~.~.~.~G" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2 , 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [b[k], `|`, 3/Pi, -1/Pi, 1/Pi, 0, \+ 3/5/Pi, -1/(3*Pi), 3/7/Pi, 0, 1/(3*Pi), -1/(5*Pi), 3/11/Pi, 0]])" "6#- %'matrixG6#7$70%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\" \"*\"#5\"#6\"#770&%\"bG6#F(F)*&F,F*%#PiG!\"\",$*&F*F*F;F " 0 "" {MPLTEXT 1 0 242 "interface(showassumed=0): k := 'k': assume(k,integer):\nf := x -> pie cewise(x<0,-1,x<1,0,1):\nb[k]=1/2*Int('f(x)'*sin(k*Pi*x/2),x=-2..2);\n ``=value(rhs(%));\nbb := unapply(rhs(%),k):\nmatrix([[k,`|`,seq(k,k=1. .12)],[b[k],`|`,seq(bb(k),k=1..12)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$*&#\"\"\"\"\"#F+-%$IntG6$*&-%\"fG6#%\"xGF+-%$s inG6#,$**F,!\"\"F'F+%#PiGF+F4F+F+F+/F4;!\"#F,F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,(*&\"\"#\"\"\")!\"\",&F)F)%#k|irGF)F)F)F)F)-%$co sG6#,$*(F(F+F-F)%#PiGF)F)F)F)F-F+F3F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$70%#k|irG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"( \"\")\"\"*\"#5\"#6\"#770&%\"bG6#F(F),$*&F,F*%#PiG!\"\"F*,$*&F*F*F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "It follows that the Fou rier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x)=-1/4-1/ Pi" "6#/-%\"FG6#%\"xG,&*&\"\"\"F*\"\"%!\"\"F,*&F*F*%#PiGF,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(Pi*x/2)+3/Pi;" "6#,&-%$cosG6#*(%#PiG\"\" \"%\"xGF)\"\"#!\"\"F)*&\"\"$F)F(F,F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 " sin(Pi*x/2)-1/Pi;" "6#,&-%$sinG6#*(%#PiG\"\"\"%\"xGF)\"\"#!\"\"F)*&F)F )F(F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi*x)+1/(3*Pi);" "6#,&-%$ sinG6#*&%#PiG\"\"\"%\"xGF)F)*&F)F)*&\"\"$F)F(F)!\"\"F)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "cos(3*Pi*x/2)+1/Pi;" "6#,&-%$cosG6#**\"\"$\"\"\"%#Pi GF)%\"xGF)\"\"#!\"\"F)*&F)F)F*F-F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "si n(3*Pi*x/2)-``" "6#,&-%$sinG6#**\"\"$\"\"\"%#PiGF)%\"xGF)\"\"#!\"\"F)% !GF-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(5*Pi);" "6#*&\"\"\"F$*&\"\"&F$% #PiGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(5*Pi*x/2)+3/(5*Pi);" "6#,&-%$cosG6#**\"\"&\"\"\"%#PiGF)%\"xGF)\"\"#!\"\"F)*&\"\"$F)*&F(F)F* F)F-F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(5*Pi*x/2)-1/(3*Pi);" "6#,& -%$sinG6#**\"\"&\"\"\"%#PiGF)%\"xGF)\"\"#!\"\"F)*&F)F)*&\"\"$F)F*F)F-F -" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(3*Pi*x)+1/(7*Pi);" "6#,&-%$sinG 6#*(\"\"$\"\"\"%#PiGF)%\"xGF)F)*&F)F)*&\"\"(F)F*F)!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(7*Pi*x/2)+3/(7*Pi);" "6#,&-%$cosG6#**\"\"(\" \"\"%#PiGF)%\"xGF)\"\"#!\"\"F)*&\"\"$F)*&F(F)F*F)F-F)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "cos(7*Pi*x/2)+` . . . `;" "6#,&-%$cosG6#**\"\"(\"\" \"%#PiGF)%\"xGF)\"\"#!\"\"F)%(~.~.~.~GF)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/4;" "6#/%!G,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "Sum(1/(k*Pi),k = 1 .. infinity);" "6#-%$SumG6$*&\"\"\"F '*&%\"kGF'%#PiGF'!\"\"/F);F'%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-sin(Pi*k/2)*cos(k*Pi*x/2)+(1-2*(-1)^k+cos(k*Pi/2))*sin*k*x);" " 6#-%!G6#,&*&-%$sinG6#*(%#PiG\"\"\"%\"kGF-\"\"#!\"\"F--%$cosG6#**F.F-F, F-%\"xGF-F/F0F-F0**,(F-F-*&F/F-),$F-F0F.F-F0-F26#*(F.F-F,F-F/F0F-F-F)F -F.F-F5F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "We can define a function of two variables to co nstruct finite truncated Fourier series as follows." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "FS := (x, n)->-1/4+sum(-sin(1/2*Pi*k)/(k*Pi)*cos(k*Pi*x/2)+(1-2*(-1)^k+\n \+ cos(k*Pi/2))/(k*Pi)*sin(k*Pi*x/2),k=1..n); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&#\"\" \"\"\"%!\"\"-%$sumG6$,&**-%$sinG6#,$*(\"\"#F1%\"kGF/%#PiGF/F/F/F=F1F>F 1-%$cosG6#,$*&#F/FF/9$F/F/F/F/F1**,(F/F/*&FF1-F8FAF/F//F=;F/9%F/F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The terms as far as the terms in " } {XPPEDIT 18 0 "sin(7*Pi*x/2);" "6#-%$sinG6#**\"\"(\"\"\"%#PiGF(%\"xGF( \"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(7*Pi*x/2);" "6#-%$ cosG6#**\"\"(\"\"\"%#PiGF(%\"xGF(\"\"#!\"\"" }{TEXT -1 10 " are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "FS(x,7);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8#\"\"\"\"\"%!\"\" *&%#PiGF'-%$cosG6#,$*(\"\"#F'F)F%%\"xGF%F%F%F'*(\"\"$F%F)F'-%$sinGF,F% F%*&F)F'-F46#*&F)F%F0F%F%F'*&#F%F2F%*&F)F'-F+6#,$**F2F%F/F'F)F%F0F%F%F %F%F%*&F)F'-F4F=F%F%*&#F%\"\"&F%*&F)F'-F+6#,$**FDF%F/F'F)F%F0F%F%F%F%F '*&#F2FDF%*&F)F'-F4FGF%F%F%*&#F%F2F%*&F)F'-F46#,$*(F2F%F)F%F0F%F%F%F%F '*&#F%\"\"(F%*&F)F'-F+6#,$**FWF%F/F'F)F%F0F%F%F%F%F%*&#F2FWF%*&F)F'-F4 FZF%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture compares the graphs of some truncated Fouri er series with the graph of the function " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "f := x -> piecewise(x<0,-1, x<1,0,1):\nf_ := x -> f(x-4*floor(x/4+1/2)):\nFS := (x,n)->-1/4+sum(-s in(1/2*Pi*k)/(k*Pi)*cos(k*Pi*x/2)+(1-2*(-1)^k+\n cos(k*Pi /2))/(k*Pi)*sin(k*Pi*x/2),k=1..n):\nplot([f_(x),seq(FS(x,i),i=1..7)],x =-3..3,\n color=[black,red,blue,green,magenta,coral,brown,navy],\n \+ linestyle=[3,1$5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 661 344 344 {PLOTDATA 2 "6,-%'CURVESG6%7ip7$$!\"$\"\"!$\"\"\"F*7$$!3!******\\2 <#pG!#F0$!\"\"F*7$$!33+ vof\"eK*>F0FL7$$!3#*\\i!*eJH*)>F0FL7$$!3'***\\7e\"G`)>F0FL7$$!3%)*\\il :)Rx>F0FL7$$!3%******\\:o%p>F0FL7$$!3#***\\(=:3O&>F0FL7$$!3!****\\([\" [x$>F0FL7$$!3')****\\U\"Gg!>F0FL7$$!3/++DO\"3V(=F0FL7$$!3)***\\i&G_!4= F0FL7$$!3#******\\V'zViUC\"F0FL7$$!3-++DhkaI6F0FL7 $$!3s******\\XF`**!#=FL7$$!3u*******>#z2))F[qFL7$$!3S++]7RKvuF[qFL7$$! 3s,+++P'eH'F[qFL7$$!3q)***\\7*3=+&F[qFL7$$!3[)***\\PFcpPF[qFL7$$!3;)** **\\7VQ[#F[qFL7$$!32)***\\i6:.8F[qFL7$$!39$***\\(oKQm'!#>FL7$$!3Wb+++v `hH!#?FL7$$\"3![W7y]bB<\"Fhr$F*F*7$$\"3/X\\i:&[iI&FhrF\\s7$$\"3GXuVB:9 S%*FhrF\\s7$$\"3c%*\\7`MSd8FdrF\\s7$$\"3g%\\(oa?=%=#FdrF\\s7$$\"3l%** \\ilg4,$FdrF\\s7$$\"3w%*\\Pfy^kYFdrF\\s7$$\"3%[***\\i]2=jFdrF\\s7$$\"3 /&**\\(o%*=D'*FdrF\\s7$$\"3]****\\(QIKH\"F[qF\\s7$$\"3K******\\4+p=F[q F\\s7$$\"38****\\7:xWCF[qF\\s7$$\"3E,++vuY)o$F[qF\\s7$$\"3!z******4FL( \\F[qF\\s7$$\"3A)****\\d6.B'F[qF\\s7$$\"3s****\\(o3lW(F[qF\\s7$$\"3U** *\\iX)p@\")F[qF\\s7$$\"35*****\\A))oz)F[qF\\s7$$\"3G)**\\PCK-5*F[qF\\s 7$$\"3X(***\\iid.%*F[qF\\s7$$\"3['*\\(=F[_b*F[qF\\s7$$\"3i'**\\7G?pq*F [qF\\s7$$\"3C(\\PfGcFy*F[qF\\s7$$\"3w'*\\i!H#fe)*F[qF\\s7$$\"3_Y(oHH5l *)*F[qF\\s7$$\"3F'\\7`HGW$**F[qF\\s7$$\"3-Yil(HYB(**F[qF\\s7$$\"3e**** **Hk-,5F0F+7$$\"3%)****\\FL!e1\"F0F+7$$\"36+++D-eI6F0F+7$$\"3u***\\(=_ (zC\"F0F+7$$\"3M+++b*=jP\"F0F+7$$\"3g***\\(3/3(\\\"F0F+7$$\"33++vB4JB; F0F+7$$\"3u*****\\KCnu\"F0F+7$$\"3t**\\(=_D8\"=F0F+7$$\"3s***\\(=n#f(= F0F+7$$\"3*)*\\iS.Oq!>F0F+7$$\"30+]P\\`9Q>F0F+7$$\"3C]7.2+q`>F0F+7$$\" 3A+vokYDp>F0F+7$$\"35Dc^$*>.x>F0F+7$$\"3=]PMA$4[)>F0F+7$$\"3u7yv')zp)) >F0F+7$$\"3Gv=<^me#*>F0F+7$$\"3#y$fe:`Z'*>F0F+7$$\"3P+++!)RO+?F0FL7$$ \"3A++D;:*R1#F0FL7$$\"30++]_!>w7#F0FL7$$\"3O++v)Q?QD#F0FL7$$\"3G+++5jy pBF0FL7$$\"3<++]Ujp-DF0FL7$$\"3++++gEd@EF0FL7$$\"39++v3'>$[FF0FL7$$\"3 7++D6EjpGF0FL7$$\"\"$F*FL-%'COLOURG6&%$RGBGF*F*F*-%*LINESTYLEG6#Fa]l-F $6%7jo7$F($\"3=?P^&e'H\\qF[q7$$!3&*****\\P&3Y$HF0$\"3i4ga*eZ`K(F[q7$F. $\"3[Jrbs_#y\\(F[q7$$!3l*\\iSpo2%GF0$\"3(pe&oE)H+a(F[q7$$!3')**\\78.K7 GF0$\"3_ycT[))=ivF[q7$$!31+v=K>(Qy#F0$\"3'3#z!f4eUc(F[q7$F2$\"31&Q$yPM BYvF[q7$$!3'***\\(=>P9p#F0$\"3nG2!*Q&=DV(F[q7$F5$\"32Yrn=\"[&=sF[q7$$! 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You can advance one frame at a time by using the ->| button in the context bar. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 349 "f := x -> piecewise(x<0,-1,x<1,0,1):\nf_ := x -> f(x-4*floor(x/4+ 1/2)):\nFS := (x,n)->-1/4+sum(-sin(1/2*Pi*k)/(k*Pi)*cos(k*Pi*x/2)+(1-2 *(-1)^k+\n cos(k*Pi/2))/(k*Pi)*sin(k*Pi*x/2),k=1..n): \nf rames := [seq(plot([f_(x),FS(x,i)],x=-2..5.9,\n color=[black,red],li nestyle=[3,1],numpoints=125),i=1..30)]:\nplots[display](frames,inseque nce=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 633 330 330 {PLOTDATA 2 "6#-% (ANIMATEG6@7&-%'CURVESG6%7`v7$$!\"#\"\"!$!\"\"F.7$$!3_A.H'*GML>!#N7F 4F/7$$!3k!eA`7k*o6F4F/7$$!3'4(Q['3S^5\"F4F/7$$!39HhwW<=Z5F4F/7$$!39L!H hqOEy*!#=F/7$$!3/*F\\oF/7$$!3,b$>*>7o>&)F\\oF/7$$!3?Y1e#)p`= zF\\oF/7$$!3'3(Q)zZ')*esF\\oF/7$$!3S****\\F%Q4j'F\\oF/7$$!3u(4(QthjvfF \\oF/7$$!3'o4()39kQP&F\\oF/7$$!3Rm!esqrZs%F\\oF/7$$!3'GeAG4Z00%F\\oF/7 $$!3m9;&ReKOY$F\\oF/7$$!3\"3eA`$)[(HGF\\oF/7$$!3)zN>u*[)[<#F\\oF/7$$!3 YgA.%HGU`\"F\\oF/7$$!3*)RX1$)3gV\"*!#>F/7$$!3hG;&R46Bq&FgqF/7$$!3M<(Q[ I@5E#FgqF/7$$!31*es:K$)z[\"FgqF/7$$!3)ygkIQ`%\\r!#?F/7$$!3?(G**[Ueo@&F drF/7$$!3_mRtmME%G$FdrF/7$$!3%eko&3&o;N\"FdrF/7$$\"3K[n'f\\k#4e!#@$F.F .7$$\"3=;t#fO;hW%FdrFbs7$$\"3adzD#G18J)FdrFbs7$$\"3I_!eZf1sP#FgqFbs7$$ \"3%)3.HhDGBRFgqFbs7$$\"3*Hg^kv9[A(FgqFbs7$$\"3s*Gh^pME0\"F\\oFbs7$$\" 3%=1e(p/(4l\"F\\oFbs7$$\"3MX1e7&4^I#F\\oFbs7$$\"36F!HOP.1#HF\\oFbs7$$ \"3sa$>\\!=(Rc$F\\oFbs7$$\"3K$Q[N/#)H>%F\\oFbs7$$\"3uj!e(pu\\^[F\\oFbs 7$$\"3e.Hh\"4Ed[&F\\oFbs7$$\"3\\;**F\\o Fbs7$$\"3S\"R'[]mkP**F\\oFbs7$$\"3!3vVBG,\"f**F\\oFbs7$$\"3A56?9fb!)** F\\oFbs7$$\"3&o%ega5?+5F4$\"\"\"F.7$$\"3t=t(4)>\\/5F4F^x7$$\"3R!z[t!Hy 35F4F^x7$$\"3$Rt\"4gZO<5F4F^x7$$\"3ZxY$GhYf-\"F4F^x7$$\"3ck0K=.6V5F4F^ x7$$\"3k^k!Q-u-1\"F4F^x7$$\"3S$>*RR!****3\"F4F^x7$$\"3*\\0C(>6F4F^x 7$$\"3]'4P@P+s=\"F4F^x7$$\"3wqQ)*4xp[7F4F^x7$$\"32h^*G8mfJ\"F4F^x7$$\" 3c****\\0u+u8F4F^x7$$\"3*>un*R'4,W\"F4F^x7$$\"3HK!H^8hR]\"F4F^x7$$\"3u DKl<4xn:F4F^x7$$\"31++]bfMJ;F4F^x7$$\"3G1e(Rm@Cp\"F4F^x7$$\"3O^kIAhWe< F4F^x7$$\"3b1eZ=AP@=F4F^x7$$\"3!Qx'fr*)f()=F4F^x7$$\"3$oQ)H^Xd<>F4F^x7 $$\"3')*****48]v%>F4F^x7$$\"3#yDKXW9T'>F4F^x7$$\"3y:X1e(y1)>F4F^x7$$\" 3kW1$[\"4'*))>F4F^x7$$\"3stnfrIC(*>F4F^x7$$\"3/1$)y5OJ**>F4F^x7$$\"3;Q )z*\\TQ,?F4F/7$$\"3Eq8<*oaM+#F4F/7$$\"3\"G!HOG__0?F4F/7$$\"3Znfu1jm4?F 4F/7$$\"3oJ!H^Q2Q,#F4F/7$$\"31.z)p$)Qb/#F4F/7$$\"3Wun%))Gqs2#F4F/7$$\" 3qW13A(y09#F4F/7$$\"3'=unfi/n?#F4F/7$$\"3gx'fu**3wE#F4F/7$$\"3>$>uZS$) *HBF4F/7$$\"3oDK!>;_))R#F4F/7$$\"3o7;X!=27Y#F4F/7$$\"3=EKlmo(\\_#F4F/7 $$\"3O>u#\\**G)*e#F4F/7$$\"3!p4P1sS%\\EF4F/7$$\"3kn4()*)[)Hr#F4F/7$$\" 3'Q1enNXgx#F4F/7$$\"3nk!eZX+N%GF4F/7$$\"3&Rx'fcm-.HF4F/7$$\"3-$[Nk`2<( HF4F/7$$\"3>a$>at'oLIF4F/7$$\"3UDK:8>,&4$F4F/7$$\"3#H!H6!=c4;$F4F/7$$ \"3p1eA0s;FKF4F/7$$\"3)4;&RM%4uG$F4F/7$$\"3+\"eAe+%Q^LF4F/7$$\"3W$[N\\ $p\"QT$F4F/7$$\"3Y***\\())[\\\"[$F4F/7$$\"3nA.HN^0SNF4F/7$$\"3)**G6uq8 tg$F4F/7$$\"3'y$[5,.aqOF4F/7$$\"3>Q[&[>cJt$F4F/7$$\"3SGh,+>#ez$F4F/7$$ \"3'*4()e_ptfQF4F/7$$\"32!esv$e[FRF4F/7$$\"3!G!zB%QX&eRF4F/7$$\"3aDK!4 $\\g*)RF4F/7$$\"3??D+,uV$*RF4F/7$$\"3U9=5r)ps*RF4F/7$$\"3uh9:1h=**RF4F /7$$\"3k36?TB5,SF4Fbs7$$\"3Tc2Dw&=I+%F4Fbs7$$\"3I./I6[$\\+%F4Fbs7$$\"3 t\"**)\\^(*f7SF4Fbs7$$\"3;!e(p\"pk--%F4Fbs7$$\"3.dZ4sXfNSF4Fbs7$$\"3yM >\\_W#40%F4Fbs7$$\"3e#[&=K#*p$3%F4Fbs7$$\"3GJ!z=,uk6%F4Fbs7$$\"3+Q[gv3 )==%F4Fbs7$$\"3q7;&*>ueSUF4Fbs7$$\"3'e$>Cb&H,J%F4Fbs7$$\"3)on4#*=>$oVF 4Fbs7$$\"3\\g^9-&yjV%F4Fbs7$$\"34;X1BXL,XF4Fbs7$$\"3w3()e7DLfXF4Fbs7$$ \"3mK!HwJdXi%F4Fbs7$$\"3uGh,vW@!p%F4Fbs7$$\"3sg^Rv&fbv%F4Fbs7$$\"3?7;? 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}}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 29 " be the function defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < \+ 1;" "6#2%!G\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 28 " is periodic with period 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " can be drawn as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "f := x -> x:\n'f(x)'=f(x);\nf_ := x -> f(x-2*floor((x+1)/2)): \n'f_(x)'='f(x-2*floor((x+1)/2))';\nplot(f_(x),x=-1.5..5.5,y,color=COL OR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"f G6#%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,& F'\"\"\"*&\"\"#F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,F.F,F,F4" }}{PARA 13 " " 1 "" {GLPLOT2D 592 211 211 {PLOTDATA 2 "6'-%'CURVESG6#7ip7$$!3++++++ ++:!#<$\"3++++++++]!#=7$$!3GLL$3#*>uM\"F*$\"3;nmm\"z+e_'F-7$$!3om;z43m 97F*$\"3;LL3->R`yF-7$$!3/+v=d+,S6F*$\"3i**\\7G%**)*f)F-7$$!3QLLe/$f`1 \"F*$\"37mm;apSY$*F-7$$!3[L3-3HdY5F*$\"3Il;z>4FM&*F-7$$!3LL$e9^'yF5F*$ \"3rmmT&)[8A(*F-7$$!3E$3xJJ$R=5F*$\"3Tn\"H#oo1;)*F-7$$!3>Le*[6+!45F*$ \"36o;/^))**4**F-7$$!3E3_v:NI/5F*$\"3OeRZ<.\"\\**F-Fin7$$!3qKLL$=P@!**F-F\\o7$$!37KL3_#4k_*F-F_o7$$ !3kKL$3K\"o]\"*F-Fbo7$$!3P**\\(o*pz-%)F-Feo7$$!35mm\"Hn7\\l(F-Fho7$$!3 GLLekO9oiF-F[p7$$!3:***\\7oCA$[F-F^p7$$!3qJLe9_>ZLF-Fap7$$!3-++Dc#Gp'= F-Fdp7$$!3`Ymm;u\"HW$!#>Fgp7$$\"3izmm\"Ht%o**FipF[q7$$\"34.++vAl1DF-F^ q7$$\"3l+++DmlASF-Faq7$$\"3[,++Dxg$[&F-Fdq7$$\"3^nmT&G\"H5oF-Fgq7$$\"3 9NLLej%yQ)F-Fjq7$$\"3)eLLeRWg0*F-F]r7$$\"3gOLLLCCC(*F-F`r7$$\"3&y(oHz: S@)*F-Fcr7$$\"35>/ED2c=**F-Ffr7$$\"3s*=U#)HSr'**F-Fir7$$\"3.'RAr)>d,5F *$!3mRgxG,G%)**F-7$$\"35t0UW*Hk+\"F*$!3.pUzb0qN**F-7$$\"3;](=2.\"F*$!3!fT&)3p-Gp*F-7$$\"3)o;/,c^,0\" F*$!3=J$e*)R%[)\\*F-7$$\"3g$e*[=_,*3\"F*$!3(R;/^\"y%)4\"*F-7$$\"35+](o ()yy7\"F*$!3(*)**\\7B67s)F-7$$\"3omTgA2o'>\"F*$!34L$eRx#>L!)F-7$$\"3GL LLoD[l7F*$!3?nmm;VByR%F-7$$\"3qLL3-$=-r\"F*$!3-jm;zp\"y*GF-7$$\"3kL$3 xplz%=F*$!3ojm\"H-V._\"F-7$$\"3gmm\"H([a'*>F*$!3-)RLL3F^X$!#?7$$\"3wm; ayo(3:#F*$\"3fnmT&yo(3:F-7$$\"3?+]7VLA&G#F*$\"3'>+]7VLA&GF-7$$\"3'pm;a ?@.V#F*$\"3jpm;a?@.VF-7$$\"3)******\\\\@-e#F*$\"3w******\\\\@-eF-7$$\" 3Q++v$opos#F*$\"3%Q++v$oposF-7$$\"3Y+DJ:X\"yz#F*$\"3q/]7`^9yzF-7$$\"3c +](oMf(oGF*$\"3g0+voMf(o)F-7$$\"3Q](=nTX\"3HF*$\"3w.v=nTX\"3*F-7$$\"3? +Dc'[Jv%HF*$\"3#>+Dc'[Jv%*F-7$$\"35vV[@XAnHF*$\"3+^P%[@XAn*F-7$$\"3-]i Scv\"p)HF*$\"33+D1kbLF*$!3P\"****\\P(*)4oF-7$$\"3 Mm;a)3rfX$F*$!3`OLe9\"*GSaF-7$$\"3*4++vW0dg$F*$!32!****\\_XH%RF-7$$\"3 ;L$3-\"QfYPF*$!3Zom\"z*=1MDF-7$$\"3C+]PWF'Q*QF*$!3g(**\\ibs81\"F-7$$\" 3[LL$e/Xy.%F*$\"3%zMLLe/Xy$Fip7$$\"3m**\\(=<\"e)=%F*$\"3w'**\\(=<\"e)= F-7$$\"3%ymmm(zvLVF*$\"3Wymmm(zvL$F-7$$\"3-nm\"zAAA[%F*$\"3Bqm;zAAA[F- 7$$\"3LM$3-7d%HYF*$\"3IVL3-7d%H'F-7$$\"3#4++]p]Zw%F*$\"3C4++]p]ZwF-7$$ \"3#ym\"HZ:GU[F*$\"3?ym\"HZ:GU)F-7$$\"3$QL$e*R7)>\\F*$\"3GQL$e*R7)>*F- 7$$\"3mmT&)eY[a\\F*$\"3om;a)eY[a*F-7$$\"3S+]7=p:*)\\F*$\"3%R+]7=p:*)*F -7$$\"3%Q&Q`]4\\$*\\F*$\"3MQ&Q`]4\\$**F-7$$\"3;3F%H)\\#y*\\F*$\"3g\"3F %H)\\#y**F-7$$\"3\\i:N:!f@+&F*$!37vV[Y)4%y**F-7$$\"3\"oTgx/$\\1]F*$!3' =$eRA&p]$**F-7$$\"3ZD\"yD6h^,&F*$!3KX(=U())Q[)*F-7$$\"37MeRx\"HQ-&F*$! 3ye;/E#3 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The constant coefficient \+ in the Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 4 " is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " c = 1/2;" "6#/%\"cG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = -1 .. 1);" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$\"\"\"!\"\"F -" }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x,x = -1 .. 1);" "6#-%$IntG6$%\"xG/F&;,$\"\" \"!\"\"F*" }{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 51 "The coefficients of the cosine terms are \+ given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = \+ Int(f(x)*cos*k*Pi*x,x = 1 .. 3);" "6#/&%\"aG6#%\"kG-%$IntG6$*,-%\"fG6# %\"xG\"\"\"%$cosGF0F'F0%#PiGF0F/F0/F/;F0\"\"$" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(x*cos*k*Pi*x,x =-1..1)" "6#/%!G-%$IntG6$*,%\"xG\"\"\"%$cosGF*%\"kGF*%#PiGF*F)F*/F);,$ F*!\"\"F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= 0" "6#/%!G\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " k =1,2,3,` . . . `" "6&/%\"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 8 "because " }{XPPEDIT 18 0 "x*cos*k*Pi *x" "6#*,%\"xG\"\"\"%$cosGF%%\"kGF%%#PiGF%F$F%" }{TEXT -1 7 " is an " }{TEXT 263 3 "odd" }{TEXT -1 11 " function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "The coefficients of the sine terms " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 49 "The coefficients of the sine terms are given by: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b[k] = Int(f(x)*sin*k*Pi*x,x = -1 .. 1);" "6#/&%\"bG6#%\"kG-%$IntG6$*,-%\"fG6#%\"xG\"\"\"%$sinGF0F'F 0%#PiGF0F/F0/F/;,$F0!\"\"F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(x*sin*k*Pi*x,x = -1 .. 1);" "6 #/%!G-%$IntG6$*,%\"xG\"\"\"%$sinGF*%\"kGF*%#PiGF*F)F*/F);,$F*!\"\"F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = 2*Int(x*sin*k*Pi*x,x = 0 .. 1);" "6#/%!G*&\"\"#\"\"\"-%$IntG6$*,%\"x GF'%$sinGF'%\"kGF'%#PiGF'F,F'/F,;\"\"!F'F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "because " }{XPPEDIT 18 0 "x*sin*k*Pi*x;" "6#*,% \"xG\"\"\"%$sinGF%%\"kGF%%#PiGF%F$F%" }{TEXT -1 7 " is an " }{TEXT 263 4 "even" }{TEXT -1 10 " function." }}{PARA 0 "" 0 "" {TEXT -1 4 "N ow " }{XPPEDIT 18 0 "Int(x*sin*k*Pi*x,x) = Int(u*``(dv/dx),x);" "6#/-% $IntG6$*,%\"xG\"\"\"%$sinGF)%\"kGF)%#PiGF)F(F)F(-F%6$*&%\"uGF)-%!G6#*& %#dvGF)%#dxG!\"\"F)F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x" " 6#/%\"uG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*k*Pi*x/(k* Pi);" "6#/%\"vG,$*,%$cosG\"\"\"%\"kGF(%#PiGF(%\"xGF(*&F)F(F*F(!\"\"F- " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the inte gration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-I nt(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\" \"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 10 ", we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " b[k] = 2*(-x*cos*k*Pi*x/(k*Pi));" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\",$*.% \"xGF*%$cosGF*F'F*%#PiGF*F-F**&F'F*F/F*!\"\"F1F*" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6 %7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-2*Int(-c os*k*Pi*x/(k*Pi),x = 0 .. 1);" "6#,$*&\"\"#\"\"\"-%$IntG6$,$*,%$cosGF& %\"kGF&%#PiGF&%\"xGF&*&F-F&F.F&!\"\"F1/F/;\"\"!F&F&F1" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = -2*x*cos*k*Pi*x/(k*Pi)+2*sin*k*Pi*x/(k^2*Pi^2); " "6#/%!G,&*0\"\"#\"\"\"%\"xGF(%$cosGF(%\"kGF(%#PiGF(F)F(*&F+F(F,F(!\" \"F.*.F'F(%$sinGF(F+F(F,F(F)F(*&F+F'F,F'F.F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6 %7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -2*cos *k*Pi/(k*Pi)+0;" "6#/%!G,&*,\"\"#\"\"\"%$cosGF(%\"kGF(%#PiGF(*&F*F(F+F (!\"\"F-\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(-2/(k*Pi))*(- 1)^k;" "6#/%!G*&-F$6#,$*&\"\"#\"\"\"*&%\"kGF+%#PiGF+!\"\"F/F+),$F+F/F- F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (-1)^(k+1)*``(2/(k*Pi));" "6#/%! G*&),$\"\"\"!\"\",&%\"kGF(F(F(F(-F$6#*&\"\"#F(*&F+F(%#PiGF(F)F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([-2/(k*Pi), `if k is eve n`],[2/(k*Pi), `if k is odd`]);" "6#/%!G-%*PIECEWISEG6$7$,$*&\"\"#\"\" \"*&%\"kGF,%#PiGF,!\"\"F0%-if~k~is~evenG7$*&F+F,*&F.F,F/F,F0%,if~k~is~ oddG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6 , 7, 8, 9, 10], [b[k], `|`, 2/Pi, -1/Pi, 2/3/Pi, -1/(2*Pi), 2/5/Pi, -1 /(3*Pi), 2/7/Pi, -1/(4*Pi), 2/9/Pi, -1/(5*Pi)]])" "6#-%'matrixG6#7$7.% \"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"bG6 #F(F)*&F+F*%#PiG!\"\",$*&F*F*F9F:F:*(F+F*F,F:F9F:,$*&F*F**&F+F*F9F*F:F :*(F+F*F.F:F9F:,$*&F*F**&F,F*F9F*F:F:*(F+F*F0F:F9F:,$*&F*F**&F-F*F9F*F :F:*(F+F*F2F:F9F:,$*&F*F**&F.F*F9F*F:F:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "f := x -> x:\ninterface(showassumed=0): k := 'k': assume(k,integer):\nb[k]=I nt('f(x)'*sin(k*Pi*x),x=-1..1);\n``=value(rhs(%));\nbb := unapply(rhs( %),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..10)],[b[k],`|`,seq(bb(k),k =1..10)]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG-%$In tG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#*(F'F0%#PiGF0F/F0F0/F/;!\"\"F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\")!\"\",&F(F(%#k|irG F(F(F,F*%#PiGF*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7.% \"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"bG6 #F(F),$*&F+F*%#PiG!\"\"F*,$*&F*F*F:F;F;,$*(F+F*F,F;F:F;F*,$*&F*F**&F+F *F:F*F;F;,$*(F+F*F.F;F:F;F*,$*&F*F**&F,F*F:F*F;F;,$*(F+F*F0F;F:F;F*,$* &F*F**&F-F*F:F*F;F;,$*(F+F*F2F;F:F;F*,$*&F*F**&F.F*F:F*F;F;Q)pprint356 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 38 "It follows that the Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = Sum(``(2*(-1)^(k+1)/(k*Pi))*sin* k*Pi*x,k = 1 .. infinity);" "6#/-%\"FG6#%\"xG-%$SumG6$*,-%!G6#*(\"\"# \"\"\"),$F1!\"\",&%\"kGF1F1F1F1*&F6F1%#PiGF1F4F1%$sinGF1F6F1F8F1F'F1/F 6;F1%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 2*sin*Pi*x/Pi-s in*2*Pi*x/Pi+2*sin*3*Pi*x/(3*Pi)-sin*4*Pi*x/(2*Pi)+2*sin*5*Pi*x/(5*Pi) -sin*6*Pi*x/(3*Pi)+2*sin*7*Pi*x/(7*Pi)+` . . . `;" "6#/%!G,2*,\"\"#\" \"\"%$sinGF(%#PiGF(%\"xGF(F*!\"\"F(*,F)F(F'F(F*F(F+F(F*F,F,*.F'F(F)F( \"\"$F(F*F(F+F(*&F/F(F*F(F,F(*,F)F(\"\"%F(F*F(F+F(*&F'F(F*F(F,F,*.F'F( F)F(\"\"&F(F*F(F+F(*&F5F(F*F(F,F(*,F)F(\"\"'F(F*F(F+F(*&F/F(F*F(F,F,*. F'F(F)F(\"\"(F(F*F(F+F(*&F;F(F*F(F,F(%(~.~.~.~GF(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The foll owing picture compares the graphs of some truncated Fourier series wit h the graph of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 217 "f_ := x -> x-2*floor((x+1)/2):\nFS := (x,n) - > sum((-1)^(k+1)*(2/(k*Pi))*sin(k*Pi*x),k=1..n);\nplot([f_(x),seq(FS(x ,i),i=1..7)],x=-1..3.5,y,\n color=[black,red,blue,green,magenta,co ral,brown,navy],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF)-%$sumG6$,$*,\"\"#\"\" \")!\"\",&%\"kGF3F3F3F3F7F5%#PiGF5-%$sinG6#*(F7F3F8F39$F3F3F3/F7;F39%F )F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 735 407 407 {PLOTDATA 2 "6,-%'CURV ESG6%7co7$$!\"\"\"\"!F(7$$!3K++]i!G\">!*!#=F,7$$!3!)**\\PMmnl\")F.F07$ $!3s***\\PC\")e?(F.F37$$!3&****\\iqB(RiF.F67$$!3]+](o9e\"y_F.F97$$!3/+ ]P%yjmQ%F.F<7$$!3c+]P4IdjMF.F?7$$!3!***\\P47\"*3DF.FB7$$!3#3+v=-6tb\"F .FE7$$!3k))***\\iKZy&!#>FH7$$\"3k&***\\(ovo$GFJFL7$$\"3O,+]ikFa7F.FO7$ $\"3:****\\(o])GAF.FR7$$\"3E,+]PN.oJF.FU7$$\"3F**\\iS:!4-%F.FX7$$\"3@+ +](3W].&F.Fen7$$\"3?+++]e:%*eF.Fhn7$$\"3q**\\ilq]$*oF.F[o7$$\"3q)**** 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(Q#F.7$F]cl$!3+*yQe#\\tkUF.7$Fbcl$!3s%Qj$z[3wfF.7$$\"3Wj]!*f5AgIF_s$!3 %*)e4>1*>$[(F.7$Fgcl$!3Www/a662()F.7$$\"3!4ip!)z7#*3$F_s$!3n&pAO(3@='* F.7$F\\dl$!3a^FE)*zp?5F_s7$$\"3]M()RH4P5JF_s$!3w(*4!Q\">;P5F_s7$$\"3!) pbk\">Lq6$F_s$!3/!))z5#pCZ5F_s7$$\"360C*QX&pBJF_s$!3)pu^Q!\\E^5F_s7$$ \"3'3CRhrd.8$F_s$!3g6T@.vf\\5F_s7$$\"3\">\"HjSAoVJF_s$!34^/\"p$R.J5F_s 7$Fadl$!3eZ:)[oao&**F.7$$\"3-wgQ)y8a=$F_s$!3)*y(=\"GG**f))F.7$Ffdl$!3g re$43g1n(F.7$$\"3Douz)f6?C$F_s$!3-#pRqjp(fnF.7$F[el$!3I'=&*y4['*H'F.7$ $\"3S-ipv+G&G$F_s$!39xg\")Hc%*RiF.7$$\"3$y.Vawd.I$F_s$!3_'*Q;ex3%G'F.7 $$\"3Et)*=baV:LF_s$!3#z/**y%eW*R'F.7$F`el$!3o%QqgF)=[lF.7$$\"3b$ouHb=Q M$F_s$!3K=>T?'4kn'F.7$$\"3)zl75'R7dLF_s$!3))e\\\"*z3SwnF.7$$\"3SK10p$H /P$F_s$!3\\*)H\"y-T\"GoF.7$Feel$!3(ov\"ztKH " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 " f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 29 " be the function defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = cos*x;" "6#/-%\"fG6#%\"xG*& %$cosG\"\"\"F'F*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-Pi <= x;" "6#1,$% #PiG!\"\"%\"xG" }{XPPEDIT 18 0 "`` < Pi;" "6#2%!G%#PiG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " an d " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " can be drawn as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "f := x -> cos(x):\n'f(x)'=f(x);\nf_ := x -> f(x-Pi*f loor(x/Pi)):\n'f_(x)'='f(x-Pi*floor(x/Pi))';\nplot(f_(x),x=-3..7,y,col or=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6#%\"xG-%$cosGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#% \"xG-%\"fG6#,&F'\"\"\"*&%#PiGF,-%&floorG6#*&F'F,F.!\"\"F,F3" }}{PARA 13 "" 1 "" {GLPLOT2D 580 207 207 {PLOTDATA 2 "6'-%'CURVESG6#7`q7$$!\"$ \"\"!$\"3EbW+m\\#***)*!#=7$$!3YLLLe%G?y#!#<$\"3W/]zvD]g$*F-7$$!3#om;ae sBf#F1$\"3oU!48B>$H&)F-7$$!3ALL$3s%3zBF1$\"3&))oUQkf5B(F-7$$!31LL$e/$Q k@F1$\"3ah&>h)yP$f&F-7$$!3ommT5=q]>F1$\"3v\")QQPpK3PF-7$$!3)****\\7)ok ^=F1$\"3[CX'[yHf_7$$!35++vo1YZ:F1$!3=s4d=SNLBFW7$$!3_mmTNrQT9F1$!31$RKr' H[!H\"F-7$$!3;LL3-OJN8F1$!3\"H$>q]T7LBF-7$$!3lmm\"zC!eH7F1$!3s&RgI4LjM $F-7$$!3\"****\\P*o%Q7\"F1$!3)pmAJSm@K%F-7$$!3+L$3F9(3:5F1$!3o#4#4N!fa F&F-7$$!3;jmm;RFj!*F-$!3aAer=9TmhF-7$$!3'3LL$e4OZrF-$!3'[hJ3NiEb(F-7$$ !3[(*****\\n\\!*\\F-$!3/um;cxP!y)F-7$$!3'z******\\Ow!RF-$!3q/u;$f%=Y#* F-7$$!3V)*****\\ixCGF-$!3kNDpNyn.'*F-7$$!3y)*****\\(R7y\"F-$!3?A-V_#y< %)*F-7$$!3Y\"******\\KqP(FW$!3YX'>N.-G(**F-7$$!3*olTNraz+&FW$!3#40``\" GY()**F-7$$!3MAL3Fp()QEFW$!3mi,Lm$=l***F-7$$!31bT&Q.QVX\"FW$!3E&p\"3pC %*)***F-7$$!3wx)\\iS\"*zp#!#?$!3#o$RB/O'*****F-7$$\"3jef*)fIqCKFes$\"3 IKf\\1![*****F-7$$\"3-&zTg_(RZ\"*Fes$\"3]!y4&G;e****F-7$$\"39j(=#*>4q] \"FW$\"33it7$[k))***F-7$$\"3yYL$eky#*4#FW$\"3+_[]&f'z(***F-7$$\"3*e,]( =UVPoFW$\"3!f#4%>&Qjw**F-7$$\"3]om;z*ev:\"F-$\"3)[2gtjxI$**F-7$$\"3!>+ ]7.%Q%G#F-$\"3V-@lm@@S(*F-7$$\"3GNLL$347T$F-$\"3^^$3*[E!QU*F-7$$\"3tOL LLjM?`F-$\"3TbbU.rw<')F-7$$\"3$3++D\"o7TvF-$\"3kr4\"*eKz)G(F-7$$\"3$>L LL$Q*o]*F-$\"3!)['R>'>A6eF-7$$\"3#pm\"H#GF&e5F1$\"3q(>jg%Gc,\\F-7$$\"3 m++D\"=lj;\"F1$\"3(*zKa6'e\\$RF-7$$\"3a++]iB0p7F1$\"3-FHW\"He=(HF-7$$ \"3S++vV&RF-7$$\"3Kn;zWG))y9F1$\"3)=p`C67%y\"*FW7 $$\"3CML$e9Ege\"F1$!3yvuCHI#H_\"FW7$$\"3'QL3F9F1$!3tso%4)e`;TF-7$$\"3)omT5D,`5#F1$!37rXD)y]T4&F-7$$\"3= nm\"zRQb@#F1$!3B\"*y6wI#*4gF-7$$\"3:++v=>Y2CF1$!3DXU36R8CuF-7$$\"3Znm; zXu9EF1$!3i2^6b='Rk)F-7$$\"34+++]y))GGF1$!3N)f1ej\\]^*F-7$$\"3>++DcljL HF1$!3+$\\8HV\\Xy*F-7$$\"3H++]i_QQIF1$!3YVdQI%)yY**F-7$$\"3O+v=U,1*3$F 1$!3)Q$f;\\[?')**F-7$$\"3U+](=-N(RJF1$!2I-Fgu#)*****F17$$\"3wPf$=Qpg9$ F1$\"3s-%)o%z**)****F-7$$\"3avozTPS_JF1$\"33\"*3b2cT****F-7$$\"3M8yv, \"Q(eJF1$\"35-?//-`)***F-7$$\"3m](=OC(***F-7$$\"3OD1k \"=Tx<$F1$\"3%o,H58nM***F-7$$\"3[+Dc,*4/>$F1$\"3?c*zwu'3))**F-7$$\"3H] iSTtu:KF1$\"3[t*y-/2w(*F-7$$\"3+++]P![hY$F1$\"3SM]mDg#zZ*F-7$$\"3iKLL $Qx$oOF1$\"398#**>M!GW')F-7$$\"3Y+++v.I%)QF1$\"3%3a>,q+kO(F-7$$\"3?mm \"zpe*zSF1$\"3vvMs7v16fF-7$$\"3;,++D\\'QH%F1$\"3ms&*32M7kSF-7$$\"3gm;z p%*\\%R%F1$\"3H-BV-^iDJF-7$$\"3%HL$e9S8&\\%F1$\"35afpB$*\\b@F-7$$\"3%o m;/6E.g%F1$\"3*\\m**)zYG=6F-7$$\"3s++D1#=bq%F1$\"3[tc4w+sqoFes7$$\"3#o m\"H2FO3[F1$!39)pF-7$$\"3yl ;zpe()=]F1$!3J7Gmw56g,&\\F-7$$\"3enmmm*RRL&F1$!39<\"e\\vnH#eF-7$$\"3%zmmTvJ ga&F1$!36#y>I*)[QS(F-7$$\"3]MLe9tOcdF1$!3*[6Qg58Tk)F-7$$\"31,++]Qk\\fF 1$!3K3!3$)G)))[%*F-7$$\"3%zm;/@-/1'F1$!3_oU!\\;jGv*F-7$$\"3![LL3dg6<'F 1$!3YIb.')yJP**F-7$$\"31o;zpBp?iF1$!3w/9,=&z/)**F-7$$\"3K,+voTAqiF1$!3 aN<@c+;****F-7$$\"35$z>Ok:kF'F1$!3IL::f3x****F-7$$\"3'[e*[=rg#G'F1$!2- C*[G$)******F17$$\"3kw$fLf)z)G'F1$\"3S\"*\\]bC%)****F-7$$\"3So\"H#o+* \\H'F1$\"3_gSvoz,ttI'F1$\"3Qxp%e\"\\2(***F-7$$\"3fM$ 3x'fv>jF1$\"3g0,>zOJ$***F-7$$\"3x+v=n=_WjF1$\"3ux;xG]>\")**F-7$$\"3%ym mmw(GpjF1$\"3-)R5vsaH'**F-7$$\"3/M$eRA5\\Z'F1$\"3Z%4_]3qn\")*F-7$$\"3C ++D\"oK0e'F1$\"3-ar%pJq6c*F-7$$\"3m+++]oi\"o'F1$\"3x " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The constant coefficient in the Fourier series of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c = 1/Pi;" "6#/%\"cG*&\"\"\"F& %#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. Pi);" "6 #-%$IntG6$-%\"fG6#%\"xG/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*x,x = 0 .. Pi);" "6#-%$ IntG6$*&%$cosG\"\"\"%\"xGF(/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/Pi" "6#/%!G*&\"\"\"F&% #PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\" xGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi,``],[0,``])" "6#-% *PIECEWISEG6$7$%#PiG%!G7$\"\"!F(" }{XPPEDIT 18 0 " ``=0" "6#/%!G\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "The coefficients of \+ the cosine terms are given by: " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a[ k] = 2/Pi" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x)*cos*2*k*x,x = 0 .. Pi);" "6#-%$IntG6$*,-%\"fG6 #%\"xG\"\"\"%$cosGF+\"\"#F+%\"kGF+F*F+/F*;\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2/Pi" "6#/%!G*& \"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*x*cos*2 *k*x,x = 0 .. Pi);" "6#-%$IntG6$*.%$cosG\"\"\"%\"xGF(F'F(\"\"#F(%\"kGF (F)F(/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=1/Pi" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(cos((2*k+1)*x)+cos((2*k-1)*x),x=0..Pi)" "6#-%$In tG6$,&-%$cosG6#*&,&*&\"\"#\"\"\"%\"kGF.F.F.F.F.%\"xGF.F.-F(6#*&,&*&F-F .F/F.F.F.!\"\"F.F0F.F./F0;\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 21 "( using the formula: " }{XPPEDIT 18 0 "cos*alpha*cos*be ta=1/2" "6#/**%$cosG\"\"\"%&alphaGF&F%F&%%betaGF&*&F&F&\"\"#!\"\"" } {XPPEDIT 18 0 "``(cos(alpha+beta)+cos(alpha-beta) )" "6#-%!G6#,&-%$cos G6#,&%&alphaG\"\"\"%%betaGF,F,-F(6#,&F+F,F-!\"\"F," }{TEXT -1 2 " )" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/Pi" "6#/%!G*& \"\"\"F&%#PiG!\"\"" }{XPPEDIT 18 0 " ``(sin((2*k+1)*x)/(2*k+1)+sin((2* k-1)*x)/(2*k-1) )" "6#-%!G6#,&*&-%$sinG6#*&,&*&\"\"#\"\"\"%\"kGF/F/F/F /F/%\"xGF/F/,&*&F.F/F0F/F/F/F/!\"\"F/*&-F)6#*&,&*&F.F/F0F/F/F/F4F/F1F/ F/,&*&F.F/F0F/F/F/F4F4F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([P i, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F( " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=0" "6#/%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 263 4 "Note" }{TEXT -1 29 ": Apart from its valu e at 0, " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 48 " is es sentially an odd function on the interval " }{XPPEDIT 18 0 "[-Pi/2,Pi/ 2]" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 29 "Thus we see immediately that " }{XPPEDIT 18 0 "a[k]=2/Pi" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*2*k*x,x=-Pi/2..Pi/2)=0" "6#/-%$IntG 6$*,-%\"fG6#%\"xG\"\"\"%$cosGF,\"\"#F,%\"kGF,F+F,/F+;,$*&%#PiGF,F.!\" \"F5*&F4F,F.F5\"\"!" }{TEXT -1 10 ", because " }{XPPEDIT 18 0 "f(x)*co s*2*k*x" "6#*,-%\"fG6#%\"xG\"\"\"%$cosGF(\"\"#F(%\"kGF(F'F(" }{TEXT -1 37 " is ( essentially ) an odd function. " }}{PARA 0 "" 0 "" {TEXT -1 51 "The coefficients of the cosine terms are given by: " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[k] = 2/Pi;" "6#/&%\"bG6#%\"kG*&\"\"#\"\" \"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin*2*k*x,x = 0 .. Pi);" "6#-%$IntG6$*,-%\"fG6#%\"xG\"\"\"%$sinGF+\"\"#F+%\"kGF+F*F+/ F*;\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=2/Pi" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(cos*x*sin*2*k*x,x = 0 .. Pi);" "6#-%$IntG6$*.%$c osG\"\"\"%\"xGF(%$sinGF(\"\"#F(%\"kGF(F)F(/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/Pi" "6#/%! G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin((2*k+1) *x)+sin((2*k-1)*x),x = 0 .. Pi);" "6#-%$IntG6$,&-%$sinG6#*&,&*&\"\"#\" \"\"%\"kGF.F.F.F.F.%\"xGF.F.-F(6#*&,&*&F-F.F/F.F.F.!\"\"F.F0F.F./F0;\" \"!%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "( using the f ormula: " }{XPPEDIT 18 0 "sin*alpha*cos*beta = 1/2;" "6#/**%$sinG\"\" \"%&alphaGF&%$cosGF&%%betaGF&*&F&F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(sin( alpha+beta)+sin(alpha-beta));" "6#-%!G6#,&-%$sinG6#,&%&alphaG\"\"\"%%b etaGF,F,-F(6#,&F+F,F-!\"\"F," }{TEXT -1 3 " )" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/Pi" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{XPPEDIT 18 0 "``(-cos((2*k+1)*x)/(2*k+1)-cos((2*k-1)*x)/(2*k-1));" " 6#-%!G6#,&*&-%$cosG6#*&,&*&\"\"#\"\"\"%\"kGF/F/F/F/F/%\"xGF/F/,&*&F.F/ F0F/F/F/F/!\"\"F4*&-F)6#*&,&*&F.F/F0F/F/F/F4F/F1F/F/,&*&F.F/F0F/F/F/F4 F4F4" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, \+ ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/Pi " "6#/%!G*&\" \"\"F&%#PiG!\"\"" }{XPPEDIT 18 0 "``(-cos((2*k+1)*Pi)/(2*k+1)-cos((2*k -1)*Pi)/(2*k-1)+1/(2*k+1)+1/(2*k-1) )" "6#-%!G6#,**&-%$cosG6#*&,&*&\" \"#\"\"\"%\"kGF/F/F/F/F/%#PiGF/F/,&*&F.F/F0F/F/F/F/!\"\"F4*&-F)6#*&,&* &F.F/F0F/F/F/F4F/F1F/F/,&*&F.F/F0F/F/F/F4F4F4*&F/F/,&*&F.F/F0F/F/F/F/F 4F/*&F/F/,&*&F.F/F0F/F/F/F4F4F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/ Pi " "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{XPPEDIT 18 0 "``(2/(2*k+1)+2/(2*k- 1));" "6#-%!G6#,&*&\"\"#\"\"\",&*&F(F)%\"kGF)F)F)F)!\"\"F)*&F(F),&*&F( F)F,F)F)F)F-F-F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(4*k-2+4*k+2)/((4* k^2-1)*Pi)" "6#/%!G*&,**&\"\"%\"\"\"%\"kGF)F)\"\"#!\"\"*&F(F)F*F)F)F+F )F)*&,&*&F(F)*$F*F+F)F)F)F,F)%#PiGF)F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = 8*k/((4*k^2-1)*Pi);" "6#/%!G*(\"\")\"\"\"%\"kGF'*&,&*&\"\"%F'*$F( \"\"#F'F'F'!\"\"F'%#PiGF'F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8], [b[k], `|`, 8/3/Pi, 16/15/Pi, 24/35/Pi, 32/63/Pi, 40/99/Pi, 48/143/Pi, 56/195/Pi, 64/255/Pi]])" "6#-%'matrixG 6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&%\"bG6#F (F)*(F1F*F,!\"\"%#PiGF7*(\"#;F*\"#:F7F8F7*(\"#CF*\"#NF7F8F7*(\"#KF*\"# jF7F8F7*(\"#SF*\"#**F7F8F7*(\"#[F*\"$V\"F7F8F7*(\"#cF*\"$&>F7F8F7*(\"# kF*\"$b#F7F8F7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "f := x -> cos(x):\ninterfac e(showassumed=0): k := 'k': assume(k,integer):\nb[k]=2/Pi*Int('f(x)'*s in(2*k*x),x=0..Pi);\n``=value(rhs(%));\nbb := unapply(rhs(%),k): k := \+ 'k':\nmatrix([[k,`|`,seq(k,k=1..8)],[b[k],`|`,seq(bb(k),k=1..8)]]);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$,$-%$IntG6$*&-% \"fG6#%\"xG\"\"\"-%$sinG6#,$*(\"\"#F2F'F2F1F2F2F2/F1;\"\"!%#PiG*&F8F2F F8F9F8F*,$*(\"#kF*\"$b#F8F9F8 F*Q)pprint366\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "It foll ows that the Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = Sum(``(8*k/((4*k^2-1)*Pi))*sin*2*k*x,k = 1 .. infinity); " "6#/-%\"FG6#%\"xG-%$SumG6$*,-%!G6#*(\"\")\"\"\"%\"kGF1*&,&*&\"\"%F1* $F2\"\"#F1F1F1!\"\"F1%#PiGF1F9F1%$sinGF1F8F1F2F1F'F1/F2;F1%)infinityG " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 8/(3*Pi)" "6#/%!G*&\"\")\"\"\"*&\" \"$F'%#PiGF'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*x+16/(15*Pi); " "6#,&*(%$sinG\"\"\"\"\"#F&%\"xGF&F&*&\"#;F&*&\"#:F&%#PiGF&!\"\"F&" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*4*x+24/(35*Pi);" "6#,&*(%$sinG\"\" \"\"\"%F&%\"xGF&F&*&\"#CF&*&\"#NF&%#PiGF&!\"\"F&" }{TEXT 289 0 "" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*6*x+32/(63*Pi);" "6#,&*(%$sinG\"\" \"\"\"'F&%\"xGF&F&*&\"#KF&*&\"#jF&%#PiGF&!\"\"F&" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sin*8*x+40/(99*Pi);" "6#,&*(%$sinG\"\"\"\"\")F&%\"xGF&F &*&\"#SF&*&\"#**F&%#PiGF&!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin* 10*x+` . . . `;" "6#,&*(%$sinG\"\"\"\"#5F&%\"xGF&F&%(~.~.~.~GF&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture compares the graphs of some trunca ted Fourier series with the graph of the function " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 118 "Plot the graphs of some truncated Fourie r series and investigate the convergence of the Fourier series experim entally." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "f := x -> piecewise(x<0,x+2,2):\n'f(x)'=f(x);\nf_ := x -> f(x-4*floor((x+2)/4)):\n'f_(x)'='f(x-4*floor((x+2)/4))';\nplot(f _(x),x=-2..10,color=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&F'\"\"\"\"\"#F-2F'\" \"!7$F.%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG- %\"fG6#,&F'\"\"\"*&\"\"%F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,\"\"#F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 569 205 205 {PLOTDATA 2 "6'-%'CURVESG6#7f q7$$!\"#\"\"!$F*F*7$$!3!******\\2<#p=!#<$\"30,++]#HyI\"!#=7$$!3,+++]TV QR()F27$$!37 ******\\lfs**F2$\"34+++X.u-5F/7$$!3')****\\P\"40p)F2$\"3,++D'3\\48\"F/ 7$$!3i+++DiF2$\"3#*****\\UP-y8F/ 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P$F/$\"3M++v8%fYP\"F/7$$\"3P+++v8B.NF/$\"3P+++v8B.:F/7$$\"3Q++Dr0I@OF/ $\"3Q++Dr0I@;F/7$$\"3R++]n(p$RPF/$\"3R++]n(p$RF/7$$\"3!****\\7] hj7%F/F\\q7$$\"3!*****\\xgkeUF/F\\q7$$\"3%)****\\-V&*)[%F/F\\q7$$\"3E+ ++&\\$pPZF/F\\q7$$\"3e******>am%*\\F/F\\q7$$\"3k*****\\JigC&F/F\\q7$$ \"3%*****\\PA&zh*F/F\\q7$$\"3F++]A_ER(*F /F\\q7$$\"3v**\\(o\"*[W!)*F/F\\q7$$\"3-,+D6Ejp)*F/F\\q7$$\"3l,vVeWA-** F/F\\q7$$\"3^+]i0j\"[$**F/F\\q7$$\"3#3v=#HA6^**F/F\\q7$$\"3Q*\\7G:3u'* *F/F\\q7$$\"3Uv$4Y6cb(**F/F\\q7$$\"3o\\iSwSq$)**F/F\\q7$$\"3'R7.#Q?&=* **F/F\\q7$$\"#5F*F+-%*THICKNESSG6#F]q-%+AXESLABELSG6$Q\"x6\"Q!Fbfl-%&C OLORG6&%$RGBG$\"\"%!\"\"F+$\"\"\"F*-%%VIEWG6$;F(Fiel%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 42.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Solution 1" }{TEXT 272 53 " \+ .. using the integral formulas for the coefficients " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 100 ": \+ For the purposes of finding the coefficients via the integral formulas it is sufficient to define " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[-2, 2];" "6#7$,$\" \"#!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f := x -> piecewise(x<0,x+2,2):\n'f (x)'=f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWI SEG6$7$,&F'\"\"\"\"\"#F-2F'\"\"!7$F.%*otherwiseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "First we find the constan t coefficient using the formula " }{XPPEDIT 18 0 "c = 1/4;" "6#/%\"cG* &\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = -2 . . 2);" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$\"\"#!\"\"F-" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "c=1/4*Int('f(x)',x=-2..2);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*&#\"\"\"\"\"%F(-%$IntG6$-%\"fG6#%\"xG/F0;!\"# \"\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"\"\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The cosin e coefficients can be calculated using the formula " }{XPPEDIT 18 0 " a[k] = 1/2;" "6#/&%\"aG6#%\"kG*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x)*cos(k*Pi*x/2),x = -2 .. 2);" "6#-%$IntG6$*&-% \"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+\"\"#!\"\"F+/F*;,$F2F3F2 " }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "f := x \+ -> piecewise(x<0,x+2,2):\ninterface(showassumed=0): k := 'k': assume(k ,integer):\na[k]=1/2*Int('f(x)'*cos(k*Pi*x/2),x=-2..2);\n``=value(rhs( %));\naa := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..10 )],[a[k],`|`,seq(aa(k),k=1..10)]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%#k|irG,$*&#\"\"\"\"\"#F+-%$IntG6$*&-%\"fG6#%\"xGF+-%$cosG 6#,$**F,!\"\"F'F+%#PiGF+F4F+F+F+/F4;!\"#F,F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\",&)!\"\"%#k|irGF(F(F+F(F,!\"#%#PiGF- F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7.%\"kG%\"|grG\"\" \"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"aG6#F(F),$*&F-F*%# PiG!\"#F*\"\"!,$*(F-F*F2!\"\"F:F;F*F<,$*(F-F*\"#DF?F:F;F*F<,$*(F-F*\"# \\F?F:F;F*F<,$*(F-F*\"#\")F?F:F;F*F " 0 "" {MPLTEXT 1 0 248 "f := x -> piecewise(x<0,x+2,2):\ninterface(showassumed=0): k := ' k': assume(k,integer):\nb[k]=1/2*Int('f(x)'*sin(k*Pi*x/2),x=-2..2);\n` `=value(rhs(%));\nbb := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,s eq(k,k=1..10)],[b[k],`|`,seq(bb(k),k=1..10)]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$*&#\"\"\"\"\"#F+-%$IntG6$*&-%\"fG6#% \"xGF+-%$sinG6#,$**F,!\"\"F'F+%#PiGF+F4F+F+F+/F4;!\"#F,F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\")!\"\",&F(F(%#k|irGF(F(F, F*%#PiGF*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7.%\"kG% \"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"bG6#F(F) ,$*&F+F*%#PiG!\"\"F*,$*&F*F*F:F;F;,$*(F+F*F,F;F:F;F*,$*&F*F**&F+F*F:F* F;F;,$*(F+F*F.F;F:F;F*,$*&F*F**&F,F*F:F*F;F;,$*(F+F*F0F;F:F;F*,$*&F*F* *&F-F*F:F*F;F;,$*(F+F*F2F;F:F;F*,$*&F*F**&F.F*F:F*F;F;Q)pprint386\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The Fourier series is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c + ``" "6#,&%\"cG \"\"\"%!GF%" }{XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/2)+b[k]*sin(k*Pi*x/2) ,k = 1 .. infinity);" "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+ F,%#PiGF,%\"xGF,\"\"#!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F 4F,F,/F+;F,%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``= 3/2 +`` " "6#/%!G,&*&\"\"$\"\"\"\"\"#!\"\"F (F$F(" }{XPPEDIT 18 0 "Sum(``(-2*(-1+(-1)^k)/(k^2*Pi^2))*cos(k*Pi*x/2) +``(-2*(-1)^k/(k*Pi))*sin(k*Pi*x/2),k = 1 .. infinity)" "6#-%$SumG6$,& *&-%!G6#,$*(\"\"#\"\"\",&F.!\"\"),$F.F0%\"kGF.F.*&F3F-%#PiGF-F0F0F.-%$ cosG6#**F3F.F5F.%\"xGF.F-F0F.F.*&-F)6#,$*(F-F.),$F.F0F3F.*&F3F.F5F.F0F 0F.-%$sinG6#**F3F.F5F.F:F.F-F0F.F./F3;F.%)infinityG" }{TEXT -1 4 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We ca n define a function " }{TEXT 266 2 "FS" }{TEXT -1 46 " to construct fi nite truncated Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "FS := (x,n) -> 3/2+sum(-2*( -1+(-1)^k)/(k^2*Pi^2)*cos(k*Pi*x/2)\n +(-2*( -1)^k/k/Pi)*sin(k*Pi*x/2),k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&#\"\"$\"\"#\"\"\"-%$su mG6$,&*,F0F1,&F1!\"\")F8%\"kGF1F1F:!\"#%#PiGF;-%$cosG6#,$*&#F1F0F1*(F: F1F " 0 "" {MPLTEXT 1 0 8 "FS(x,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4#\"\"$\"\"#\"\"\"*(\"\"%F'%#PiG!\"#-%$cosG6#,$*(F&!\" \"F*F'%\"xGF'F'F'F'*(F&F'F*F1-%$sinGF.F'F'*&F*F1-F56#*&F*F'F2F'F'F1*&# F)\"\"*F'*&F*F+-F-6#,$**F%F'F&F1F*F'F2F'F'F'F'F'*&#F&F%F'*&F*F1-F5F?F' F'F'*&#F'F&F'*&F*F1-F56#,$*(F&F'F*F'F2F'F'F'F'F1*&#F)\"#DF'*&F*F+-F-6# ,$**\"\"&F'F&F1F*F'F2F'F'F'F'F'*&#F&FUF'*&F*F1-F5FRF'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Here are a few gr aphs of truncated Fourier series along with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "f := x -> piecewi se(x<0,x+2,2):\nf_ := x -> f(x-4*floor((x+2)/4)):\nplot([f_(x),FS(x,1) ,FS(x,2),FS(x,3),FS(x,5),FS(x,6)],x=-2..10,\n color=[black,red,blue, green,magenta,coral],linestyle=[3,1$5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 667 301 301 {PLOTDATA 2 "6*-%'CURVESG6%7fq7$$!\"#\"\"!$F*F*7 $$!3!******\\2<#p=!#<$\"30,++]#HyI\"!#=7$$!3,+++]TVQR()F27$$!37******\\lfs**F2$\"3 4+++X.u-5F/7$$!3')****\\P\"40p)F2$\"3,++D'3\\48\"F/7$$!3i+++DiF2$\"3#*****\\UP-y8F/7$$!3&3++]U.6.&F2$ \"3#*****\\d'*)o\\\"F/7$$!3\"3++]s:.!QF2$\"3#*****\\F%o*>;F/7$$!3y+++D 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(3'y')F0$\"3))ev%ek/e(>F07$Fifl$\"3aL8^1+%R,#F07$$\"3v**\\(o\"*[W!))F0 $\"3]W7koKH_?F07$$\"3-,+D6Ejp))F0$\"3UGh'*f![f2#F07$$\"3l,vVeWA-*)F0$ \"3!H9_LFv$z?F07$$\"3^+]i0j\"[$*)F0$\"3YJ9'pd4k2#F07$$\"3Q*\\7G:3u'*)F 0$\"3k4WQwp.n?F07$F\\gl$\"37/>0Ort^?F0-F_gl6&Fagl$\"*++++\"!\")$F*F*Fe ^p-Fcgl6#\"\"\"-%+AXESLABELSG6$Q\"x6\"Q!F]_p-%%VIEWG6$;F(F\\gl%(DEFAUL TG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "A truncated Fourier series with only a few terms does not appr oximate the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 139 " well. A truncated Fourier series with more terms provides a b etter approximation but there is still a problem near the discontinuit ies of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " which \+ occur where " }{XPPEDIT 18 0 "x=` . . . `,-4,-2,2,6,10,14,` . . . `" " 6*/%\"xG%(~.~.~.~G,$\"\"%!\"\",$\"\"#F(F*\"\"'\"#5\"#9F%" }{TEXT -1 6 " etc. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "f := x -> piecewise(x<0,x+2,2):\n'f(x)'=f(x);\nFouri erSeries(f(x),x=-2..2,numterms=50):\nF := unapply(%,x):\nplot(F(x),x=- 3..9,numpoints=120);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG -%*PIECEWISEG6$7$,&F'\"\"\"\"\"#F-2F'\"\"!7$F.%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 508 204 204 {PLOTDATA 2 "6%-%'CURVESG6$7c\\m7$$! \"$\"\"!$\"3D8HJcOO1?!#<7$$!3=`B)e!=\"o)HF-$\"3bAhMCvg3?F-7$$!3#fqkF-7$$!3#=THNAZs%HF -$\"3[MTjwm4\"*>F-7$$!3*\\w6%H!fS$HF-$\"3)o'))[t\"*o(*>F-7$$!3tI)GF-$\"3]M%[$H***R*>F-7$$!3'e2*o#\\ W:(GF-$\"37$zj'Hi3!*>F-7$$!3=-U]:&p+'GF-$\"39`U.Gh$Q*>F-7$$!33G$>$QXf[ GF-$\"3)4vdjl#\\-?F-7$$!3'RXM6c>r$GF-$\"3?/$pFY,%4?F-7$$!3&)z&\\ReWc#G F-$\"3F)\\cti=\"4?F-7$$!3t0Zw1'pT\"GF-$\"3-4:hgDp,?F-7$$!31K)z&HYp-GF- $\"3ipX]Y\"eF*>F-7$$!3=U]g<)*y*y#F-$\"3)>#yfP#H%*)>F-7$$!3)=DIc+&)ox#F -$\"3cMKh'p)H'*>F-7$$!3+ial$>!)Rw#F-$\"3>YIA\"Gxp+#F-7$$!37s1o\"Qv5v#F -$\"3Rg9G(=z6,#F-7$$!3C#)eqp0F-7$$!3\\-jvX4O7FF-$\"3GFyst1?))>F-7$$!3g7:yLhX*p#F-$\"3!>aR +!pS%*>F-7$$!3[MhL!ylko#F-$\"3!*QZ3TbN1?F-7$$!3Mc2*oUvMn#F-$\"3wQ`p/pV 7?F-7$$!3Ay`Wt][gEF-$\"3'RT8StLk+#F-7$$!33+++?Z\\ZEF-$\"31qV(RuWQ*>F-7 $$!3&>ialO/Xj#F-$\"3=*GG2Vyo)>F-7$$!3\"QC4J,9:i#F-$\"3SMRx_F-7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 12 "Note that: " }{XPPEDIT 18 0 "(Limit(f(x),x = 2,left)+Limit(f(x),x = \+ 2,right))/2 = (2+0)/2;" "6#/*&,&-%&LimitG6%-%\"fG6#%\"xG/F,\"\"#%%left G\"\"\"-F'6%-F*6#F,/F,F.%&rightGF0F0F.!\"\"*&,&F.F0\"\"!F0F0F.F7" } {XPPEDIT 18 0 "`` = 1;" "6#/%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "f := x -> piecewise(x<0,x+2,2):\nFourierSeries(f(x),x=-2..2,numterms=1000): \nF := unapply(%,x):\nevalf(F(2));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+UE?+5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 " }}{PARA 0 "" 0 "" {TEXT 293 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 64 "Find the first few terms of the Fourier s eries for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 13 " defined by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([2*x+2, -1 <= x and x < 0],[2-x, 0 <= x and x < 2]);" "6#/-%\"fG6#%\"xG-%*PIECEW ISEG6$7$,&*&\"\"#\"\"\"F'F/F/F.F/31,$F/!\"\"F'2F'\"\"!7$,&F.F/F'F331F5 F'2F'F." }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 28 " is periodic with period 3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "Plot the graphs of some trunca ted Fourier series and investigate the convergence of the Fourier seri es experimentally." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 172 "f := x -> piecewise(x<0,2*x+2,2-x):\n'f(x)' =f(x);\nf_ := x -> f(x-3*floor((x+1)/3)):\n'f_(x)'='f(x-3*floor((x+1)/ 3))';\nplot(f_(x),x=-1..8,color=COLOR(RGB,.4,0,1),thickness=2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&*&\" \"#\"\"\"F'F/F/F.F/2F'\"\"!7$,&F.F/F'!\"\"%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*&\"\"$F,-%&floorG 6#,&*&F.!\"\"F'F,F,#F,F.F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 565 182 182 {PLOTDATA 2 "6'-%'CURVESG6#7es7$$!\"\"\"\"!$F*F*7$$!3g***\\7.k&4&* !#=$\"3!z++]P>(3)*!#>7$$!3K++]i!G\">!*F/$\"3O*****\\(Quh>F/7$$!3/,+v$4 #pG&)F/$\"3$z***\\7ehUHF/7$$!3k+++DhDQ!)F/$\"3s)*****\\x[BRF/7$$!37+]( 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\\7.ik!Q2'Fen$\"3/](oz`$>E>Fen7$$\"31++](['3?hFen$\"3%*****\\7N\"*z=Fe n7$$\"3Q*\\7GG))p@'Fen$\"3i+v=<<,$y\"Fen7$$\"3e**\\7y+*QJ'Fen$\"3U+](= #*4ho\"Fen7$$\"3&)*\\iS-=sS'Fen$\"39+v$f(>y#f\"Fen7$$\"3,,++qfa+lFen$ \"3+******HSX*\\\"Fen7$$\"35+]Pus)ff'Fen$\"3!***\\iDF,/9Fen7$$\"31++vy &G9p'Fen$\"3#****\\7Ur&38Fen7$$\"3')*\\P4ezgy'Fen$\"39+D1>/#R@\"Fen7$$ \"3_+]7$eI2)oFen$\"3[**\\(oTp#>6Fen7$$\"3/+D1C]qnpFen$\"3(**\\Pf(\\HK5 Fen7$$\"3a*****\\YzY0(Fen$\"3i/++]`?`%*F/7$$\"3Y**\\P*)>OarFen$\"3Y0+D 1,Qc%)F/7$$\"3Q***\\P^WSD(Fen$\"3K1+]i[bfuF/7$$\"37+](=v,KM(Fen$\"3u)* *\\7[#)zc'F/7$$\"3*3+++**eBV(Fen$\"37\"******45kn&F/7$$\"36+Dc,#>u_(Fe n$\"3%*)*\\P%)z!es%F/7$$\"3L**\\78%zCi(Fen$\"3u1+voe?vPF/7$$\"3a***** \\;kMr(Fen$\"3g/++]$e`'GF/7$$\"3v**\\(o\"*[W!yFen$\"3[-+DJ3^b>F/7$$\"3 ))*\\P%eWA-zFen$\"3O7+DcTbx(*F27$$\"\")F*F+-%*THICKNESSG6#\"\"#-%+AXES LABELSG6$Q\"x6\"Q!Fifm-%&COLORG6&%$RGBG$\"\"%F)F+$\"\"\"F*-%%VIEWG6$;F (F_fm%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Solu tion 1" }{TEXT 292 53 " .. using the integral formulas for the coeffic ients " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 100 ": For the purposes of finding the coefficients v ia the integral formulas it is sufficient to define " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[-1, 2];" "6#7$,$\"\"\"!\"\"\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f := x \+ -> piecewise(x<0,2*x+2,2-x):\n'f(x)'=f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&*&\"\"#\"\"\"F'F/F/F.F /2F'\"\"!7$,&F.F/F'!\"\"%*otherwiseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "First we find the constant coefficien t using the formula " }{XPPEDIT 18 0 "c = 1/3;" "6#/%\"cG*&\"\"\"F&\" \"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = -1 .. 2);" "6#- %$IntG6$-%\"fG6#%\"xG/F);,$\"\"\"!\"\"\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "c= 1/3*Int('f(x)',x=-1..2);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*&#\"\"\"\"\"$F(-%$IntG6$-%\"fG6#%\"xG/F0;!\"\" \"\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"#6\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The cosine coef ficients can be calculated using the formula " }{XPPEDIT 18 0 "a[k] = 2/3;" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x)*cos(2*k*Pi*x/3),x = -1 .. 2);" "6#-%$IntG6$*&- %\"fG6#%\"xG\"\"\"-%$cosG6#*,\"\"#F+%\"kGF+%#PiGF+F*F+\"\"$!\"\"F+/F*; ,$F+F4F0" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "f := x -> piecewise(x<0,2*x+2,2-x):\ninterface(showassumed=0): k \+ := 'k': assume(k,integer):\na[k]=2/3*Int('f(x)'*cos(2*k*Pi*x/3),x=-1.. 2);\n``=value(rhs(%));\naa := unapply(rhs(%),k): k := 'k':\nmatrix([[k ,`|`,seq(k,k=1..10)],[a[k],`|`,seq(aa(k),k=1..10)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%#k|irG,$*&#\"\"#\"\"$\"\"\"-%$IntG6$*&-% \"fG6#%\"xGF--%$cosG6#,$*,F+F-F,!\"\"F'F-%#PiGF-F5F-F-F-/F5;F;F+F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"$\"\"#\"\"\"*(,(*&F)F*- %$cosG6#,$**F)F*F(!\"\"%#k|irGF*%#PiGF*F*F*F*-F/6#,$**\"\"%F*F(F3F4F*F 5F*F*F*F(F3F*F4!\"#F5F;F*F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7$7.%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"# 57.&%\"aG6#F(F),$*(\"#FF*F-!\"\"%#PiG!\"#F*,$*(F:F*\"#;F;FF;F " 0 "" {MPLTEXT 1 0 253 "f := x -> piecewise(x<0,2*x+2,2-x):\ninterface(showassumed=0): k := 'k': assume (k,integer):\nb[k]=2/3*Int('f(x)'*sin(2*k*Pi*x/3),x=-1..2);\n``=value( rhs(%));\nbb := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1 ..10)],[b[k],`|`,seq(bb(k),k=1..10)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$*&#\"\"#\"\"$\"\"\"-%$IntG6$*&-%\"fG6#%\"xGF-- %$sinG6#,$*,F+F-F,!\"\"F'F-%#PiGF-F5F-F-F-/F5;F;F+F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"$\"\"#\"\"\"*(,&*&F)F*-%$sinG6#,$**F) F*F(!\"\"%#k|irGF*%#PiGF*F*F*F3-F/6#,$**\"\"%F*F(F3F4F*F5F*F*F*F*F4!\" #F5F;F*F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7.%\"kG%\"| grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"bG6#F(F),$* *F2F*F-!\"\"F,#F*F+%#PiG!\"#F*,$**F2F*\"#;F:F,F;FF:F,F;F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The Fourier series is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c + ``" "6#,&%\"cG\" \"\"%!GF%" }{XPPEDIT 18 0 "Sum(a[k]*cos(2*k*Pi*x/3)+b[k]*sin(2*k*Pi*x/ 3),k = 1 .. infinity);" "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#*, \"\"#F,F+F,%#PiGF,%\"xGF,\"\"$!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#*,F1F,F+ F,F2F,F3F,F4F5F,F,/F+;F,%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 1+``;" "6#/%!G,&\"\"\"F&F$F&" }{XPPEDIT 18 0 "Sum(``((9-3*cos(4/3*k*Pi)-6*cos(2/3*k*Pi))/(2*k^2*Pi^2 ))*cos(2*k*Pi*x/3)+``((6*sin(2/3*k*Pi)-3*sin(4/3*k*Pi))/(2*k^2*Pi^2))* sin(2*k*Pi*x/3),k = 1 .. infinity);" "6#-%$SumG6$,&*&-%!G6#*&,(\"\"*\" \"\"*&\"\"$F.-%$cosG6#**\"\"%F.F0!\"\"%\"kGF.%#PiGF.F.F6*&\"\"'F.-F26# **\"\"#F.F0F6F7F.F8F.F.F6F.*(F>F.*$F7F>F.F8F>F6F.-F26#*,F>F.F7F.F8F.% \"xGF.F0F6F.F.*&-F)6#*&,&*&F:F.-%$sinG6#**F>F.F0F6F7F.F8F.F.F.*&F0F.-F L6#**F5F.F0F6F7F.F8F.F.F6F.*(F>F.*$F7F>F.F8F>F6F.-FL6#*,F>F.F7F.F8F.FD F.F0F6F.F./F7;F.%)infinityG" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can define a function \+ " }{TEXT 266 2 "FS" }{TEXT -1 50 " for constructing finite truncated F ourier series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "FS := (x,n)->1+sum((-3*(cos(4/3*k*Pi)-3+2*cos(2 /3*k*Pi))/(2*k^2*Pi^2))*cos(2*k*Pi*x/3)+\n(3*(-sin(4/3*k*Pi)+2*sin(2/3 *k*Pi))/(2*k^2*Pi^2))*sin(2*k*Pi*x/3),k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&\"\"\"F .-%$sumG6$,&*&#\"\"$\"\"#F.**,(-%$cosG6#,$**\"\"%F.F5!\"\"%\"kGF.%#PiG F.F.F.F5F?*&F6F.-F:6#,$**F6F.F5F?F@F.FAF.F.F.F.F.F@!\"#FAFG-F:6#,$*&#F 6F5F.*(F@F.FAF.9$F.F.F.F.F.F?*&#F5F6F.**,&-%$sinGF;F?*&F6F.-FTFDF.F.F. F@FGFAFG-FTFIF.F.F./F@;F.9%F.F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The truncated Fourier series " } {XPPEDIT 18 0 "c + ``" "6#,&%\"cG\"\"\"%!GF%" }{XPPEDIT 18 0 "Sum(a[k] *cos(2*k*Pi*x/3)+b[k]*sin(2*k*Pi*x/3),k = 1 .. 5);" "6#-%$SumG6$,&*&&% \"aG6#%\"kG\"\"\"-%$cosG6#*,\"\"#F,F+F,%#PiGF,%\"xGF,\"\"$!\"\"F,F,*&& %\"bG6#F+F,-%$sinG6#*,F1F,F+F,F2F,F3F,F4F5F,F,/F+;F,\"\"&" }{TEXT -1 12 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "FS(x,5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4 \"\"\"F$*&#\"#F\"\"%F$*&%#PiG!\"#-%$cosG6#,$**\"\"#F$\"\"$!\"\"F*F$%\" xGF$F$F$F$F$*&#\"\"*F(F$*(F2#F$F1F*F+-%$sinGF.F$F$F$*&#F'\"#;F$*&F*F+- F-6#,$**F(F$F2F3F*F$F4F$F$F$F$F$*&#F7F>F$*(F2F9F*F+-F;FAF$F$F3*&#F'\"# kF$*&F*F+-F-6#,$**\"\")F$F2F3F*F$F4F$F$F$F$F$*&#F7FJF$*(F2F9F*F+-F;FMF $F$F$*&#F'\"$+\"F$*&F*F+-F-6#,$**\"#5F$F2F3F*F$F4F$F$F$F$F$*&#F7FWF$*( F2F9F*F+-F;FZF$F$F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 74 "Here are a few graphs of truncated Fourier series along with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "f := x -> piecewise(x<0,2*x+2,2-x):\nf_ := x -> f(x- 3*floor((x+1)/3)):\nplot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,5),FS(x,6 )],x=-1..8,\n color=[black,red,blue,green,magenta,coral],linestyle=[ 3,1$5]);" }}{PARA 13 "" 1 "" 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#[7GQU'Q/gFfn$\"3SC]3F_/!)>Ffn7$$\"3M7`>Ik&f,'Ffn$\"3ck>MsS#3)>Ffn7$$ \"3()*\\ilVEv-'Ffn$\"3#yAj]n7l(>Ffn7$$\"3'*\\7.ik!Q2'Ffn$\"32.4JAq4G>F fn7$$\"31++](['3?hFfn$\"3;e^\\Jd^w=Ffn7$$\"3Q*\\7GG))p@'Ffn$\"3u`D>&R' =$y\"Ffn7$$\"3e**\\7y+*QJ'Ffn$\"3Y&oh4nRto\"Ffn7$$\"3&)*\\iS-=sS'Ffn$ \"3Q8\"eC&Q&=f\"Ffn7$$\"3,,++qfa+lFfn$\"3%f?XJU/&*\\\"Ffn7$$\"35+]Pus) ff'Ffn$\"3')3/W.Qm/9Ffn7$$\"31++vy&G9p'Ffn$\"3kKK4jr(zI\"Ffn7$$\"3')* \\P4ezgy'Ffn$\"3CCY%etGR@\"Ffn7$$\"3_+]7$eI2)oFfn$\"3TOGCioy>6Ffn7$$\" 3/+D1C]qnpFfn$\"3&4h04=1<.\"Ffn7$$\"3a*****\\YzY0(Ffn$\"3alfE%)G=d%*F1 7$$\"3Y**\\P*)>OarFfn$\"3kBDEwIZf%)F17$$\"3Q***\\P^WSD(Ffn$\"3/j6&))\\ jJX(F17$$\"37+](=v,KM(Ffn$\"3+'>AA9xu_(Ffn$\"3k![)*y![$or%F17$$\"3L**\\78%zCi(Ffn$ \"3'H![4w@J$y$F17$$\"3a*****\\;kMr(Ffn$\"3;v3N3#*>nGF17$$\"3v**\\(o\"* [W!yFfn$\"3_5'e$\\KMO>F17$$\"3))*\\P%eWA-zFfn$\"32aQ0@$ym+\"F17$$\"\") F*$\"3UZFAz3so@F--%'COLOURG6&%$RGBG$\"#5F)$F*F*F_fm-%+AXESLABELSG6$Q\" x6\"Q!Fdfm-%%VIEWG6$;F(Feem%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 8 " }}{PARA 0 "" 0 "" {TEXT 274 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 64 "Find the first few term s of the Fourier series for the function " }{XPPEDIT 18 0 "f(x) = abs( cos(x/2))" "6#/-%\"fG6#%\"xG-%$absG6#-%$cosG6#*&F'\"\"\"\"\"#!\"\"" } {TEXT -1 68 " and investigate the convergence of this Fourier series g raphically." }}{PARA 0 "" 0 "" {TEXT -1 118 "Plot the graphs of some t runcated Fourier series and investigate the convergence of the Fourier series experimentally." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot(abs(cos(x/2)),x=-2*Pi..4*Pi,co lor=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 633 196 196 {PLOTDATA 2 "6'-%'CURVESG6#7]s7$$!3)****>YH&=$G'!#<$\"\"\"\"\" 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uxbF37$$\"3%*GIgE;BN%)F*$\"3[Lb#=L?$[ZF37$$\"3UwedV*)3H')F*$\"3%)=BX,5 KuQF37$$\"3Us`o)eR?$))F*$\"3rn*=L<'\\?HF37$$\"3Vo[zL-*\\.*F*$\"3#Q\\F* )3Cm$>F37$$\"3=^;HEeXI#*F*$\"3*Rx^8#o#3q*Fbr7$$\"3#RV)y=9#fU*F*$\"3E]( )z$y*G')=\"fds*F*$\"3q))3W[=B*\\\"F37$$\"3^%['))=]qD)* F*$\"3'RtRx)fB\"*>F37$$\"3=\"z!RhTR-5!#;$\"3gg:(\\%o?^HF37$$\"3#R$H*3# y@A5Fe^m$\"3w-?V'[2A)QF37$$\"3\"=xp^qL//\"Fe^m$\"3)y$obCOJ/ZF37$$\"3q4 mW*e\\'e5Fe^m$\"3'H!3<#GAu[&F37$$\"39_$*[[q_z5Fe^m$\"32TnhBqjGjF37$$\" 3c%4Kv]//5\"Fe^m$\"354FNdI&45(F37$$\"3R>>ybv2>6Fe^m$\"35fv,%)4YExF37$$ \"3AW<./1vP6Fe^m$\"3sY.iMdm%G)F37$$\"3s+[9o*fw:\"Fe^m$\"3v/8B66B+))F37 $$\"3CdyDK$pv<\"Fe^m$\"3w@\")Gx>mG#*F37$$\"3'=n\"4M^i'>\"Fe^m$\"3g#*eq N')=`&*F37$$\"3]'[Df$4o:7Fe^m$\"3S$=q)zh0\"z*F37$$\"3$[6um'*>fA\"Fe^m$ \"3?K*p%p\"*G#))*F37$$\"3;VFU(**ehB\"Fe^m$\"3!)\\tW\"*oiZ**F37$$\"3Tdq z7&y7C\"Fe^m$\"3-[@:g)G0(**F37$$\"3mr8r&f*Rs'***F37$$\"3+++#*eqjc7Fe^m$\"2))******* ********F*-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fedm-%&COLORG6& %$RGBG$\"\"%!\"\"$F-F-F+-%%VIEWG6$;$!+3`=$G'!\"*$\"+iqjc7!\")%(DEFAULT G" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Solution 1" }{TEXT 275 52 " .. using the integral formulas for the coefficients" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 15 ": The function " }{XPPEDIT 18 0 "f(x) = abs(cos (x/2))" "6#/-%\"fG6#%\"xG-%$absG6#-%$cosG6#*&F'\"\"\"\"\"#!\"\"" } {TEXT -1 12 " has period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiG F%" }{TEXT -1 104 ", but for the purposes of finding the coefficients \+ via the integral formulas it is sufficient to define " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 57 ", in which case we \+ can omit the absolute value operation." }}{PARA 0 "" 0 "" {TEXT -1 57 "First we find the constant coefficient using the formula " }{XPPEDIT 18 0 "c=1/(2*Pi)" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = -Pi .. Pi);" "6#-%$IntG6$-%\"fG6#% \"xG/F);,$%#PiG!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "f := x -> cos(x/2):\n'f( x)'=f(x);\nc=1/(2*Pi)*Int('f(x)',x=-Pi..Pi);\n``=value(rhs(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$cosG6#,$*&\"\"#!\"\"F '\"\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*&#\"\"\"\"\"#F(, $-%$IntG6$-%\"fG6#%\"xG/F1;,$%#PiG!\"\"F5*$F5F6F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,$*&\"\"#\"\"\"%#PiG!\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The cosine coefficients c an be calculated using the formula " }{XPPEDIT 18 0 "a[k] = 1/Pi;" "6 #/&%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(f(x)*cos(k*x),x = -Pi .. Pi);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$ cosG6#*&%\"kGF+F*F+F+/F*;,$%#PiG!\"\"F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "f := x -> cos(x/2):\ninterface(showassumed=0): k := 'k': assume(k,integer): \n'f(x)'=f(x);\na[k]=1/Pi*Int('f(x)'*cos(k*x),x=-Pi..Pi);\n``=value(rh s(%));\naa := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1.. 10)],[a[k],`|`,seq(aa(k),k=1..10)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$cosG6#,$*&\"\"#!\"\"F'\"\"\"F/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6#%#k|irG,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$co sG6#*&F'F1F0F1F1/F0;,$%#PiG!\"\"F9*$F9F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"%\"\"\"%#PiG!\"\")F*,&F(F(%#k|irGF(F(,&F(F**&F'F()F- \"\"#F(F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7.%\"kG %\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"aG6#F(F ),$*(F-F*F,!\"\"%#PiGF:F*,$*(F-F*\"#:F:F;F:F:,$*(F-F*\"#NF:F;F:F*,$*(F -F*\"#jF:F;F:F:,$*(F-F*\"#**F:F;F:F*,$*(F-F*\"$V\"F:F;F:F:,$*(F-F*\"$& >F:F;F:F*,$*(F-F*\"$b#F:F;F:F:,$*(F-F*\"$B$F:F;F:F*,$*(F-F*\"$*RF:F;F: F:Q)pprint416\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The sine coefficients can be calculated using the formula " }{XPPEDIT 18 0 "b[k] = 1/Pi;" "6#/&%\"bG6#%\"kG*&\"\"\"F)%#PiG!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin(k*x),x = -Pi .. Pi); " "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#*&%\"kGF+F*F+F+/F*;,$%#PiG !\"\"F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "The sine coeff icients are all 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 154 "f := x -> cos(x/2):\ninterface(showassumed= 0): k := 'k': assume(k,integer):\n'f(x)'=f(x);\n1/Pi*Int('f(x)'*sin(k* x),x=-Pi..Pi);\nb[k]=value(rhs(%));\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$cosG6#,$*&\"\"#!\"\"F'\"\"\"F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sin G6#*&%#k|irGF,F+F,F,/F+;,$%#PiG!\"\"F5*$F5F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series is " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c+Sum(a[k]*cos(k*x),k = 1 .. infinity);" "6#,&%\"cG \"\"\"-%$SumG6$*&&%\"aG6#%\"kGF%-%$cosG6#*&F-F%%\"xGF%F%/F-;F%%)infini tyGF%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "Sum(``(-4*(-1)^k/(Pi*(2*k-1)*(2*k+1)))*cos(k*x),k = 1 . . infinity);" "6#-%$SumG6$*&-%!G6#,$*(\"\"%\"\"\"),$F-!\"\"%\"kGF-*(%# PiGF-,&*&\"\"#F-F1F-F-F-F0F-,&*&F6F-F1F-F-F-F-F-F0F0F--%$cosG6#*&F1F-% \"xGF-F-/F1;F-%)infinityG" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can define a function " } {TEXT 266 2 "FS" }{TEXT -1 46 " to construct finite truncated Fourier \+ series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "FS := (x,n) -> 2/Pi+sum(-4*(-1)^k/(Pi*(4*k^2-1))*cos( k*x),k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6 \"6$%)operatorG%&arrowGF),&*&\"\"#\"\"\"%#PiG!\"\"F0-%$sumG6$,$*,\"\"% F0)F2%\"kGF0F1F2,&*&F8F0)F:F/F0F0F0F2F2-%$cosG6#*&F:F09$F0F0F2/F:;F09% F0F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The truncated Fourier series " }{XPPEDIT 18 0 "c + ``" "6#,&%\"cG \"\"\"%!GF%" }{XPPEDIT 18 0 "Sum(a[k]*cos(k*x),k = 1 .. 5);" "6#-%$Sum G6$*&&%\"aG6#%\"kG\"\"\"-%$cosG6#*&F*F+%\"xGF+F+/F*;F+\"\"&" }{TEXT -1 11 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "FS(x,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, .*&\"\"#\"\"\"%#PiG!\"\"F&*&#\"\"%\"\"$F&*&F'F(-%$cosG6#%\"xGF&F&F&*&# F+\"#:F&*&F'F(-F/6#,$*&F%F&F1F&F&F&F&F(*&#F+\"#NF&*&F'F(-F/6#,$*&F,F&F 1F&F&F&F&F&*&#F+\"#jF&*&F'F(-F/6#,$*&F+F&F1F&F&F&F&F(*&#F+\"#**F&*&F'F (-F/6#,$*&\"\"&F&F1F&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 74 "Here are a few graphs of truncated Fourie r series along with the graph of " }{XPPEDIT 18 0 "f_(x)=abs(cos(x/2)) " "6#/-%#f_G6#%\"xG-%$absG6#-%$cosG6#*&F'\"\"\"\"\"#!\"\"" }{TEXT -1 4 ". ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "f_ := x -> abs(cos(x/2));\nplot([f_(x),FS(x,1),FS(x, 2),FS(x,3),FS(x,5),FS(x,6)],x=-2*Pi..4*Pi,\n color=[black,red,blue,g reen,magenta,coral],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f_Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$absG6#-%$cosG6#,$*&# \"\"\"\"\"#F59$F5F5F(F(F(" }}{PARA 13 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A truncated Fourier series with more terms prov ides a better approximation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "FourierSeries(cos(x/2),x=-P i..Pi,numterms=15):\nF := unapply(%,x):\n'F(x)'=F(x);\nplot(F(x),x=-3* Pi..5*Pi,thickness=2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\" xG,B*&\"\"#\"\"\"%#PiG!\"\"F+*&#\"\"%\"\"$F+*&F,F--%$cosGF&F+F+F+*&#F0 \"#:F+*&F,F--F46#,$*&F*F+F'F+F+F+F+F-*&#F0\"#NF+*&F,F--F46#,$*&F1F+F'F +F+F+F+F+*&#F0\"#jF+*&F,F--F46#,$*&F0F+F'F+F+F+F+F-*&#F0\"#**F+*&F,F-- F46#,$*&\"\"&F+F'F+F+F+F+F+*&#F0\"$V\"F+*&F,F--F46#,$*&\"\"'F+F'F+F+F+ F+F-*&#F0\"$&>F+*&F,F--F46#,$*&\"\"(F+F'F+F+F+F+F+*&#F0\"$b#F+*&F,F--F 46#,$*&\"\")F+F'F+F+F+F+F-*&#F0\"$B$F+*&F,F--F46#,$*&\"\"*F+F'F+F+F+F+ F+*&#F0\"$*RF+*&F,F--F46#,$*&\"#5F+F'F+F+F+F+F-*&#F0\"$$[F+*&F,F--F46# ,$*&\"#6F+F'F+F+F+F+F+*&#F0\"$v&F+*&F,F--F46#,$*&\"#7F+F'F+F+F+F+F-*&# F0\"$v'F+*&F,F--F46#,$*&\"#8F+F'F+F+F+F+F+*&#F0\"$$yF+*&F,F--F46#,$*& \"#9F+F'F+F+F+F+F-*&#F0\"$**)F+*&F,F--F46#,$*&F7F+F'F+F+F+F+F+" }} {PARA 13 "" 1 "" {GLPLOT2D 665 220 220 {PLOTDATA 2 "6&-%'CURVESG6$7gu7 $$!3^***H>%zxC%*!#<$\"3Oux#*o@h`?!#>7$$!3U2dS$QAyG*F*$\"3Zr1OQfvWhF-7$ $!3M:9)[#o'3:*F*$\"3?I\"4mX-MR\"!#=7$$!3DBrNm7\"R,*F*$\"3i#3yhLlb/#F87 $$!3QHG$yqbp())F*$\"3)4l&)F*$\"33)*)puj(4fVF87$$ !3q\\W86&)H+%)F*$\"3BWgmGW5-\\F87$$!3]Fwq$z&Gm#)F*$\"3C=9-BA$R[&F87$$! 3a.3GwIFK\")F*$\"3I!oc*p0F7gF87$$!3czR&)e.E)*zF*$\"3xZ3n\\L[VlF87$$!3g brUTwCkyF*$\"3nM-O%*))fUqF87$$!3WN'4+cXWf(F*$\"3+_Z0Vx-FzF87$$!3H:@fyM kCtF*$\"3;m\\,&z&yn')F87$$!3#GSr$>P7cqF*$\"3))HQ8>0]o#*F87$$!3M!p],'Rg (y'F*$\"3e'e9J7kPo*F87$$!3I85Hly7jmF*$\"379HKMIME)*F87$$!3RN8VqN'F*$\"3o!\\=,E!G(***F87$$!3W!)>r!e*p* G'F*$\"3;ymvob_A'F*$\"3M\\*z+!y\"*****F87$$!3W$e@Wz63 ;'F*$\"3)4b5N/T\"z**F87$$!3#eQw7!zO'4'F*$\"3MU[Zkj&*\\**F87$$!3M(=J\"3 S#>.'F*$\"3y!HMmsjj\"**F87$$!3C\"zS=AOI!fF*$\"3[#\\=S7ch#)*F87$$!3E%R] bV[Tx&F*$\"3u*y#*=S'[x'*F87$$!3srbjplb2bF*$\"3))fAR)HyNE*F87$$!3I[2s.Z '4C&F*$\"3iwdmM&Gfm)F87$$!3+P4I,yAv\\F*$\"3A'y&[0)4v$zF87$$!3gE6)))*3 \\4ZF*$\"3SGYO>[3oqF87$$!3u\"4._O>Gd%F*$\"3iR(QTB-?c'F87$$!3(o0D:$y9OW F*$\"3R'eL#*z6.-'F87$$!3+Aq%yHw%*H%F*$\"39z2B$>_9[&F87$$!3/))*oTw/G;%F *$\"3]ih%feyr)[F87$$!3N3N\"HSCC/%F*$\"3)4iQI[S\"RVF87$$!36H!e;/W?#RF*$ \"3-SY

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\"\"%\"kGF-*&%#PiGF-,&*&F,F-*$F1\"\"#F-F-F-F0F-F0F0F-),$F-F0F1F-/F1;F- %)infinityG" }{TEXT -1 3 " " }}{PARA 256 "" 0 "" {TEXT -1 7 " = \+ " }{XPPEDIT 18 0 "2/Pi-4/Pi;" "6#,&*&\"\"#\"\"\"%#PiG!\"\"F&*&\"\"%F&F 'F(F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(1/(4*k^2-1),k = 1 .. infin ity);" "6#-%$SumG6$*&\"\"\"F',&*&\"\"%F'*$%\"kG\"\"#F'F'F'!\"\"F./F,;F '%)infinityG" }{TEXT -1 8 " = " }{XPPEDIT 18 0 "2/Pi-4/Pi" "6#,&* &\"\"#\"\"\"%#PiG!\"\"F&*&\"\"%F&F'F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/2) = 0" "6#/-%!G6#*&\"\"\"F(\"\"#!\"\"\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "since " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(1/(4*k^2-1),k = 1 .. infinity) = 1/3+ 1/15+1/3 5+1/63+1/99+1/143+` . . . `" "6#/-%$SumG6$*&\"\"\"F(,&*&\"\"%F(*$%\"kG \"\"#F(F(F(!\"\"F//F-;F(%)infinityG,0*&F(F(\"\"$F/F(*&F(F(\"#:F/F(*&F( F(\"#NF/F(*&F(F(\"#jF/F(*&F(F(\"#**F/F(*&F(F(\"$V\"F/F(%(~.~.~.~GF(" } {XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 14 " \+ -------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Sum(1/(4*k^2 -1),k=1..infinity);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$SumG6$*&\"\"\"F',&*&\"\"%F')%#k|irG\"\"#F'F'F'!\"\"F./F,;F'%)infinit yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "However, the converg ence of the series (i) is rather slow." }}{PARA 0 "" 0 "" {TEXT -1 49 "In fact the convergence of the Fourier series at " }{XPPEDIT 18 0 "x= Pi" "6#/%\"xG%#PiG" }{TEXT -1 152 " is slower than at nearby points as is indicated by the following error curve for the truncated Fourier s eries including terms up as far as the term in " }{XPPEDIT 18 0 "cos(2 00*x)" "6#-%$cosG6#*&\"$+#\"\"\"%\"xGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "f_ := \+ x -> abs(cos(x/2)):\n'f_(x)'=f_(x);\nFS := (x,n)->2/Pi+sum(-4*(-1)^k/( Pi*(2*k-1)*(2*k+1))*cos(k*x),k=1..n);\nplot(FS(x,200)-f_(x),x=Pi-0.1.. Pi+0.1,color=blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG- %$absG6#-%$cosG6#,$*&\"\"#!\"\"F'\"\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&*&\"\"# \"\"\"%#PiG!\"\"F0-%$SumG6$,$*.\"\"%F0)F2%\"kGF0F1F2,&*&F/F0F:F0F0F0F2 F2,&*&F/F0F:F0F0F0F0F2-%$cosG6#*&F:F09$F0F0F2/F:;F09%F0F)F)F)" }} {PARA 13 "" 1 "" {GLPLOT2D 544 230 230 {PLOTDATA 2 "6&-%'CURVESG6#7gu7 $$\"3!)***Hol#fTI!#<$!3nDWu*[49+(!#A7$$\"3e(3Av3_f/$F*$!3m:!H%eWS@=F-7 $$\"3)R_d[?X(\\IF*$\"3GDp@\\G:)Q%F-7$$\"3#=DMM3y=0$F*$\"3qNKpqss+sF-7$ $\"35!)4,i4,aIF*$\"3UxZwE`\\/))F-7$$\"3+*3\"=;xaaIF*$\"3%zYw$)>K*p*)F- 7$$\"3M)>^.Z%3bIF*$\"3ZJhswQmJ!*F-7$$\"3o28_C7ibIF*$\"3%*>vS3]o()*)F-7 $$\"3e;9pyz:cIF*$\"3Ciri,g>P))F-7$$\"3GN;.([Js0$F*$\"3=[+)zC]'>#)F-7$$ \"3_`=P&*\\IeIF*$\"3!)fbOdBK(>(F-7$$\"3k(ycw!=WgIF*$\"3@i$=$*)4kCTF-7$ $\"3K@<%*>'yD1$F*$\"3)[)3!))RAP8\"!#B7$$\"3%o'3Y=(fX1$F*$!3;#ofe%oGOQF -7$$\"3!G,!)p\"3amIF*$!3_83L(*\\EntF-7$$\"3C[MOtkcnIF*$!3V(=NNY#zJ))F- 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GCuF-7$$\"34***)=UN7/KF*$\"3T7t9UE(3C#F-7$$\"3jpfZ'\\(>1KF*$!3&[\\%H;, \">$GF-7$$\"3=SHw]9F3KF*$!3/Od73lb?rF-7$$\"3G\"3m'4>L4KF*$!3%4DyTi*y$y )F-7$$\"3RA#p&oBR5KF*$!3KU&40yg(****F-7$$\"3%Hz?!)fA4@$F*$!38*yT4@$F*$!3%Ht8-1!= !4\"F^q7$$\"3f/bP'G8D@$F*$!3W`@hti!\\4\"F^q7$$\"3u3&f_7RI@$F*$!3!\\EB8 k\"=(3\"F^q7$$\"3M8N9k\\c8KF*$!3)*R\"=!**GOn5F^q7$$\"3$z^FI!349KF*$!3l ea\\P]%e.\"F^q7$$\"3_A:\">k;Y@$F*$!3M&)*4A-j9$**F-7$$\"3rJ&z'>$oc@$F*$ !3!Rohjx'\\o()F-7$$\"3XSvW(**>n@$F*$!3YX;4MctZsF-7$$\"3u&3RFw_'=KF*$!3 snZwBr]$y$F-7$$\"3eI1.Gbe?KF*$\"3$y()G9F>)=r!#C7$$\"3f#>n^p+GA$F*$\"39 w-_y['=A%F-7$$\"3faPIie,DKF*$\"3'3[$R!eBwM(F-7$$\"3`C3D)\\1gA$F*$\"3' \\ed:tD'f#)F-7$$\"3Z%*y>Mr*pA$F*$\"3.F>Ux*ob#))F-7$$\"3WH9<_C\\FKF*$\" 3gNs8,Qwt*)F-7$$\"3Tk\\9qx)zA$F*$\"3^A=3jMBJ!*F-7$$\"3#)*\\=\")3$[GKF* $\"3Ui/uv8W)**)F-7$$\"3zM?41%y*GKF*$\"3W/m(R5In())F-7$$\"32D%p^&34JKF* $\"3H*3qladjU(F-7$$\"3!\\\"oC/L?LKF*$\"3SHWh%)43WZF-7$$\"3x'zl32ZsB$F* $!3Dg/`cXWV=F-7$$\"3))***\\.l#fTKF*$!3#)R[4!\\49+(F--%+AXESLABELSG6$Q \"x6\"Q!Fban-%'COLOURG6&%$RGBG$\"\"!FianFhan$\"*++++\"!\")-%%VIEWG6$;$ \"+aEfTI!\"*$\"+aEfTKFcbn%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 9 " }}{PARA 0 "" 0 "" {TEXT 270 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 64 "Find the first few terms of the Fourier s eries for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 13 " defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x) = 1/8;" "6#/-%\"fG6#%\"xG*&\"\"\"F)\"\")!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "(x+3)^2*(x-1)^2" "6#*&,&%\"xG\"\"\"\"\" $F&\"\"#,&F%F&F&!\"\"F(" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "-3<=x" "6# 1,$\"\"$!\"\"%\"xG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " a nd " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " is periodi c with period 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "Plot the graphs of some truncated Fourier series and inv estigate the convergence of the Fourier series experimentally." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f := x -> (x+3)^2*(x-1)^2/8:\n'f(x)'=f(x);\nf_ := x -> f(x-4*floo r((x+3)/4)):\n'f_(x)'='f(x-4*floor((x+3)/4))';\nplot(f_(x),x=-3..9,col or=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6#%\"xG,$*(\"\")!\"\",&F'\"\"\"\"\"$F-\"\"#,&F'F-F-F+F/F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*&\"\" %F,-%&floorG6#,&*&F.!\"\"F'F,F,#\"\"$F.F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 589 188 188 {PLOTDATA 2 "6'-%'CURVESG6#7_t7$$!\"$\"\"!$F*F*7 $$!3)****\\(oUInH!#<$\"38GXxSC@.@!#?7$$!3&*****\\P&3Y$HF/$\"3M`')fubvu #)F27$$!3#****\\i!G\">!HF/$\"3!e(GJQf+J=!#>7$$!3!******\\2<#pGF/$\"35) >c+A)z+KF=7$$!3%)****\\7c#Q!GF/$\"3y?I;'*fUgpF=7$$!3z******\\TVQFF/$\" 39d>#eAI_>\"!#=7$$!3s***\\iiSYi#F/$\"33cs;4g%QJ#FM7$$!34++]-r%3^#F/$\" 3Y)Qi*y_d'o$FM7$$!3%****\\PQuGQ#F/$\"33c$*>'>(*yW&FM7$$!3A+++l;!\\D#F/ $\"3+y.yTg7_tFM7$$!3'*******f13E@F/$\"3$)))HNNiQH$*FM7$$!3#******\\lfs *>F/$\"3Vr]%)H'3\"H6F/7$$!3)****\\P\"40p=F/$\"32t'eMt]gJ\"F/7$$!31++]s @%3u\"F/$\"3k]>@@q!))[\"F/7$$!32++]di(>i\"F/$\"3+=BB#e_=j\"F/7$$!34++] U.6.:F/$\"3!oBG6q))[v\"F/7$$!33++]s:.!Q\"F/$\"3+\\UB!H$=e=F/7$$!33++]- G&pD\"F/$\"3zy'=c;?X$>F/7$$!35++]#o3L>\"F/$\"3MsI],j!G'>F/7$$!3#***** \\iXmH6F/$\"3/3tUNCA$)>F/7$$!3#)****\\-D%y4\"F/$\"3GEtO+$Q/*>F/7$$!3%* ****\\U/-m5F/$\"3!)QS&fnVc*>F/7$$!32++]#Q)>M5F/$\"3s]#4;kI))*>F/7$$!3( *****\\AjP-5F/$\"3)=O&4`V****>F/7$$!3H****\\iJc1(*FM$\"3I]-!4/R\"**>F/ 7$$!3'********4j$*Q*FM$\"3-ZKtjHF'*>F/7$$!3k++]PI;s!*FM$\"3!>iq'\\/S\" *>F/7$$!3?+++vH'\\v)FM$\"3a;_&Q')GX)>F/7$$!3W+++]Gc?\")FM$\"3cx-(G7L[' >F/7$$!3o+++DF;'[(FM$\"3r#e_KS0t$>F/7$$!3c****\\7d/\"='FM$\"31*[i[p9o& =F/7$$!3Y)******pGf([FM$\"3gp\\NAi0Yc2#p[\"F/7$$!3w)****\\i9FG\"FM$\"3*> l$4!QsAJ\"F/7$$\"3=8-+++RS6F2$\"3u0%>14*GB6F/7$$\"3-9+++0?6mF=$\"3$\\e lA;')[-\"F/7$$\"3n++++i$3J\"FM$\"3G#yK4Z>CD*FM7$$\"3%*******\\Bbg>FM$ \"33S%G8n.ED)FM7$$\"3@*******\\o-h#FM$\"3Q-83G!))*esFM7$$\"3c++++B^iQF M$\"3IL=mAF?*R&FM7$$\"3#>+++5cZ6&FM$\"3i5&R2MC%yOFM7$$\"3s++]PM\">D'FM $\"31$fT0ZixI#FM7$$\"3a*****\\xq!*Q(FM$\"3OEc4gL@\">\"FM7$$\"3!)***\\i !e;l!)FM$\"3[@KOEMO!y'F=7$$\"30++]P3ET()FM$\"3s(>n(p)[D(HF=7$$\"3i**\\ 7`$3$z!*FM$\"31iaL\\)*>=;F=7$$\"3I++voeN<%*FM$\"33ylym:8$f'F27$$\"3)4+ vVQ.av*FM$\"3c-6/C7'>=\"F27$$\"31+++!4X$45F/$\"3[P1IUDYQm1\"F/$\"35u-Sqd.$e)F27$$\"3/++D miD&4\"F/$\"3$Q\\=4?Z$H<#oStAFM7$$\"3#*****\\<_$\\]\"F/$\"3)H.b*yi1$*QF M7$$\"3z***\\(Q7)Gi\"F/$\"3;Jzgl77JbFM7$$\"3m******fs#3u\"F/$\"3'))*yD )HmrG(FM7$$\"3'****\\(QFBq=F/$\"33g;.&zsEF*FM7$$\"3E++]<#Q'**>F/$\"3]M ?b$HdW7\"F/7$$\"3G+++NG'G7#F/$\"3*yY[Ef@YI\"F/7$$\"3I++]_u3YAF/$\"3%3% R_N%)*>Z\"F/7$$\"3M++v8%fYP#F/$\"3@II)pCk!G;F/7$$\"3P+++v8B.DF/$\"3c\" )4p@M$3w\"F/7$$\"3Q++Dr0I@EF/$\"3-u='Hmd\"f=F/7$$\"3R++]n(p$RFF/$\"3P` /cg'[E$>F/7$$\"3[+](obYI!GF/$\"3;v?kIuRh>F/7$$\"39++DYLsmGF/$\"3'=10Ex wA)>F/7$$\"3u*\\P4uh&)*GF/$\"357;R9Ns*)>F/7$$\"3z**\\iN,SIHF/$\"3S\")= 4#ze^*>F/7$$\"3%)*\\7.`QA'HF/$\"3;%3mbKu&)*>F/7$$\"3))*****\\#p2%*HF/$ \"3#\\ROt\"\\'***>F/7$$\"36+D1p![r-$F/$\"3]bbC]IE**>F/7$$\"3!***\\78#> -1$F/$\"3g'ov,Hvj*>F/7$$\"3m*\\(=d.H$4$F/$\"3I\"RHsP18*>F/7$$\"3!**** \\7]hj7$F/$\"3;\"3_+kkS)>F/7$$\"3))**\\P*y.D>$F/$\"3G+,I]R6j>F/7$$\"3! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "f := x -> (x+3)^2*(x-1)^2/8:\ninterface(showass umed=0): k := 'k': assume(k,integer):\n1/2*Int('f(x)'*cos(k*Pi*x/2),x= -3..1);\na := unapply(value(%),k):\n'a(k)'=a(k);\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&-%$IntG6$*&-%\"fG6#%\"xGF& -%$cosG6#,$**F'!\"\"%#k|irGF&%#PiGF&F/F&F&F&/F/;!\"$F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#%\"kG,$**\"#;\"\"\",.*&\"\"$F+-%$si nG6#,$**F.F+\"\"#!\"\"%#k|irGF+%#PiGF+F+F+F5**F.F+F6F+F7F+-%$cosGF1F+F +*()F6F4F+)F7F4F+F/F+F+*&F.F+-F06#,$*(F4F5F6F+F7F+F+F+F5**F.F+F6F+F7F+ -F:F@F+F+*(F " 0 "" {MPLTEXT 1 0 171 "f := x -> (x+3)^2*(x-1)^2/8:\ninterface(showassumed= 0): k := 'k': assume(k,integer):\n1/2*Int('f(x)'*sin(k*Pi*x/2),x=-3..1 );\nb := unapply(value(%),k):\n'b(k)'=b(k);\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&-%$IntG6$*&-%\"fG6#%\"xGF&-%$sin G6#,$**F'!\"\"%#k|irGF&%#PiGF&F/F&F&F&/F/;!\"$F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"bG6#%\"kG,$**\"#;\"\"\",.*&\"\"$F+-%$cosG6#,$** F.F+\"\"#!\"\"%#k|irGF+%#PiGF+F+F+F5*()F6F4F+)F7F4F+F/F+F+**F.F+F6F+F7 F+-%$sinGF1F+F5*&F.F+-F06#,$*(F4F5F6F+F7F+F+F+F+**F.F+F6F+F7F+-F=F@F+F +*(F9F+F:F+F?F+F5F+F6!\"&F7FFF5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The Fourier series is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c + ``" "6#,&%\"cG\"\"\"%!GF%" }{XPPEDIT 18 0 "Sum(a[k] *cos(k*Pi*x/2)+b[k]*sin(k*Pi*x/2),k = 1 .. infinity);" "6#-%$SumG6$,&* &&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#PiGF,%\"xGF,\"\"#!\"\"F,F,*&&%\"b G6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F,%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We c an use the functions " }{TEXT 266 1 "a" }{TEXT -1 5 " and " }{TEXT 266 1 "b" }{TEXT -1 22 " to define a function " }{TEXT 266 2 "FS" } {TEXT -1 50 " for constructing finite truncated Fourier series." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "FS := (x,n)->16/15+sum(a(k)*cos(k*Pi*x/2)+b(k)*sin(k*Pi*x/2),k=1.. n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)oper atorG%&arrowGF),&#\"#;\"#:\"\"\"-%$sumG6$,&*&-%\"aG6#%\"kGF1-%$cosG6#, $*&#F1\"\"#F1*(F:F1%#PiGF19$F1F1F1F1F1*&-%\"bGF9F1-%$sinGF=F1F1/F:;F19 %F1F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The truncated Fourier series " }{XPPEDIT 18 0 "c + ``" "6#,&% \"cG\"\"\"%!GF%" }{XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/2)+b[k]*sin(k*Pi* x/2),k = 1 .. 5);" "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,% #PiGF,%\"xGF,\"\"#!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F, F,/F+;F,\"\"&" }{TEXT -1 12 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "FS(x,5);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,.#\"#;\"#:\"\"\"*(\"#'*F'%#PiG!\"%-%$sinG6#,$*( \"\"#!\"\"F*F'%\"xGF'F'F'F2*(\"\"'F'F*F+-%$cosG6#*&F*F'F3F'F'F'*&#\"#K \"#FF'*&F*F+-F-6#,$**\"\"$F'F1F2F*F'F3F'F'F'F'F'*&#FC\"\")F'*&F*F+-F76 #,$*(F1F'F*F'F3F'F'F'F'F2*&#F)\"$D'F'*&F*F+-F-6#,$**\"\"&F'F1F2F*F'F3F 'F'F'F'F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Here are two graphs of truncated Fourier series along with the \+ graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "f := x -> (x+3)^2*(x-1)^2/8:\nf_ := x -> f(x-4*floor((x+3)/4)):\n plot([f_(x),FS(x,1),FS(x,2)],x=-3..9,color=[black,red,blue],linestyle= [3,1$2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 607 206 206 {PLOTDATA 2 "6'-%' CURVESG6%7_t7$$!\"$\"\"!$F*F*7$$!3)****\\(oUInH!#<$\"38GXxSC@.@!#?7$$! 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Fdgm$\"3f\\&=>K)y07FM7$Figm$\"3mKMK**4#Hp(F=7$F^hm$\"3Pn5+x`GAXF=7$Fch m$\"3Vr$)Qp7/-MF=7$Fhhm$\"3M`G())G.&)f#F=7$F]im$\"3JN\\eQ1.:@F=7$FbimF ^jo-Feim6&FgimF+F+FdioFgio-%+AXESLABELSG6$Q\"x6\"Q!Ffgq-%%VIEWG6$;F(Fb im%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "Because the periodic function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 266 25 "differentiable everywhere" }{TEXT -1 47 ", we can obtain a good vi sual approximation to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 57 " with only a small number of terms of the Fourier series. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Solution 2" }{TEXT 268 24 " .. using the procedure " }{TEXT 260 13 "FourierSeries" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 16 ": The procedure " }{TEXT 0 13 "Fourie rSeries" }{TEXT -1 57 " can be used to obtain specific truncated Fouri er series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 134 "f := x -> (x+3)^2*(x-1)^2/8:\n'f(x)'=f(x);\nFourie rSeries(f(x),x=-3..1,numterms=5):\nF := unapply(%,x):\n'F(x)'=F(x);\np lot(F(x),x=-3..9);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG ,$*(\"\")!\"\",&F'\"\"\"\"\"$F-\"\"#,&F'F-F-F+F/F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"FG6#%\"xG,.#\"#;\"#:\"\"\"*(\"#'*F,%#PiG!\"%-%$s inG6#,$*(\"\"#!\"\"F/F,F'F,F,F,F7*(\"\"'F,F/F0-%$cosG6#*&F/F,F'F,F,F,* &#\"#K\"#FF,*&F/F0-F26#,$**\"\"$F,F6F7F/F,F'F,F,F,F,F,*&#FG\"\")F,*&F/ F0-F;6#,$*(F6F,F/F,F'F,F,F,F,F7*&#F.\"$D'F,*&F/F0-F26#,$**\"\"&F,F6F7F /F,F'F,F,F,F,F7" }}{PARA 13 "" 1 "" {GLPLOT2D 604 200 200 {PLOTDATA 2 "6%-%'CURVESG6$7]t7$$!\"$\"\"!$\"3n@@T*Q)yU>!#?7$$!3)****\\(oUInH!#<$ \"30e'R\"3kqVQF-7$$!3&*****\\P&3Y$HF1$\"3Dihi.8SH&*F-7$$!3#****\\i!G\" >!HF1$\"3#eb[_r=\\*=!#>7$$!3!******\\2<#pGF1$\"3+G'f>I???$F>7$$!3%)*** *\\7c#Q!GF1$\"3314;%oXQ'oF>7$$!3z******\\TVQFF1$\"3m^%f'fH&==\"!#=7$$! 3s***\\iiSYi#F1$\"3s&ymcAipI#FN7$$!34++]-r%3^#F1$\"3#G,kQVs6p$FN7$$!3% ****\\PQuGQ#F1$\"3a)HA\"R]vcaFN7$$!3A+++l;!\\D#F1$\"33H)\\Vp-VN(FN7$$! 3'*******f13E@F1$\"3mzfd=vtB$*FN7$$!3#******\\lfs*>F1$\"3ax#*)o;I&G6F1 7$$!3)****\\P\"40p=F1$\"3i\"F1$\"3EpMLD@EK;F17$$!34++]U.6.:F1$\"3]\\U#oVmZv\"F 17$$!33++]s:.!Q\"F1$\"3]1q?7'zw&=F17$$!33++]-G&pD\"F1$\"33EgU1n=M>F17$ $!35++]#o3L>\"F1$\"3a%)*G%y7ti>F17$$!3#*****\\iXmH6F1$\"3MSrJ)eDM)>F17 $$!3#)****\\-D%y4\"F1$\"39`;%*\\3w!*>F17$$!3%*****\\U/-m5F1$\"3%)))>JS (fg*>F17$$!32++]#Q)>M5F1$\"3QayxErI**>F17$$!3(*****\\AjP-5F1$\"3:^PX8N \\+?F17$$!3H****\\iJc1(*FN$\"3EG\\7.:i**>F17$$!3'********4j$*Q*FN$\"3+ .^Xl2q'*>F17$$!3k++]PI;s!*FN$\"3Sv'>\"4)R<*>F17$$!3?+++vH'\\v)FN$\"3#3 CMWx_Z)>F17$$!3W+++]Gc?\")FN$\"33GG^9?yk>F17$$!3o+++DF;'[(FN$\"3*Q_&)) *e\"*p$>F17$$!3c****\\7d/\"='FN$\"3/ZHNfHJc=F17$$!3Y)******pGf([FN$\"3 WKLo,t(fu\"F17$$!32*****\\#4QEPFN$\"3$*=GSK$3ii\"F17$$!3o******\\J$od# FN$\"39-(4V$zY([\"F17$$!3w)****\\i9FG\"FN$\"3$eTS!)39BJ\"F17$$\"3=8-++ +RS6F-$\"3AUxov$)pA6F17$$\"3n++++i$3J\"FN$\"3q%e*3'HgpC*FN7$$\"3@***** **\\o-h#FN$\"3-.w'4l0;E(FN7$$\"3c++++B^iQFN$\"3M3<]-\\63aFN7$$\"3#>+++ 5cZ6&FN$\"3Ex%)>b,(Ho$FN7$$\"3s++]PM\">D'FN$\"3%ert!>I#3I#FN7$$\"3a*** **\\xq!*Q(FN$\"3o[=8p)Ry<\"FN7$$\"3!)***\\i!e;l!)FN$\"3o_'GD04no'F>7$$ \"30++]P3ET()FN$\"3+#)*fb)e)G)HF>7$$\"3i**\\7`$3$z!*FN$\"3q(**[omJQp\" F>7$$\"3I++voeN<%*FN$\"3)3_E&H$)\\pzF-7$$\"3)4+vVQ.av*FN$\"3PmjpBU*p+$ F-7$$\"31+++!4X$45F1$\"3ab5WWr=)4#F-7$$\"31++v[@)z.\"F1$\"33YU-.!Hu]%F -7$$\"31++]2#>m1\"F1$\"3m]&))fhlg\")*F-7$$\"3/++DmiD&4\"F1$\"3cOBK\\-y )z\"F>7$$\"31+++DL*Q7\"F1$\"3KgpOCKu'*GF>7$$\"3/++]Uu;\"=\"F1$\"3N!\\i (*)QD/fF>7$$\"30+++g:WQ7F1$\"3N1vG7$$\"3)****\\()Q)or8F1$\"3OP R03U:mAFN7$$\"3#*****\\<_$\\]\"F1$\"3UV@_!>p))*QFN7$$\"3z***\\(Q7)Gi\" F1$\"3gT>awr')RbFN7$$\"3m******fs#3u\"F1$\"3!)3#>5v]'*G(FN7$$\"3'**** \\(QFBq=F1$\"3'[PrX\\irE*FN7$$\"3E++]<#Q'**>F1$\"3$[XOP)*oQ7\"F17$$\"3 G+++NG'G7#F1$\"3jFJWCLj/8F17$$\"3I++]_u3YAF1$\"3')=&eZcK`[G;F17$$\"3P+++v8B.DF1$\"3M%oMC=#ogF17$$\"3[+](obYI !GF1$\"3%\\j'>RiIh>F17$$\"39++DYLsmGF1$\"31Z9SK^Y#)>F17$$\"3u*\\P4uh&) *GF1$\"3EyUU8P.!*>F17$$\"3z**\\iN,SIHF1$\"3gcZwwec&*>F17$$\"3%)*\\7.`Q A'HF1$\"39+$)pge/**>F17$$\"3))*****\\#p2%*HF1$\"3W^<0+NY+?F17$$\"36+D1 p![r-$F1$\"3c&=8C#zu**>F17$$\"3!***\\78#>-1$F1$\"373\\S4]!o*>F17$$\"3m *\\(=d.H$4$F1$\"3+OLr$3W;*>F17$$\"3!****\\7]hj7$F1$\"3Ek@yJ6G%)>F17$$ \"3))**\\P*y.D>$F1$\"3UsFF2D/j>F17$$\"3!*****\\xgkeKF1$\"3%>E,=MAL$>F1 7$$\"3')*******=+QP$F1$\"3m-n0,Z?i=F17$$\"3%)****\\-V&*)[$F1$\"3=4c^VA )yw\"F17$$\"3/++v)*QK8OF1$\"3$=2V:u**=k\"F17$$\"3E+++&\\$pPPF1$\"39YX. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "numapprox [infnorm](F(x)-f(x),x=-3..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+F mm3Q!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{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 29 " be the function defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([0, 0 <= x and x < 2],[3, 2 <= x a nd x < 4]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"!31F,F'2F'\"\"#7$\" \"$31F0F'2F'\"\"%" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 28 " is periodic with period 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "(a) Find formulas for the Fourier coefficients of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 53 " and give the first few terms of the Fourier s eries. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "(b) Find the fundamental wave and plot its graph along with the gr aph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(c) Compa re the graphs of some truncated Fourier series with the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 23 "Const ant coefficient: " }{XPPEDIT 18 0 "c = 3/2;" "6#/%\"cG*&\"\"$\"\"\"\" \"#!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "Cosine coef ficients: " }{XPPEDIT 18 0 "a[k] = 0;" "6#/&%\"aG6#%\"kG\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "k=1,2,3, ` . . . `" "6&/%\"kG\"\"\"\" \"#\"\"$%(~.~.~.~G" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 " Sine coefficients: " }{XPPEDIT 18 0 "b[k]=3*((-1)^k-1)/(k*Pi)" "6#/&% \"bG6#%\"kG*(\"\"$\"\"\",&),$F*!\"\"F'F*F*F.F**&F'F*%#PiGF*F." }{TEXT -1 2 ", " }{XPPEDIT 18 0 "k=1,2,3, ` . . . `" "6&/%\"kG\"\"\"\"\"#\"\" $%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "Fourier s eries: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = 3/2 -6/Pi;" "6#/-%\"FG6#%\"xG,&*&\"\"$\"\"\"\"\"#!\"\"F+*&\"\"'F+%#PiGF-F- " }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi*x/2)-2/Pi;" "6#,&-%$sinG6#*(% #PiG\"\"\"%\"xGF)\"\"#!\"\"F)*&F+F)F(F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(3/2*Pi*x)-6/(5*Pi);" "6#,&-%$sinG6#**\"\"$\"\"\"\"\"#!\"\"%# PiGF)%\"xGF)F)*&\"\"'F)*&\"\"&F)F,F)F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(5/2*Pi*x)-6/(7*Pi);" "6#,&-%$sinG6#**\"\"&\"\"\"\"\"#!\"\"%# PiGF)%\"xGF)F)*&\"\"'F)*&\"\"(F)F,F)F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(7/2*Pi*x)+` . . . `;" "6#,&-%$sinG6#**\"\"(\"\"\"\"\"#!\"\"% #PiGF)%\"xGF)F)%(~.~.~.~GF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 19 "Fundamental wave: " } {XPPEDIT 18 0 "y = 3/2-6/Pi;" "6#/%\"yG,&*&\"\"$\"\"\"\"\"#!\"\"F(*&\" \"'F(%#PiGF*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi*x/2);" "6#-%$si nG6#*(%#PiG\"\"\"%\"xGF(\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "f := x -> p iecewise(x<2,0,3):\n'f(x)'=f(x);\nf_ := x -> f(x-4*floor(x/4)):\nFouri erSeries(f(x),x=0..4,numterms=1,info=1):\nF := unapply(%,x):\n'F(x)'=F (x);\nplot([f_(x),F(x)],x=-1..9,linestyle=[3,1],color=[black,red],nump oints=80);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWI SEG6$7$\"\"!2F'\"\"#7$\"\"$%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%_ )!#>F+7$$!3[Ot)*=:Z*H&FIF+7$$!3aPNouzA)o$FIF+7$$!3eQ(z.V%)p2#FIF+7$$!3 5RyAeEOr7FIF+7$$!35'Rf2')3ul%!#?F+7$$!3#p)*)*******HH'!#@F+7$$\"3s)ff2 ')3))R$FY$F*F*7$$\"35'4>:s:sx9&)QF0F [o7$$\"3g)z.VaZc>&F0F[o7$$\"3O!QIWNG:^'F0F[o7$$\"3)\\:sF]qF[o7$$\"38KD?@9DQ>F]qF[o7$$\"3;/VaLG\\q>F]qF[o7$$\"3+ Z(Hm=`&y>F]qF[o7$$\"31!>:(RNh')>F]qF[o7$$\"3e6zD;Pk!*>F]qF[o7$$\"37L1! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "FS := (x,n)->3/2+sum(3*(-1+(-1)^k)/k/Pi*sin(k*Pi*x/2 ),k=1..n):\nFS(x,n);\nf := x -> piecewise(x<2,0,3):\nf_ := x -> f(x-4* floor(x/4)):\nplot([f_(x),FS(x,1),FS(x,3),FS(x,5),FS(x,7)],x=-1..9,lin estyle=[3,1$4],\n color=[black,red,magenta,tan,blue],numpoints=80); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"$\"\"#\"\"\"-%$sumG6$,$*,F% F',&F'!\"\")F.%\"kGF'F'F0F.%#PiGF.-%$sinG6#,$**F&F.F0F'F1F'%\"xGF'F'F' F'/F0;F'%\"nGF'" }}{PARA 13 "" 1 "" {GLPLOT2D 784 232 232 {PLOTDATA 2 "6)-%'CURVESG6%7cs7$$!\"\"\"\"!$\"\"$F*7$$!39)3n$\\fhv')!#=F+7$$!3aHWN okFBvF0F+7$$!3=X;`--NFiF0F+7$$!3)4`-ipLG#\\F0F+7$$!3KCR6Ht^COF0F+7$$!3 1$\\f2'=\"3U#F0F+7$$!39t)*=:#Gwz\"F0F+7$$!3C`-ipXWu6F0F+7$$!3SM\\f2'e> _)!#>F+7$$!3[Ot)*=:Z*H&FIF+7$$!3aPNouzA)o$FIF+7$$!3eQ(z.V%)p2#FIF+7$$! 35RyAeEOr7FIF+7$$!35'Rf2')3ul%!#?F+7$$!3#p)*)*******HH'!#@F+7$$\"3s)ff 2')3))R$FY$F*F*7$$\"35'4>:s:sx9&)QF0 F[o7$$\"3g)z.VaZc>&F0F[o7$$\"3O!QIWNG:^'F0F[o7$$\"3)\\:sF]qF[o7$$\"38KD?@9DQ>F]qF[o7$$\"3;/VaLG\\q>F]qF[o7$$\"3+ Z(Hm=`&y>F]qF[o7$$\"31!>:(RNh')>F]qF[o7$$\"3e6zD;Pk!*>F]qF[o7$$\"37L1! 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]el%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Since the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 88 " is discontinuous , the convergence is rather slow, especially near the discontinuities. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "f := x -> piecewi se(x<2,0,3):\n'f(x)'=f(x);\nFourierSeries(f(x),x=0..4,numterms=17):\nF := unapply(%,x):\n'F(x)'=F(x);\nf_ := x -> f(x-4*floor(x/4)):\nplot([ f_(x),F(x)],x=-1..9,linestyle=[3,1],\n color=[black,COLOR(RGB,0,.7,. 3)],numpoints=120);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG- %*PIECEWISEG6$7$\"\"!2F'\"\"#7$\"\"$%*otherwiseG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,6#\"\"$\"\"#\"\"\"*(\"\"'F,%#PiG!\"\"-%$ sinG6#,$*(F+F0F/F,F'F,F,F,F0*(F+F,F/F0-F26#,$**F*F,F+F0F/F,F'F,F,F,F0* &#F.\"\"&F,*&F/F0-F26#,$**F=F,F+F0F/F,F'F,F,F,F,F0*&#F.\"\"(F,*&F/F0-F 26#,$**FEF,F+F0F/F,F'F,F,F,F,F0*&#F+F*F,*&F/F0-F26#,$**\"\"*F,F+F0F/F, F'F,F,F,F,F0*&#F.\"#6F,*&F/F0-F26#,$**FUF,F+F0F/F,F'F,F,F,F,F0*&#F.\"# 8F,*&F/F0-F26#,$**FgnF,F+F0F/F,F'F,F,F,F,F0*&#F+F=F,*&F/F0-F26#,$**\"# :F,F+F0F/F,F'F,F,F,F,F0*&#F.\"#$)z&)yb$)F0F +7$$!3s/E^\"yna\\(F0F+7$$!3u%R([=UWHmF0F+7$$!3![r&G9n`ndF0F+7$$!3c>/0E 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Q\")FR7$$\"39PXMh=o_%)Fer$\"3[*y5EM87%zFR7$$\"3A35__N\"o'3!e>'FR7$$\"3m+e\\R'z2b)Fer$!34j?)Q<)y7oFR7$$\"3Ak alQ#*ph&)Fer$!37*4IdI)3\\HKFR7$$\"3Wq, #zQLbv)Fer$!3%4Hx-5VU`%FR7$$\"3s]=iHj*ow)Fer$!3WvedYx$yN&FR7$$\"3-JNKr #f#y()Fer$!3;dXtu)QNi&FR7$$\"3I6_-8Ai*y)Fer$!3mEX(yf*z6`FR7$$\"3(=dGk4 [B\"))Fer$!3oxezHO:lJFR7$F_\\m$\"3gSBmaWd'*=F\\o7$$\"3-;O.%yDV&))Fer$ \"3]O!fn&z()yHFR7$$\"33'HN#)exN())Fer$\"3aM]vH_=r[FR7$$\"37wpV#RHG*))F er$\"3f98LM&eiL&FR7$Fb\\m$\"3)*)R0cw%pcUFR7$$\"33G$>$)fSg&*)Fer$!3]k*[ 1F'>)o\"FR7$Fe\\m$!3y10EwX.*G&FR-%&COLORG6&Fj\\mFdo$\"\"(F)$F,F)-F\\]m 6#\"\"\"-%+AXESLABELSG6$Q\"x6\"Q!Fc_v-%%VIEWG6$;F(Fe\\m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________ ________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________ ______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{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 29 " be the function defined by: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE( [1/2, -1 <= x and x < 0],[-x, 0 <= x and x < 1]);" "6#/-%\"fG6#%\"xG-% *PIECEWISEG6$7$*&\"\"\"F-\"\"#!\"\"31,$F-F/F'2F'\"\"!7$,$F'F/31F4F'2F' F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " is periodic with period 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 63 "(a) Find formulas for the Fourier coeffi cients of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 53 " and give the first few terms of the Fourier series. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "(b) Find \+ the fundamental wave and plot its graph along with the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(c) Compare the grap hs of some truncated Fourier series with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Constant coefficient: " }{XPPEDIT 18 0 "c = 0;" "6#/%\"c G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "Cosine coeffic ients: " }{XPPEDIT 18 0 "a[k]=(1-(-1)^k)/(k^2*Pi^2)" "6#/&%\"aG6#%\"k G*&,&\"\"\"F*),$F*!\"\"F'F-F**&F'\"\"#%#PiGF/F-" }{XPPEDIT 18 0 "`` = \+ PIECEWISE([2/(k^2*Pi^2), `if k is odd`],[0, `if k is even`]);" "6#/%!G -%*PIECEWISEG6$7$*&\"\"#\"\"\"*&%\"kGF*%#PiGF*!\"\"%,if~k~is~oddG7$\" \"!%-if~k~is~evenG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 20 "Sine coefficients: " }{XPPEDIT 18 0 "b[k ]= (3*(-1)^k-1)/(2*k*Pi)" "6#/&%\"bG6#%\"kG*&,&*&\"\"$\"\"\"),$F,!\"\" F'F,F,F,F/F,*(\"\"#F,F'F,%#PiGF,F/" }{XPPEDIT 18 0 "``=PIECEWISE([-2/( k*Pi), `if k is odd`],[1/(k*Pi),`if k is even`])" "6#/%!G-%*PIECEWISEG 6$7$,$*&\"\"#\"\"\"*&%\"kGF,%#PiGF,!\"\"F0%,if~k~is~oddG7$*&F,F,*&F.F, F/F,F0%-if~k~is~evenG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 17 " Fourier series: " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "2*cos*Pi*x/(Pi^2)-2*sin*Pi*x/Pi+sin* 2*Pi*x/(2*Pi)+2*cos*3*Pi*x/(9*Pi^2)-2*sin*3*Pi*x/(3*Pi)+sin*4*Pi*x/(4* Pi)+2*cos*5*Pi*x/(25*Pi^2)-2*sin*5*Pi*x/(5*Pi)+` . . . `;" "6#,4*,\"\" #\"\"\"%$cosGF&%#PiGF&%\"xGF&*$F(F%!\"\"F&*,F%F&%$sinGF&F(F&F)F&F(F+F+ *,F-F&F%F&F(F&F)F&*&F%F&F(F&F+F&*.F%F&F'F&\"\"$F&F(F&F)F&*&\"\"*F&*$F( F%F&F+F&*.F%F&F-F&F1F&F(F&F)F&*&F1F&F(F&F+F+*,F-F&\"\"%F&F(F&F)F&*&F8F &F(F&F+F&*.F%F&F'F&\"\"&F&F(F&F)F&*&\"#DF&*$F(F%F&F+F&*.F%F&F-F&F;F&F( F&F)F&*&F;F&F(F&F+F+%(~.~.~.~GF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 19 "Fundamental wave: " }{XPPEDIT 18 0 "y = 2*cos*Pi*x/(Pi^2)-2*sin*Pi*x/Pi;" "6#/%\"yG,&*,\" \"#\"\"\"%$cosGF(%#PiGF(%\"xGF(*$F*F'!\"\"F(*,F'F(%$sinGF(F*F(F+F(F*F- F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "f := x -> piecewise(x<0,1/2,-x):\n'f(x)'=f (x);\nf_ := x -> f(x-2*floor((x+1)/2)):\nFourierSeries(f(x),x=-1..1,nu mterms=1,info=1):\nF := unapply(%,x):\n'F(x)'=F(x);\nplot([f_(x),F(x)] ,x=-1.5..3.5,linestyle=[3,1],color=[black,red],numpoints=80);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$#\"\"\" \"\"#2F'\"\"!7$,$F'!\"\"%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%9constant~coefficient~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %icF-7$$!3e@xTBQ;w8F*$!37%yAewh$QiF-7$$!3E#eE,^n8J\"F*$!3SxTt)*[K' )oF-7$$!3cE,\"[oThC\"F*$!3^M()*=:$eQvF-7$$!3E'pbk'eA\"=\"F*$!3GPIWN8u( =)F-7$$!3auz.$fS57\"F*$!3ea-ipSf*y)F-7$$!3i$\\f2T\"))*3\"F*$!3qj]S#*e= ,\"*F-7$$!3s75[GAse5F*$!3$G()*=:xx7%*F-7$$!3uuz.$z4E/\"F*$!3e_-ip?!Rd* F-7$$!3xO\\fdt\\E5F*$!3NK10Ck-N(*F-7$$!3*yTt)R6W=5F*$!36@eE,')e:)*F-7$ $!3z)*=:A\\Q55F*$!3575[y2:'*)*F-7$$!38R6H8oN15F*$!3r3')3n=VO**F-7$$!3p z.V/(GB+\"F*$!35.ipbHrw**F-7$$!3(*******\\YJ+5F*$!3G++++N&o***F-7$$!3_ -ipbf+$)**F-$\"3++++++++]F-7$$!3M0CR6a'G'**F-Ffp7$$!3.2')3n[sU**F-Ffp7 $$!3!30CRh4:i*F-Ffp7$$!3X$\\f2O%H+$*F-Ffp7$$!3=kJD?cZR')F-Ffp7$$!3%*\\ S#R6Eu0)F-Ffp7$$!3q+\"[yAw@S(F-Ffp7$$!3#)4[yAeBWnF-Ffp7$$!3^AR6H$)=5hF -Ffp7$$!3X:`-iCTMbF-Ffp7$$!3%RyAeEf(\\[F-Ffp7$$!3CX;`-#o(pUF-Ffp7$$!3% R()*=:25&f$F-Ffp7$$!3Gwg)3n0z*HF-Ffp7$$!3VB?'p0)oUBF-Ffp7$$!3.sz.Vfw=< F-Ffp7$$!3Stg)3nox1\"F-Ffp7$$!3%*p8\"H8$f)o(!#>Ffp7$$!3s0?'pb*\\*p%F^t Ffp7$$!3'*)Rt)R*Gu3$F^tFfp7$$!3=#z%yA$e`Z\"F^tFfp7$$!3]SE^o1Ms5F^tFfp7 $$!3!z)[SU,B$p'!#?Ffp7$$!3WHT/r=9yYF^uFfp7$$!3)4P$o*f`Im#F^uFfp7$$!3+D hA$G`'zk!#@Ffp7$$\"3)f9QI%H7n8F^u$!3)f9QI%H7n8F^u7$$\"3$)z6[GgZF%*F^u$ !3$)z6[GgZF%*F^u7$$\"3Q@CR6Hy[t)*=:s/M!*F-$!3>t)*=:s/M!*F-7$$ \"3Q#R6H)=9h$*F-$!3Q#R6H)=9h$*F-7$$\"3c6Hj]lB)o*F-$!3c6Hj]lB)o*F-7$$\" 3kAx\"4J%*pw*F-$!3kAx\"4J%*pw*F-7$$\"3gKD?r?vX)*F-$!3gKD?r?vX)*F-7$$\" 39Q\\M^48&))*F-$!39Q\\M^48&))*F-7$$\"3cUt[J)4X#**F-$!3cUt[J)4X#**F-7$$ \"3xW&e:F*>W**F-$!3xW&e:F*>W**F-7$$\"35[(H;r))Q'**F-$!35[(H;r))Q'**F-7 $$\"3U^4q^\"yN)**F-$!3U^4q^\"yN)**F-7$$\"3O:sZ\"F*Ffp7$$\"3rn$\\ fnp,a\"F*Ffp7$$\"3wX;`_\"H#*f\"F*Ffp7$$\"3%4')3n=iim\"F*Ffp7$$\"3ukJD? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 304 "FS := (x,n) -> \n sum((1-(-1)^k)/(k^2*Pi^2)*cos(k*Pi *x)+(3*(-1)^(k)-1)/(2*k*Pi)*sin(k*Pi*x),k=1..n):\nFS(x,n);\nf := x -> \+ piecewise(x<0,1/2,-x):\nf_ := x -> f(x-2*floor((x+1)/2)):\nplot([f_(x) ,FS(x,1),FS(x,3),FS(x,5),FS(x,7)],x=-1.5..3.5,linestyle=[3,1$4],\n c olor=[black,red,magenta,tan,blue],numpoints=80);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sumG6$,&**,&\"\"\"F))!\"\"%\"kGF+F)F,!\"#%#PiGF--%$c osG6#*(F,F)F.F)%\"xGF)F)F)*&#F)\"\"#F)**,&*&\"\"$F)F*F)F)F)F+F)F,F+F.F +-%$sinGF1F)F)F)/F,;F)%\"nG" }}{PARA 13 "" 1 "" {GLPLOT2D 784 232 232 {PLOTDATA 2 "6)-%'CURVESG6%7cs7$$!3++++++++:!#<$!3++++++++]!#=7$$!3_a$ ouz!yL9F*$!3#[X;`-#>icF-7$$!3e@xTBQ;w8F*$!37%yAewh$QiF-7$$!3E#eE,^n8J \"F*$!3SxTt)*[K')oF-7$$!3cE,\"[oThC\"F*$!3^M()*=:$eQvF-7$$!3E'pbk'eA\" =\"F*$!3GPIWN8u(=)F-7$$!3auz.$fS57\"F*$!3ea-ipSf*y)F-7$$!3i$\\f2T\"))* 3\"F*$!3qj]S#*e=,\"*F-7$$!3s75[GAse5F*$!3$G()*=:xx7%*F-7$$!3uuz.$z4E/ \"F*$!3e_-ip?!Rd*F-7$$!3xO\\fdt\\E5F*$!3NK10Ck-N(*F-7$$!3*yTt)R6W=5F*$ !36@eE,')e:)*F-7$$!3z)*=:A\\Q55F*$!3575[y2:'*)*F-7$$!38R6H8oN15F*$!3r3 ')3n=VO**F-7$$!3pz.V/(GB+\"F*$!35.ipbHrw**F-7$$!3(*******\\YJ+5F*$!3G+ +++N&o***F-7$$!3_-ipbf+$)**F-$\"3++++++++]F-7$$!3M0CR6a'G'**F-Ffp7$$!3 .2')3n[sU**F-Ffp7$$!3!30CRh4:i*F-Ffp7$$!3X$\\f2O%H+$*F-Ffp7$$!3=kJD?cZ R')F-Ffp7$$!3%*\\S#R6Eu0)F-Ffp7$$!3q+\"[yAw@S(F-Ffp7$$!3#)4[yAeBWnF-Ff p7$$!3^AR6H$)=5hF-Ffp7$$!3X:`-iCTMbF-Ffp7$$!3%RyAeEf(\\[F-Ffp7$$!3CX;` -#o(pUF-Ffp7$$!3%R()*=:25&f$F-Ffp7$$!3Gwg)3n0z*HF-Ffp7$$!3VB?'p0)oUBF- Ffp7$$!3.sz.Vfw=Ffp7$$ !3s0?'pb*\\*p%F^tFfp7$$!3'*)Rt)R*Gu3$F^tFfp7$$!3=#z%yA$e`Z\"F^tFfp7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "FS := (x,n) -> \n sum((1- (-1)^k)/(k^2*Pi^2)*cos(k*Pi*x)+(3*(-1)^(k)-1)/(2*k*Pi)*sin(k*Pi*x),k=1 ..n):\nFS(x,21);\nf := x -> piecewise(x<0,1/2,-x):\nf_ := x -> f(x-2*f loor((x+1)/2)):\nplot([f_(x),FS(x,21)],x=-1.5..3.5,linestyle=[3,1],\n \+ color=[black,COLOR(RGB,.7,.2,0)],numpoints=150);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#,\\o*&#\"\"\"\"\"#F&*&%#PiG!\"\"-%$sinG6#,$*(F'F&F)F& %\"xGF&F&F&F&F&*(F'F&F)!\"#-%$cosG6#*&F)F&F0F&F&F&*&#F'\"\"$F&*&F)F*-F ,6#,$*(F9F&F)F&F0F&F&F&F&F**&#F'\"\"*F&*&F)F2-F4FF&F)F&F0F&F&F&F&F&*&#F'F]rF&*&F)F*-F,FjqF&F&F**&#F'\"$ *GF&*&F)F2-F46#,$*(\"#8 +gF-7$$!3q7p`!)RSl8F*$!3-t3j%>gfM'F-7$$!3>n+^Eb)4L\"F*$!39G$**[tW,p'F- 7$$!3=*QeP@v!*H\"F*$!343hTiyC4qF-7$$!3]w9ALR.m7F*$!3'\\B&yn1mRtF-7$$!3 &4;COfi=B\"F*$!3[!ReP1u8o(F-7$$!3k3j%>%3!y>\"F*$!3_8p`!e\"*>-)F-7$$!39 hTiVTwi6F*$!3g)QePceBP)F-7$$!3$39r1!Q!>8\"F*$!3u\"f)G$*>'4o)F-7$$!3;*Q ePOir4\"F*$!3T3hTijPG!*F-7$$!3G&yn^FyA1\"F*$!3?Z@K[s@x$*F-7$$!3/ZqbL'p a/\"F*$!3sH&HWm.`a*F-7$$!3y3j%>*4mG5F*$!3A7p`!3!R8(*F-7$$!3T6n]j!H5-\" F*$!3!f)G$\\O4(*y*F-7$$!3/9r1NrR85F*$!3gf)G$\\'Gg')*F-7$$!3D:t%3<\"e45 F*$!3aZo_\"H)=/**F-7$$!3n;vi1_w05F*$!3GL[sLzMU**F-7$$!3R+\"F*$!3+>G#fd20)**F-7$$!3YoFN^_*4+\"F*$!3 W:BZ'[Z+***F-7$$!3\")=yHg7/+5F*$!3'=\"=-(R(e****F-7$$!3o\"pGCps3***F-$ \"3++++++++]F-7$$!3;%>z=yK8)**F-F^s7$$!3=9r1g2$)*z*F-F^s7$$!34L]DQ(G$= '*F-F^s7$$!3'pd')zp;3J*F-F^s7$$!3mESg!42J&*)F-F^s7$$!3.n+^wMZO')F-F^s7 $$!3_037=i2*G)F-F^s7$$!3#f#*Qe(=FezF-F^s7$$!3&p9A$[A68wF-F^s7$$!3AZs3j z9'H(F-F^s7$$!3!e#Q2h\"fU&pF-F^s7$$!3]%>z=yK\"*f'F-F^s7$$!3#\\@K[A%*** G'F-F^s7$$!3#*eR49h6cfF-F^s7$$!3t)p/dN(=6cF-F^s7$$!3[./1f=ut_F-F^s7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "FS := (x,n) -> 2/Pi+sum(-4/((4*k^2-1)*Pi)*cos(2*k*x) ,k=1..n):\nFS(x,n);\nf := x -> sin(x):\nf_ := x -> f(x-Pi*floor((x/Pi) )):\nplot([f_(x),FS(x,1),FS(x,2),FS(x,3)],x=-0.5..5.5,linestyle=[3,1$4 ],\n color=[black,red,magenta,blue],numpoints=80);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"%\"\"\"%#PiG!\"\"F&*(\"\"#F&,&*&F*F&%\"nGF&F &F&F&F(F'F(F(-%$sumG6$,$**\"\")F&,&*&F%F&)%\"kGF*F&F&F&F(F(F'F(-%$cosG 6#*&F7F&%\"xGF&F*F(/F7;F&F-F&" }}{PARA 13 "" 1 "" {GLPLOT2D 622 227 227 {PLOTDATA 2 "6(-%'CURVESG6%7_q7$$!3++++++++]!#=$\"3;J?/'QbUz%F*7$$ !35`-ip&p`?%F*$\"3Mq9;Po]#3%F*7$$!3ReE,\")e'R^$F*$\"3Nv]y$*Q4UMF*7$$!3 L()*=:75kt#F*$\"3!*G]dCwQ-FF*7$$!3[=:s<-q`>F*$\"3)G5loI&HT>F*7$$!3Ea$o uR5Z<\"F*$\"3]]HoN0,s6F*7$$!3$QD?'pv)f8)!#>$\"3-[JT2Y,F\")FI7$$!3.lpbk 6([_%FI$\"32U&H6DFL_%FI7$$!3=/\"[yA?`l#FI$\"3A5Q=$FY7$$\"33r=:s<1! \\\"FY$\"3/E&e2Ah+\\\"FY7$$\"3CBS#R6RR;'FY$\"3u%oJ<3+R;'FY7$$\"3a<'pbk \"y$3\"FI$\"3S.ba,!pd#)pKF*7$$\"3hzAeE&)Q([i8DS*F*$\"39C4G,I1x!)F*7$$\"39dX ;Lu()=5!#<$\"3f6\\=>(*>:&)F*7$$\"3AVa$o3[P4\"Fht$\"3U(Gs(z6a$)))F*7$$ \"31rO\\fx'=<\"Fht$\"3\"QsUz+xZ@*F*7$$\"3$fX;`+1OC\"Fht$\"3#Gs>!\\1\\p %*F*7$$\"3(*3n$\\R&)4K\"Fht$\"3)\\y%*f!=f*o*F*7$$\"3/')3nL3O,9Fht$\"3Y rmmW6!o&)*F*7$$\"3D()*=:OWjV\"Fht$\"3MZ9gD%\\(4**F*7$$\"3q)3n$*)yKr9Fh t$\"3kyp4C4d]**F*7$$\"3H,\"[y06\"4:Fht$\"37'p\\*p1)4)**F*7$$\"359\"HjA %*oa\"Fht$\"3%p=iReVr***F*7$$\"3='pbk$z#fe\"Fht$\"3_`!*y$>b))***F*7$$ \"3FyAeY;'\\i\"Fht$\"37&4PoFht$\"39e7 !)o&Q#e$*F*7$$\"35Z(z.j>(4?Fht$\"35MW)>Y(4_!*F*7$$\"3'\\@xTB[53#Fht$\" 3-Ym.X]@E()F*7$$\"3-85[Q#H!f@Fht$\"3k!*e-._A>$)F*7$$\"3*Q6HjG.CB#Fht$ \"3#)Q![%z&f+*yF*7$$\"3wl753.54BFht$\"39Az*)456'R(F*7$$\"3q%yAem&3%Q#F ht$\"3w.]!3\"G7roF*7$$\"3A\\f2'Q)eiCFht$\"3_R<6!e%=!G'F*7$$\"3[n$\\f$e >QDFht$\"3o\\#f0DFWn&F*7$$\"30')3nt`^:EFht$\"37uIi(4\\9-&F*7$$\"3!*yAe mY>#p#Fht$\"3goGpWPBWVF*7$$\"3KdX;t\\liFFht$\"3'[v;o@P$*p$F*7$$\"3En$ \\fV5M%GFht$\"3vB#>bDJy$HF*7$$\"3(>'pb%3Rc\"HFht$\"34N9ISzNSAF*7$$\"3y f2'))))\\E*HFht$\"3aY(G\\&o#R[\"F*7$$\"3C4n$\\0fj1$Fht$\"3.s55>ZE;vFI7 $$\"3Qvz.L8G2JFht$\"3HUnao)e/V$FI7$$\"33T#R6h.#[JFht$\"3_>8]%R44h'FY7$ $\"3#yg)3(HRO=$Fht$\"3%oBl3aDM?%FI7$$\"3duz.$)\\2>KFht$\"34azB/D[SxFI7 $$\"3AL10CY^*H$Fht$\"33$H'\\mPms:F*7$$\"3U(z.ViEGP$Fht$\"3-Q6ya#)y\"H# F*7$$\"3!4CR6V=IX$Fht$\"3[2/R<9;kIF*7$$\"3#H,\"[)R5A_$Fht$\"3FO`Pk2%\\ r$F*7$$\"3PG#eE(>,,OFht$\"3%\\h#[:rFMWF*7$$\"3cuz../8xOFht$\"3)=w0nuNI 5&F*7$$\"3s2')32\"*>`PFht$\"3*z#)GwYM=u&F*7$$\"3Sm75oz)*GQFht$\"3/F=M> 10)RFht$\"3Qq\" =^3v\"RuF*7$$\"3)f$ouR9_bSFht$\"3,4phh\"*3>zF*7$$\"3gWNoa9ZMTFht$\"3k5 /663-w$)F*7$$\"34=M()H-%f?%Fht$\"3%o!y_*REZu)F*7$$\"3o?'pbgE\\G%Fht$\" 3Wc$os$o?+\"*F*7$$\"3W`-i*y\"egVFht$\"3Y$*enu\\^(Q*F*7$$\"3)GWNo#G0OWF ht$\"3op$3A5:1i*F*7$$\"3L>:s*HRM$\\&****F*7$$\"3x :s<%*[%Ru%Fht$\"3AFPmM:-&***F*7$$\"3K;`-A@6\"y%Fht$\"3+P<^T\\Rw**F*7$$ \"3x&Fht$\"3%H*=%RwNV'))F*7$$\"3%)yAeEv,u_Fht$ \"3C-+/m#*)QY)F*7$$\"3F5[y-(z\\M&Fht$\"3Y$QI#y[)\\1)F*7$$\"3![(z.B`&oU &Fht$\"3o$RWs3IWb(F*7$$\"3++++++++bFht$\"39=RqbKSbqF*-%'COLOURG6&%$RGB G\"\"!FhjlFhjl-%*LINESTYLEG6#\"\"$-F$6%7fq7$F($\"3i]%*>u#F*7$F=$\"3c,iqUp&>W #F*7$FB$\"3P$\\`_Vh'QAF*7$FG$\"3]-/c-&H\"y@F*7$FM$\"3(GT:2JL%R@F*7$FR$ \"3t`f8V$\\!G@F*7$FW$\"3#f4N3***eA@F*7$Ffo$\"3#45!e&)G1B@F*7$F`p$\"3E \"R*HmtYH@F*7$Fep$\"3yc:MX$*[h@F*7$Fjp$\"3()))=w\\*['=AF*7$$\"3[R6Hj%) =a9F*$\"3!=r@3Z-.I#F*7$F_q$\"3%)4`-Z\"3hS#F*7$Fdq$\"3ABV1Q*)*op#F*7$Fi q$\"3o@!\\5t5'HIF*7$F^r$\"3'o6!Hv+d\"[$F*7$Fcr$\"3r@:91!**p+%F*7$Fhr$ \"3IL!Gtd`!pXF*7$F]s$\"3`,q\"e&=!e6&F*7$Fbs$\"3[-ex^#ygz&F*7$Fgs$\"3Kr 'pu`B^Q'F*7$F\\t$\"3gM(R6(33pqF*7$Fat$\"3)>)*3lp>(fwF*7$Fft$\"37hdEg#* yw#)F*7$F\\u$\"3LT1B8Zx?))F*7$Fau$\"3^M`@5,jH$*F*7$Ffu$\"3.QL:x#)fL(*F *7$F[v$\"3**e^-%Ha\"45Fht7$F`v$\"3#HI=3#o*o.\"Fht7$Fev$\"3w9f*)y1yX5Fh t7$Fjv$\"3'o\")4ANiE0\"Fht7$F_w$\"3WTgp-s!y0\"Fht7$Fdw$\"3qf_&H5[01\"F ht7$$\"3E0CR\"36kc\"Fht$\"3)zU*zIm,h5Fht7$Fiw$\"3'p,KnhQ31\"Fht7$$\"36 ()*=:zWag\"Fht$\"3#yDY>L9+1\"Fht7$F^x$\"3Kf\")=K]ae5Fht7$Fcx$\"3%o?FW$ R\"Q0\"Fht7$Fhx$\"3Q&>fta]m/\"Fht7$F]y$\"3li&)R***[g-\"Fht7$Fby$\"3GmV B')z_K**F*7$Fgy$\"3+C*Rk1,eb*F*7$F\\z$\"3gSd#=*))Qx!*F*7$Faz$\"3c@'G:] 4ce)F*7$Ffz$\"3e:;B.9v'*zF*7$F[[l$\"3<9I&f2oiS(F*7$F`[l$\"3U(e$)=wa`w' F*7$Fe[l$\"3O&[^H$GdHhF*7$Fj[l$\"3'*yg\\g!)*)paF*7$F_\\l$\"3Q`\\&G\\7_ &[F*7$Fd\\l$\"3!f`RnUxBE%F*7$Fi\\l$\"3%f&R\"y:-Ss$F*7$F^]l$\"3%G3S8#Hp $G$F*7$Fc]l$\"3/Hta?anaGF*7$Fh]l$\"3!fN5-:4\"[DF*7$F]^l$\"3,R&3fx!)*3B F*7$$\"3^M()*=Z/&HIFht$\"3tf%)zZXEGAF*7$Fb^l$\"3;\"R/Uq>+<#F*7$$\"33Ut )R>?o3$Fht$\"3[*=^LV0v9#F*7$Fg^l$\"3/Z'=t$\\0K@F*7$$\"3B3')3sCuFJFht$ \"3c=VS'3%pB@F*7$F\\_l$\"3YP*4+)oVA@F*7$$\"3ACR6a9#f;$Fht$\"3^!>G'R!*3 F@F*7$Fa_l$\"33Y$*[RO1P@F*7$$\"3U\"Hj+9d8?$Fht$\"35I-#f:[B:#F*7$Ff_l$ \"3AzwM)RBH<#F*7$$\"3o.Va.[HfKFht$\"3`*)RTR&=\"RAF*7$F[`l$\"3!=[l!)p.? 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" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 245 "FS := (x, n) -> 2/Pi+sum(-4/((4*k^2-1)*Pi)*cos(2*k*x),k=1..n):\nFS(x,9);\nf := x -> sin(x):\nf_ := x -> f(x-Pi*floor((x/Pi))):\nplot([f_(x),FS(x,9)],x =-0.5..5.5,linestyle=[3,1],thickness=[1,2],\n color=[black,COLO R(RGB,.9,.2,0),numpoints=100]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6* &\"\"#\"\"\"%#PiG!\"\"F&*&#\"\"%\"\"$F&*&F'F(-%$cosG6#,$*&F%F&%\"xGF&F &F&F&F(*&#F+\"#:F&*&F'F(-F/6#,$*&F+F&F3F&F&F&F&F(*&#F+\"#NF&*&F'F(-F/6 #,$*&\"\"'F&F3F&F&F&F&F(*&#F+\"#jF&*&F'F(-F/6#,$*&\"\")F&F3F&F&F&F&F(* &#F+\"#**F&*&F'F(-F/6#,$*&\"#5F&F3F&F&F&F&F(*&#F+\"$V\"F&*&F'F(-F/6#,$ *&\"#7F&F3F&F&F&F&F(*&#F+\"$&>F&*&F'F(-F/6#,$*&\"#9F&F3F&F&F&F&F(*&#F+ \"$b#F&*&F'F(-F/6#,$*&\"#;F&F3F&F&F&F&F(*&#F+\"$B$F&*&F'F(-F/6#,$*&\"# =F&F3F&F&F&F&F(" }}{PARA 13 "" 1 "" {GLPLOT2D 706 218 218 {PLOTDATA 2 "6&-%'CURVESG6&7bp7$$!3++++++++]!#=$\"3;J?/'QbUz%F*7$$!3-+++v`3YVF*$\" 3x#3FZj`0@%F*7$$!3/+++]2<#p$F*$\"3SW\\A)y`)3OF*7$$!3r***\\78.K7$F*$\"3 '*\\P9>F*$ \"3Dvk_z-q->F*7$$!3++++D$3XF\"F*$\"3mB$f!o11r7F*7$$!3y(*******H./j!#>$ \"31*Q*)4Pe)*H'FN7$$\"3AW+++DC%pkh oF*7$$\"3Z++]P&4o8)F*$\"31/!4lt-#osF*7$$\"3;+++DMe6()F*$\"3%3U83a`2l(F *7$$\"36+++&>q0+\"!#<$\"3GR*RnD*y<%)F*7$$\"3'******\\U808\"Fjr$\"3C!*> xz7JY!*F*7$$\"35+++0ytb7Fjr$\"3,\\$*piCy2&*F*7$$\"3/+](=n&f78Fjr$\"3-e #))4!*4&o'*F*7$$\"3)****\\(QNXp8Fjr$\"3'Rr,bB*)zz*F*7$$\"36+](=/jqV\"F jr$\"3<_\"=vB52\"**F*7$$\"3.+++XDn/:Fjr$\"3=%>j=>Y\"y**F*7$$\"3.++v.'4 L`\"Fjr$\"3%RaY2buH***F*7$$\"3-++]im%>c\"Fjr$\"3I\\\\w>%3'****F*7$$\"3 -++D@Pe!f\"Fjr$\"3Wl$z*eB/)***F*7$$\"3.+++!y?#>;Fjr$\"33G*)f_wF))**F*7 $$\"3)***\\P%>Weo\"Fjr$\"3zV$)H0H*Q$**F*7$$\"3'****\\(3wY_1W6=Fjr$\"3A%ym@BY=r*F*7$$\"3#)******HOTq=Fjr$ \"3UE)4!Q^\\a&*F*7$$\"37++v3\">)**>Fjr$\"3aU')p#*os$4*F*7$$\"3:++DEP/B @Fjr$\"3l,^)3$oZ8&)F*7$$\"3=++](o:;D#Fjr$\"3g)z0]\\s0x(F*7$$\"3=++v$)[ opBFjr$\"3+!Q\")HzN](pF*7$$\"3&)**\\7t;OLCFjr$\"3N2-9[>\"\\]'F*7$$\"3% *****\\i%Qq\\#Fjr$\"3[w#o)H8U3gF*7$$\"3%***\\i]2=jDFjr$\"3[p^BC'[pY&F* 7$$\"3&****\\(QIKHEFjr$\"3fIIJm1GFjr$\"35G jIas.(G$F*7$$\"37++]Zn%)oGFjr$\"3cqtf@&oPp#F*7$$\"3t***\\(G(*3LHFjr$\" 33[,,5[&*p?F*7$$\"3y******4FL(*HFjr$\"3g[t?+5gP9F*7$$\"3-++vL>=gIFjr$ \"3oG]e#>#3K\")FN7$$\"3#)****\\d6.BJFjr$\"3m!>yy-Vg&=FN7$$\"3!***\\785 %Q=$Fjr$\"3'z;%eI#zNA%FN7$$\"3(****\\(o3lWKFjr$\"30&\\;(=)e(G5F*7$$\"3 <+]iX)p@J$Fjr$\"39tGJD>^(p\"F*7$$\"3!*****\\A))ozLFjr$\"3#yORKLH&eBF*7 $$\"3(****\\iid.W$Fjr$\"3EX:,\\6SVHF*7$$\"3e******Hk-,NFjr$\"3mo0t*\\U u^$F*7$$\"3%)****\\FL!ec$Fjr$\"3%f\"4Kna,;TF*7$$\"36+++D-eIOFjr$\"3/g] 8\"[Btp%F*7$$\"3!***\\(=sx#*o$Fjr$\"3D$*eBTH72_F*7$$\"3u***\\(=_(zu$Fj r$\"3C%)f]]p)*)p&F*7$$\"3/+](o3Z@\"QFjr$\"3Y39\"Ro/U@'F*7$$\"3M+++b*=j (QFjr$\"3wz@#\\#4%Qq'F*7$$\"3(***\\(=o*pORFjr$\"3*[')Q7^&QRrF*7$$\"3g* **\\(3/3(*RFjr$\"3!o6\"G.!4*[vF*7$$\"33++vB4JBTFjr$\"3Sj,S#fKXJ)F*7$$ \"3u*****\\KCnC%Fjr$\"3^hto:KBN*)F*7$$\"3s***\\(=n#fP%Fjr$\"3]l,(e`&GR %*F*7$$\"3P+++!)RO+XFjr$\"3%f\"=m$yngx*F*7$$\"3m++D;:*Rc%Fjr$\"3MCwe\" )G4!*)*F*7$$\"30++]_!>wi%Fjr$\"3/%\\0s!=4k**F*7$$\"3o*\\ilQp\"fYFjr$\" 3=)))yjrTe)**F*7$$\"3=+]i?(>2p%Fjr$\"3WmSx2Bl(***F*7$$\"3s+voa+FAZFjr$ \"3qj]#e#=^****F*7$$\"3O++v)Q?Qv%Fjr$\"3#[Ix'>%=9***F*7$$\"3K+]P\\L!= \"[Fjr$\"3yb8%>gC1&**F*7$$\"3G+++5jyp[Fjr$\"3#z=9+#fQw)*F*7$$\"3B++DE8 CO\\Fjr$\"3cHC'*R^\\](*F*7$$\"3<++]Ujp-]Fjr$\"3+<%*p$Rf:e*F*7$$\"3++++ gEd@^Fjr$\"3)>Ha75fW<*F*7$$\"39++v3'>$[_Fjr$\"3ck6%G%y$zf)F*7$$\"37++D 6Ejp`Fjr$\"3VL4:Z?z;zF*7$$\"3++++++++bFjr$\"39=RqbKSbqF*-%'COLOURG6&%$ RGBG\"\"!FgflFgfl-%*THICKNESSG6#\"\"\"-%*LINESTYLEG6#\"\"$-F$6&7cp7$F( $\"3gL'QaoY0z%F*7$F.$\"3?[X^s=NrTF*7$F3$\"3]KSCZ&Gte$F*7$F8$\"3E'yTHti R5$F*7$F=$\"3'z6cId\"z(e#F*7$FB$\"3p/lLJ6z3>F*7$FG$\"3-gYk=Z?r6F*7$FL$ \"3=)Guw:IQl&FN7$FR$\"3avy8GEw^LFN7$FX$\"33DQ?%=Du#eFN7$Fgn$\"3YjAF'eU \\>\"F*7$F\\o$\"39m\"GJm'4\")=F*7$Fao$\"3$>FKx$4D>DF*7$Ffo$\"3#eA(o[1( Q3$F*7$F[p$\"3g&e(y_.92OF*7$F`p$\"3#ohKo)GbwTF*7$Fep$\"3g'**QJg&HzZF*7 $Fjp$\"3ar7c\\\\bf`F*7$F_q$\"3yRI/Fh,yeF*7$Fdq$\"3e%\\7dsP6O'F*7$Fiq$ \"3)3u)4WFcOoF*7$F^r$\"3IeZ*eBG,E(F*7$Fcr$\"33)zHrO*RmwF*7$Fhr$\"3:*)* 4NK9eT)F*7$F^s$\"3jb6YTh.O!*F*7$Fcs$\"3(zZE\"pQdD&*F*7$F]t$\"3#e8<\\Ai Vy*F*7$Fbt$\"37v.2)\\vf*)*F*7$Fgt$\"3SX%Q!GxY$)**F*7$F\\u$\"3cjmPS9i+5 Fjr7$Fau$\"3Ed'R(H)*o,5Fjr7$Ffu$\"3%4(y3\\YV,5Fjr7$F[v$\"3i%oh)GV%))** *F*7$F`v$\"3[z7:2T`B**F*7$Fev$\"3fg&*=/=c=)*F*7$F_w$\"3)pW(Rps/q&*F*7$ Fdw$\"3)4hBdNCs3*F*7$Fiw$\"3\"*G4;Teu/&)F*7$F^x$\"3cp$Q(z-u!z(F*7$Fcx$ \"3UY\"z9GkB&pF*7$Fhx$\"3W/n^I$fj['F*7$F]y$\"3;)y&3*z%*4-'F*7$Fby$\"37 DSvrR#z\\&F*7$Fgy$\"3H!=j>2wh!\\F*7$F\\z$\"3=Mm4y7$)fVF*7$Faz$\"3dIFS; 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "f := x -> sin(x):\n'f(x)'=f(x);\nf_ := x -> f(x-Pi*floor((x/Pi))) :\nFourierSeries(f(x),x=0..Pi,numterms=9):\nF := unapply(%,x):\n'F(x)' =F(x);\nplot([f_(x),F(x)],x=-0.5..5.5,linestyle=[3,1],thickness=[1,2], \n color=[black,COLOR(RGB,.9,.2,0),numpoints=100]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,6*&\"\"#\"\"\"%#PiG!\"\"F+*&#\"\"%\"\"$F +*&F,F--%$cosG6#,$*&F*F+F'F+F+F+F+F-*&#F0\"#:F+*&F,F--F46#,$*&F0F+F'F+ F+F+F+F-*&#F0\"#NF+*&F,F--F46#,$*&\"\"'F+F'F+F+F+F+F-*&#F0\"#jF+*&F,F- -F46#,$*&\"\")F+F'F+F+F+F+F-*&#F0\"#**F+*&F,F--F46#,$*&\"#5F+F'F+F+F+F +F-*&#F0\"$V\"F+*&F,F--F46#,$*&\"#7F+F'F+F+F+F+F-*&#F0\"$&>F+*&F,F--F4 6#,$*&\"#9F+F'F+F+F+F+F-*&#F0\"$b#F+*&F,F--F46#,$*&\"#;F+F'F+F+F+F+F-* &#F0\"$B$F+*&F,F--F46#,$*&\"#=F+F'F+F+F+F+F-" }}{PARA 13 "" 1 "" {GLPLOT2D 675 223 223 {PLOTDATA 2 "6&-%'CURVESG6&7bp7$$!3++++++++]!#=$ \"3;J?/'QbUz%F*7$$!3-+++v`3YVF*$\"3x#3FZj`0@%F*7$$!3/+++]2<#p$F*$\"3SW \\A)y`)3OF*7$$!3r***\\78.K7$F*$\"3'*\\P9>F*$\"3Dvk_z-q->F*7$$!3++++D$3XF\"F*$\"3 mB$f!o11r7F*7$$!3y(*******H./j!#>$\"31*Q*)4Pe)*H'FN7$$\"3AW+++DC%pkhoF*7$$\"3Z++]P&4o8)F*$\"31/!4lt-#os F*7$$\"3;+++DMe6()F*$\"3%3U83a`2l(F*7$$\"36+++&>q0+\"!#<$\"3GR*RnD*y<% )F*7$$\"3'******\\U808\"Fjr$\"3C!*>xz7JY!*F*7$$\"35+++0ytb7Fjr$\"3,\\$ *piCy2&*F*7$$\"3/+](=n&f78Fjr$\"3-e#))4!*4&o'*F*7$$\"3)****\\(QNXp8Fjr $\"3'Rr,bB*)zz*F*7$$\"36+](=/jqV\"Fjr$\"3<_\"=vB52\"**F*7$$\"3.+++XDn/ :Fjr$\"3=%>j=>Y\"y**F*7$$\"3.++v.'4L`\"Fjr$\"3%RaY2buH***F*7$$\"3-++]i m%>c\"Fjr$\"3I\\\\w>%3'****F*7$$\"3-++D@Pe!f\"Fjr$\"3Wl$z*eB/)***F*7$$ \"3.+++!y?#>;Fjr$\"33G*)f_wF))**F*7$$\"3)***\\P%>Weo\"Fjr$\"3zV$)H0H*Q $**F*7$$\"3'****\\(3wY_1W6=Fjr$\"3 A%ym@BY=r*F*7$$\"3#)******HOTq=Fjr$\"3UE)4!Q^\\a&*F*7$$\"37++v3\">)**> Fjr$\"3aU')p#*os$4*F*7$$\"3:++DEP/B@Fjr$\"3l,^)3$oZ8&)F*7$$\"3=++](o:; 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F`im-%%VIEWG6$;$!\"&Fehm$\"#bFehm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 29 " be the function defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = sin*x;" "6#/-%\"fG6#%\"xG*&%$sin G\"\"\"F'F*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-Pi/2 <= x;" "6#1,$*&%# PiG\"\"\"\"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` < Pi/2;" "6#2%!G*&%#PiG \"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "Pi;" " 6#%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 63 "(a) Find formulas for the Fourier coefficients of \+ the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 53 " \+ and give the first few terms of the Fourier series. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "(b) Find the fundamenta l wave and plot its graph along with the graph of " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 74 "(c) Compare the graphs of some trunca ted Fourier series with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 23 "Constant coefficient: " }{XPPEDIT 18 0 "c = 0;" "6#/%\"cG\"\"!" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Cosine coefficients: " }{XPPEDIT 18 0 "a[k] = 0;" "6#/&%\"aG6 #%\"kG\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "k=1,2,3,` . . . `" "6&/% \"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Sine coefficients: " } {XPPEDIT 18 0 "b[k] = 8*(-1)^(k+1)*k/((4*k^2-1)*Pi);" "6#/&%\"bG6#%\"k G**\"\")\"\"\"),$F*!\"\",&F'F*F*F*F*F'F**&,&*&\"\"%F**$F'\"\"#F*F*F*F- F*%#PiGF*F-" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "k=1,2,3,` . . . `" "6&/% \"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Fourier series: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "8/(3*Pi)" "6#*&\"\") \"\"\"*&\"\"$F%%#PiGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*x-1 6/(15*Pi);" "6#,&*(%$sinG\"\"\"\"\"#F&%\"xGF&F&*&\"#;F&*&\"#:F&%#PiGF& !\"\"F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*4*x+24/(35*Pi);" "6#,&*(% $sinG\"\"\"\"\"%F&%\"xGF&F&*&\"#CF&*&\"#NF&%#PiGF&!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*6*x-32/(63*Pi);" "6#,&*(%$sinG\"\"\"\"\"'F&% \"xGF&F&*&\"#KF&*&\"#jF&%#PiGF&!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*8*x+40/(99*Pi);" "6#,&*(%$sinG\"\"\"\"\")F&%\"xGF&F&*&\"#SF&*& \"#**F&%#PiGF&!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*10*x+` . . \+ . `;" "6#,&*(%$sinG\"\"\"\"#5F&%\"xGF&F&%(~.~.~.~GF&" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 19 "F undamental wave: " }{XPPEDIT 18 0 "y = 8/(3*Pi);" "6#/%\"yG*&\"\")\" \"\"*&\"\"$F'%#PiGF'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*x;" " 6#*(%$sinG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "f := x -> s in(x):\n'f(x)'=f(x);\nf_ := x -> f(x-Pi*floor(x/Pi+1/2)):\nFourierSeri es(f(x),x=-Pi/2..Pi/2,numterms=1,info=1):\nF := unapply(%,x):\n'F(x)'= F(x);\nplot([f_(x),F(x)],x=-3.3..6.5,linestyle=[3,1],color=[black,red] ,numpoints=80);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$si nGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7$$!3#Qt)*Q4\"GdIF*$\"3%o4jAzq6U)F3 7$$!36,\"[)>.GIHF*$\"3'oM1%HAV(4#F-7$$!3sZyA-xV-GF*$\"3>'[nh*H!pK$F-7$ $!3EkJD)p-_n#F*$\"3)ol(H*RZm\\%F-7$$!3=IWNi&Rsb#F*$\"3RL3n6$*f;bF-7$$! 3![yAyc&4NCF*$\"3KH$fr+L<\\'F-7$$!3tm$\\R2u(3BF*$\"3u=pi%G1$)R(F-7$$!3 [g)3ZpdG=#F*$\"39aeTmQl%=)F-7$$!3'*>'p:APL0#F*$\"35Rnk%G(=e))F-7$$!3e8 \"H$=bDR>F*$\"3hL')*y=M)G$*F-7$$!3dF#e1aE3\"=F*$\"37h+%\\#zI8(*F-7$$!3 1D?'4Q[ju\"F*$\"3KsUK;F*$\"3y/&)=A\"G!))**F-7$$ !3')Q6H.%*>/;F*$\"3'\\f06p@W***F-7$$!3A-ipR_m)e\"F*$\"3/gyB_NS)***F-7$ $!3yL()*y:)*3e\"F*$\"3IG2Qp(*[****F-7$$!3Ml75w58t:F*$\"3!QwSZus*****F- 7$$!3CJD?NvCp:F*$!2UTDi+))*****F*7$$!39(z.V*ROl:F*$!3g&Q\"=\\C&)****F- 7$$!3!G10MX![h:F*$!3EJJlvgc****F-7$$!3qGj]7pfd:F*$!3i)pf**))G\"****F-7 $$!3CX;`x3<,:F*$!3GUKDu7xv**F-7$$!3yhpbU[uW9F*$!3Gc;S_-m?**F-7$$!3e8\" H$H)[wP\"F*$!3wMFj'*F-7$$!3E-ip nX(o>\"F*$!3Ad/_4,-4$*F-7$$!3u9s)F-$!3>?E$p*=x0tF-7$$!39BWN[7yop F-$!3Xa,$)*4o#=kF-7$$!3%=qO\\fEGp&F-$!3#oQFF!)z-R&F-7$$!38\\X;8>5@XF-$ !3es5F*zY'oVF-7$$!3C&3n$\\&QsD$F-$!3yJA+Qm%**>$F-7$$!3uc)3nrRW%>F-$!3] ;*3u95A$>F-7$$!3_;)3ntWk,)F3$!3LL!)RI9'y+)F37$$\"3&Rl)3np0EVF3$\"3UX>u SxqCVF37$$\"3m;s<%*oq2Ef7Mm[@'F-7$$\"3ITD?'H(3#*zF-$\"3g%*4. 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2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "Since the function " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 88 " is discontinuous, the convergence i s rather slow, especially near the discontinuities. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "FS := (x, n) -> sum(8*(-1)^(k+1)*k/((4*k^2-1)*Pi)*sin(2*k*x),k=1..n):\nFS(x,23); \nf := x -> sin(x):\nf_ := x -> f(x-Pi*floor(x/Pi+1/2)):\nplot([f_(x), FS(x,23)],x=-3.3..6.5,linestyle=[3,1],\n color=[black,COLOR(RGB ,.6,0,.8),numpoints=100]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,P*&#\" \")\"\"$\"\"\"*&%#PiG!\"\"-%$sinG6#,$*&\"\"#F(%\"xGF(F(F(F(F(*&#\"#;\" #:F(*&F*F+-F-6#,$*&\"\"%F(F2F(F(F(F(F+*&#\"#C\"#NF(*&F*F+-F-6#,$*&\"\" 'F(F2F(F(F(F(F(*&#\"#K\"#jF(*&F*F+-F-6#,$*&F&F(F2F(F(F(F(F+*&#\"#S\"#* *F(*&F*F+-F-6#,$*&\"#5F(F2F(F(F(F(F(*&#\"#[\"$V\"F(*&F*F+-F-6#,$*&\"#7 F(F2F(F(F(F(F+*&#\"#c\"$&>F(*&F*F+-F-6#,$*&\"#9F(F2F(F(F(F(F(*&#\"#k\" $b#F(*&F*F+-F-6#,$*&F5F(F2F(F(F(F(F+*&#\"#s\"$B$F(*&F*F+-F-6#,$*&\"#=F (F2F(F(F(F(F(*&#\"#!)\"$*RF(*&F*F+-F-6#,$*&\"#?F(F2F(F(F(F(F+*&#\"#)) \"$$[F(*&F*F+-F-6#,$*&\"#AF(F2F(F(F(F(F(*&#\"#'*\"$v&F(*&F*F+-F-6#,$*& 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0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_______________________________ ___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 22 "discontinuity pictur e " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 521 "p1 := plot([[[0,1],[0,-1]],[t,1/(t^2+1),t=0..1],[-t, -1/(t^2+1),t=0..1]],\n style=[point,line,line],symbol=circle, symbolsize=13,color=red):\np2 := plot([[[0,0]]$4],color=red,style=poin t,symbol=[circle$2,diamond,cross],\n symbolsize=[13,10$3]):\np3 := \+ plot([[0,-1],[0,1]],color=black,linestyle=2):\nt1 := plots[textplot]([ [0,-1.15,`x = d`],[.3,.15,`f(d-) + f(d+)`],\n [.3,.07,`_________`], [.3,-.15,`2`]],color=black):\nt2 := plots[textplot]([.7,.9,`y = f(x)`] ,color=red):\nplots[display]([p1,p2,p3,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 80 "Code for picture for extending \+ a function on an interval to a periodic function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1460 "f := x -> \+ 0.2+0.7*sin(x)+1/10*cos(5*x):\nf_ := x -> f(x-2*floor(x/2)):\nfa := .3 : fb := f(2.):\np1 := plot(f(x),x=0..2,color=red):\np2 := plot(f(x+2), x=-2..0,color=brown):\np3 := plot(f(x-2),x=2..4,color=brown):\np4 := p lot(f(x-4),x=4..6,color=brown):\np5 := plot(f(x-6),x=6..8,color=brown) :\np6 := plot([[-2,0],[8,0]],color=black):\np7 := plot([[[-2,0],[-2,f( 0)]],seq([[2*i,0],[2*i,f(2)]],i=0..4)],\n color=black,l inestyle=2):\np8 := plot([[[.7,0],[.7,f(.7)]],[[4.7,0],[4.7,f(.7)]]], \n color=blue,linestyle=3):\np9 := plot([[[-2,fa],[2,fa ],[4,fa],[6,fa]]$3],style=point,\n symbol=[circle,diamond,cross],c olor=brown):\np10 := plot([[0,fb],[4,fb],[6,fb],[8,fb]],style=point,sy mbol=circle,color=brown):\np11 := plot([[[0,fa]]$3],style=point,symbol =[circle,diamond,cross],color=red):\np12 := plot([[2,fb]],style=point, symbol=circle,color=red):\np13 := [plottools[arrow]([.7,-.4],[.7,-.15] ,.02,.15,.3,\n color=COLOR(RGB,.6,0,.9))][1]:\np 14 := [plottools[arrow]([4.7,-.4],[4.7,-.15],.02,.15,.3,\n \+ color=COLOR(RGB,.6,0,.9))][1]:\np15 := plot([[0.7,-.4],[4.7 ,-.4]],color=COLOR(RGB,.6,0,.9)):\nt1 := plots[textplot]([[0,-.07,a],[ 2,-.07,b]],\n color=COLOR(RGB,0,0,.01),font=[SYMBOL,9]) :\nt2 := plots[textplot]([[.7,-.07,`u=x-2T`],[4.7,-.07,x]],\n \+ font=[HELVETICA,9],color=blue):\nplots[display]([p1,p2,p3,p4,p5 ,p6,p7,p8,p9,p10,p11,p12,p13,p14,p15,t1,t2],\n ax es=none);" }}}{PARA 0 "" 0 "" 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