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-1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "See: " }{URLLINK 17 "Fourier series movie" 4 "http://www.csc.fi/ math_topics/Movies/Fourier/fourier.mpg" "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Introductory examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 77 "It is useful to be \+ able to express certain functions as trigonometric series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = c+a[1]*cos*x+b[1]*sin*x+a[2]*cos*2*x+b[2]*sin*2*x+a[3]*cos*3 *x+b[3]*sin*3*x+` . . . `;" "6#/-%\"fG6#%\"xG,2%\"cG\"\"\"*(&%\"aG6#F* F*%$cosGF*F'F*F**(&%\"bG6#F*F*%$sinGF*F'F*F***&F-6#\"\"#F*F/F*F8F*F'F* F***&F26#F8F*F4F*F8F*F'F*F***&F-6#\"\"$F*F/F*F?F*F'F*F***&F26#F?F*F4F* F?F*F'F*F*%(~.~.~.~GF*" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = c+``;" "6#/%!G,&%\"cG\"\"\"F$F'" } {XPPEDIT 18 0 "Sum(a[k]*cos*k*x+b[k]*sin*k*x,k = 1 .. infinity)" "6#-% $SumG6$,&**&%\"aG6#%\"kG\"\"\"%$cosGF,F+F,%\"xGF,F,**&%\"bG6#F+F,%$sin GF,F+F,F.F,F,/F+;F,%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Series of this form are called " }{TEXT 261 14 "Fourier s eries" }{TEXT -1 82 " after Jean Baptiste Fourier (1766-1830), who use d them in his study of heat flow." }}{PARA 0 "" 0 "" {TEXT -1 12 "Note that f(" }{TEXT 269 1 "x" }{TEXT -1 26 ") is periodic with period " } {XPPEDIT 18 0 "2*pi" "6#*&\"\"#\"\"\"%#piGF%" }{TEXT -1 51 ", since th e shortest period of any of the terms is " }{XPPEDIT 18 0 "2*pi" "6#*& \"\"#\"\"\"%#piGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the f unction " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 12 " defin ed by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)=sin*x+ sin*2*x/2" "6#/-%\"gG6#%\"xG,&*&%$sinG\"\"\"F'F+F+**F*F+\"\"#F+F'F+F-! \"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "The following \+ picture illustrates how the graph of " }{XPPEDIT 18 0 "g(x)=sin*x+sin* 2*x/2" "6#/-%\"gG6#%\"xG,&*&%$sinG\"\"\"F'F+F+**F*F+\"\"#F+F'F+F-!\"\" F+" }{TEXT -1 2 " (" }{TEXT 260 3 "red" }{TEXT -1 54 " curve) is forme d as the sum of the two \"sine curves\" " }{XPPEDIT 18 0 "y=sin*x" "6# /%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 2 " (" }{TEXT 256 4 "blue" } {TEXT -1 12 " curve) and " }{XPPEDIT 18 0 "y=sin*2*x/2" "6#/%\"yG**%$s inG\"\"\"\"\"#F'%\"xGF'F(!\"\"" }{TEXT -1 2 " (" }{TEXT 300 5 "green" }{TEXT -1 13 " curve). The " }{TEXT 296 1 "y" }{TEXT -1 24 " values al ong the curve " }{XPPEDIT 18 0 "y=sin*x+sin*2*x/2" "6#/%\"yG,&*&%$sinG \"\"\"%\"xGF(F(**F'F(\"\"#F(F)F(F+!\"\"F(" }{TEXT -1 28 " are obtained by adding the " }{TEXT 297 1 "y" }{TEXT -1 35 " values along the comp onent curves " }{XPPEDIT 18 0 "y=sin*x" "6#/%\"yG*&%$sinG\"\"\"%\"xGF' " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=sin*2*x/2" "6#/%\"yG**%$sinG\" \"\"\"\"#F'%\"xGF'F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "plot([sin(x),sin( 2*x)/2,sin(x)+sin(2*x)/2],x=0..4*Pi,y,color=[blue,green,red],\n thic kness=2,labelfont=[HELVETICA,9],ytickmarks=6,\n font=[SYMBOL,9],xti ckmarks=[0=`0`,evalf(Pi/2)=`p/2`,evalf(Pi)=`p`,\n evalf(3*Pi/2)=`3p /2`,evalf(2*Pi)=`2p`,evalf(5*Pi/2)=`5p/2`,evalf(3*Pi)=`3p`,\n eval f(7*Pi/2)=`7p/2`,evalf(4*Pi)=`4p`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 737 270 270 {PLOTDATA 2 "6*-%'CURVESG6$7[s7$$\"\"!F)F(7$$\"3M\"HT_eb&p 8!#=$\"3_&>k'z\"y_O\"F-7$$\"3o#e#[q66RFF-$\"3\"G)>\"z2))\\q#F-7$$\"3;4 rAiTvIRF-$\"3%e*o@I4JIQF-7$$\"3>O;(R:(RA^F-$\"3z1(G8?28!\\F-7$$\"3ss#Q #GV_ikF-$\"3?yw8by)>-'F-7$$\"383\\]-:l-yF-$\"32V(*yL*yY.(F-7$$\"3)z*3f 4>m^\"*F-$\"3Gw*)4w\"Rm#zF-7$$\"3o)on;Bn+0\"!#<$\"3'R;\\;wmXn)F-7$$\"3 e%)yF6rK%=\"FQ$\"39Itp_lXi#*F-7$$\"3^!3))3*pe=8FQ$\"3Mt$[**eMOo*F-7$$ \"33XyJQ]#3Q\"FQ$\"3_%3!Rrl4?)*F-7$$\"3W4wu&3jIW\"FQ$\"3XW%yR&>`=**F-7 $$\"3i\"\\i%4@=u9FQ$\"3/$)>idZO`**F-7$$\"3!QPxJ8,`]\"FQ$\"3Ocn)Rcf&y** F-7$$\"3>cA*o:?k`\"FQ$\"3'zZ^l(>4%***F-7$$\"3QQrg!=Rvc\"FQ$\"31a//bp%* ****F-7$$\"3?*pzr7h(*f\"FQ$\"3/ct&\\\\0e***F-7$$\"3#)fAvtI)>j\"FQ$\"3) \\IF$\\nG\")**F-7$$\"3U?[K?]?k;FQ$\"3/K&*)4z0k&**F-7$$\"3-\"Q(*o'pU'p \"FQ$\"3'[ff/X)=@**F-7$$\"3E-D/g3(3w\"FQ$\"3&4%)G7c,*>)*F-7$$\"3rBw=`Z JD=FQ$\"3q-K3#eYyn*F-7$$\"3Aw[9'o5'e>FQ$\"3'*e\\%H0ytD*F-7$$\"3%*G@5>m !>4#FQ$\"3#)=x%pipEn)F-7$$\"3gv=Jq]xCAFQ$\"31)f%fAipOzF-7$$\"3EA;_@Nkd BFQ$\"3w_+8$>931(F-7$$\"39%\\*>b]J%\\#FQ$\"3F+HRf!p,.'F-7$$\"3flt())e' )4j#FQ$\"31KIUo41()[F-7$$\"3Q,F8]pO^FFQ$\"3$p0*41;(R!QF-7$$\"3uO!)Q6tu rGFQ$\"3/Ig9)yBem#F-7$$\"3uF0:'3ns+$FQ$\"3&[ZfK%)>#R8F-7$$\"3t=I\"4'oy UJFQ$!3-ShPRq?%>\"!#?7$$\"3Sc6*G4j)yKFQ$!3%)\\I,NuRo8F-7$$\"3k$Hp[KR\\ T$FQ$!3#yPVKh`&*p#F-7$$\"3sgGp))R2YNFQ$!3q*>RC,?a$RF-7$$\"3!yU;Dl3sn$F Q$!3Z@l;s'3P5&F-7$$\"3iNN>!\\\"H'z$FQ$!3IpT+(y.#*3'F-7$$\"3)QkqyKu`\"R FQ$!3gCZT(e^%))pF-7$$\"3C>Tu^jd)>%FQ$!3dPcU/$p(3()F-7$$\"3O])*o#)4`=VF Q$!3p4XMUTNM#*F-7$$\"3e!eNOh&[QWFQ$!3qAUgiSAF'*F-7$$\"3ca<))*[`#3XFQ$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "The following picture illustrates how the graph of " }{XPPEDIT 18 0 "h(x) = sin*x+sin*2*x/2+sin*3*x/3;" "6# /-%\"hG6#%\"xG,(*&%$sinG\"\"\"F'F+F+**F*F+\"\"#F+F'F+F-!\"\"F+**F*F+\" \"$F+F'F+F0F.F+" }{TEXT -1 2 " (" }{TEXT 260 3 "red" }{TEXT -1 47 " cu rve) is formed as the sum of the two curves " }{XPPEDIT 18 0 "y = sin* x+sin*2*x/2;" "6#/%\"yG,&*&%$sinG\"\"\"%\"xGF(F(**F'F(\"\"#F(F)F(F+!\" \"F(" }{TEXT -1 2 " (" }{TEXT 256 4 "blue" }{TEXT -1 12 " curve) and \+ " }{XPPEDIT 18 0 "y = sin*3*x/3;" "6#/%\"yG**%$sinG\"\"\"\"\"$F'%\"xGF 'F(!\"\"" }{TEXT -1 2 " (" }{TEXT 300 5 "green" }{TEXT -1 13 " curve). The " }{TEXT 298 1 "y" }{TEXT -1 24 " values along the curve " } {XPPEDIT 18 0 "y = sin*x+sin*2*x/2+sin*3*x/3;" "6#/%\"yG,(*&%$sinG\"\" \"%\"xGF(F(**F'F(\"\"#F(F)F(F+!\"\"F(**F'F(\"\"$F(F)F(F.F,F(" }{TEXT -1 28 " are obtained by adding the " }{TEXT 299 1 "y" }{TEXT -1 35 " v alues along the component curves " }{XPPEDIT 18 0 "y = sin*x+sin*2*x/2 ;" "6#/%\"yG,&*&%$sinG\"\"\"%\"xGF(F(**F'F(\"\"#F(F)F(F+!\"\"F(" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y = sin*3*x/3;" "6#/%\"yG**%$sinG\" \"\"\"\"$F'%\"xGF'F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 449 "plot([sin(x)+sin( 2*x)/2+sin(3*x)/3,sin(x)+sin(2*x)/2,sin(3*x)/3],\n x=0..4*Pi,y,color =[red,blue,green],thickness=2,\n labelfont=[HELVETICA,9],ytickmarks =6,font=[SYMBOL,9],\n xtickmarks=[0=`0`,evalf(Pi/3)=`p/3`,evalf(2* Pi/3)=`2p/3`,evalf(Pi)=`p`,\n evalf(4*Pi/3)=`4p/3`,evalf(5*Pi/3)=`5 p/3`,evalf(2*Pi)=`2p`,\n evalf(7*Pi/3)=`7p/3`,evalf(8*Pi/3)=`8p/3`, evalf(3*Pi)=`3p`,\n evalf(10*Pi/3)=`10p/3`,evalf(11*Pi/3)=`11p/3`,e valf(4*Pi)=`4p`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 757 333 333 {PLOTDATA 2 "6*-%'CURVESG6$7it7$$\"\"!F)F(7$$\"3qck?EzxZo!#>$\"3G&4#>! =`o/#!#=7$$\"3M\"HT_eb&p8F0$\"3kSngm*=\"\\SF07$$\"3;P>'yPLV0#F0$\"3)3[ (32qmjfF07$$\"3o#e#[q66RFF0$\"3sy')*R!fA]xF07$$\"3;4rAiTvIRF0$\"3&pq%* p:a\\/\"!#<7$$\"3>O;(R:(RA^F0$\"3?U@@`C\\]7FE7$$\"3Ya\\5T2Y#z&F0$\"3\" fH*\\*G#>M8FE7$$\"3ss#Q#GV_ikF0$\"3gZ>S-G)RR\"FE7$$\"3%=$\\!=7cvz'F0$ \"3&=nC,1/\\T\"FE7$$\"3)4fr`\"zeKrF0$\"3E'GE(fN#*H9FE7$$\"3+\\#Q*3(>wY 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ion " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 45 " defined a s an infinite sum of sine waves by " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "f(x) = sin*x+sin*2*x/2+sin*3*x/3+sin*4*x/4+` . . . ` ;" "6#/-%\"fG6#%\"xG,,*&%$sinG\"\"\"F'F+F+**F*F+\"\"#F+F'F+F-!\"\"F+** F*F+\"\"$F+F'F+F0F.F+**F*F+\"\"%F+F'F+F2F.F+%(~.~.~.~GF+" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = Sum(sin *k*x/k,k = 1 .. infinity);" "6#/%!G-%$SumG6$**%$sinG\"\"\"%\"kGF*%\"xG F*F+!\"\"/F+;F*%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 61 "In order to get a picture, we take a finite sum of 500 terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "f := x -> Sum(sin(k*x)/k,k=1..500):\n'f(x)'=f(x);\nplot(f(x),x= -2*Pi..4*Pi,color=brown,thickness=2,\n labelfont=[HELVETICA,9],ytic kmarks=3,\n font=[SYMBOL,9],xtickmarks=[evalf(-2*Pi)=`-2p`,\n ev alf(-Pi)=`-p`,0=`0`,evalf(Pi)=`p`,evalf(2*Pi)=`2p`,\n evalf(3*Pi)=` 3p`,evalf(4*Pi)=`4p`]);" 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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 58 "It l ooks as though the series converges for all values of " }{TEXT 273 1 " x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "In fact: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([0, x = 0] ,[(Pi-x)/2, 0 < x and x < 2*Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$ \"\"!/F'F,7$*&,&%#PiG\"\"\"F'!\"\"F2\"\"#F332F,F'2F'*&F4F2F1F2" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 6 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " } {XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "x=Pi/2" "6# /%\"xG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 20 ", the series gives: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(Pi/2) = sin(Pi/2)+1 /2;" "6#/-%\"fG6#*&%#PiG\"\"\"\"\"#!\"\",&-%$sinG6#*&F(F)F*F+F)*&F)F)F *F+F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi)+1/3" "6#,&-%$sinG6#%#Pi G\"\"\"*&F(F(\"\"$!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(3/2*Pi) +1/4" "6#,&-%$sinG6#*(\"\"$\"\"\"\"\"#!\"\"%#PiGF)F)*&F)F)\"\"%F+F)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(2*Pi)+1/5" "6#,&-%$sinG6#*&\"\"#\" \"\"%#PiGF)F)*&F)F)\"\"&!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(5 /2*Pi)+1/6" "6#,&-%$sinG6#*(\"\"&\"\"\"\"\"#!\"\"%#PiGF)F)*&F)F)\"\"'F +F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(3*Pi)+1/7" "6#,&-%$sinG6#*&\" \"$\"\"\"%#PiGF)F)*&F)F)\"\"(!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 " sin(7/2*Pi)+` . . . `" "6#,&-%$sinG6#*(\"\"(\"\"\"\"\"#!\"\"%#PiGF)F)% (~.~.~.~GF)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1-1/3+1/5-1/7+1/9-1/11+` . . . `" "6#/%!G,0\"\"\"F&* &F&F&\"\"$!\"\"F)*&F&F&\"\"&F)F&*&F&F&\"\"(F)F)*&F&F&\"\"*F)F&*&F&F&\" #6F)F)%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=arctan(1)" "6#/%!G-%'arctanG6#\"\"\"" }{XPPEDIT 18 0 "``=Pi/4" "6#/%!G*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x = Pi/4;" "6#/%\"x G*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 20 ", the series gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(Pi/4) = sin(Pi/4)+1/2;" " 6#/-%\"fG6#*&%#PiG\"\"\"\"\"%!\"\",&-%$sinG6#*&F(F)F*F+F)*&F)F)\"\"#F+ F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi/2)+1/3;" "6#,&-%$sinG6#*&%# PiG\"\"\"\"\"#!\"\"F)*&F)F)\"\"$F+F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 " sin(3/4*Pi)+1/4;" "6#,&-%$sinG6#*(\"\"$\"\"\"\"\"%!\"\"%#PiGF)F)*&F)F) F*F+F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi)+1/5;" "6#,&-%$sinG6#%# PiG\"\"\"*&F(F(\"\"&!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(5/4*P i)+1/6;" "6#,&-%$sinG6#*(\"\"&\"\"\"\"\"%!\"\"%#PiGF)F)*&F)F)\"\"'F+F) " }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(3*Pi/2)+1/7;" "6#,&-%$sinG6#*(\" \"$\"\"\"%#PiGF)\"\"#!\"\"F)*&F)F)\"\"(F,F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sin(7/4*Pi)+` . . . `;" "6#,&-%$sinG6#*(\"\"(\"\"\"\"\" %!\"\"%#PiGF)F)%(~.~.~.~GF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\" \"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(1-1/3+1/5-1/7+1/9-1/11+` . . . `) +``;" "6#,&-%!G6#,0\"\"\"F(*&F(F(\"\"$!\"\"F+*&F(F(\"\"&F+F(*&F(F(\"\" (F+F+*&F(F(\"\"*F+F(*&F(F(\"#6F+F+%(~.~.~.~GF(F(F%F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/sqrt(2);" "6#*&\"\"\"F$-%%sqrtG6#\"\"#!\"\"" } {XPPEDIT 18 0 "``(``(1-1/5+1/7-1/13+1/17-1/21+` . . . `)+``(1/3-1/7+1/ 11-1/15+1/19-1/23+` . . . `))" "6#-%!G6#,&-F$6#,0\"\"\"F**&F*F*\"\"&! \"\"F-*&F*F*\"\"(F-F**&F*F*\"#8F-F-*&F*F*\"#F-F**&F*F*\"#BF-F-F6F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi/8+1/sqrt(2)" "6#/%!G,&*&%#PiG\" \"\"\"\")!\"\"F(*&F(F(-%%sqrtG6#\"\"#F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(Sum((-1)^i/(4*i+1),i=0..infinity)+Sum((-1)^(i+1)/(4*i-1),i=1. .infinity))" "6#-%!G6#,&-%$SumG6$*&),$\"\"\"!\"\"%\"iGF-,&*&\"\"%F-F/F -F-F-F-F./F/;\"\"!%)infinityGF--F(6$*&),$F-F.,&F/F-F-F-F-,&*&F2F-F/F-F -F-F.F./F/;F-F6F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 91 "Maple can sum the two infinite series in \+ terms of the values of certain special functions. " }}{PARA 0 "" 0 "" {TEXT -1 45 "Evaluating numerically seems to suggest that " }{XPPEDIT 18 0 "f(Pi/4) = 3*Pi/8" "6#/-%\"fG6#*&%#PiG\"\"\"\"\"%!\"\"*(\"\"$F)F( F)\"\")F+" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "Pi/8+1/sqrt(2)*(Sum((-1)^i/(4*i+1) ,i = 0 .. infinity)+\n Sum((-1)^(i+1)/(4*i-1),i = 1 .. i nfinity));\nvalue(%);\nevalf(%,25);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&*&\"\")!\"\"%#PiG\"\"\"F(*&#F(\"\"#F(*&F+F*,&-%$SumG6$*&)F&%\"iGF(, &*&\"\"%F(F3F(F(F(F(F&/F3;\"\"!%)infinityGF(-F/6$*&)F&,&F3F(F(F(F(,&*& F6F(F3F(F(F(F&F&/F3;F(F:F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& *&\"\")!\"\"%#PiG\"\"\"F(*&#F(\"\"#F(*&F+F*,&*&#F(\"\"%F(-%)LerchPhiG6 %F&F(F/F(F(*&F/F(-F26%F&F(#\"\"$F0F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\":\"\\BWYsh4Xs4y6!#C" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "3*Pi/8;\nevalf(%,25);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"$\"\"\"\"\")!\"\"%#PiGF&F&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\":\"\\BWYsh4Xs4y6!#C" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6# %\"xG" }{TEXT -1 12 " defined by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = cos*x+cos*2*x/2+cos*3*x/3+cos*4*x/4+` . . . `; " "6#/-%\"gG6#%\"xG,,*&%$cosG\"\"\"F'F+F+**F*F+\"\"#F+F'F+F-!\"\"F+**F *F+\"\"$F+F'F+F0F.F+**F*F+\"\"%F+F'F+F2F.F+%(~.~.~.~GF+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = Sum(cos(k *x)/k,k = 1 .. infinity);" "6#/%!G-%$SumG6$*&-%$cosG6#*&%\"kG\"\"\"%\" xGF.F.F-!\"\"/F-;F.%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 61 "In order to get a picture, we take a finite sum of 500 te rms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "g := x -> Sum(cos(k*x)/k,k=1..500):\n'g(x)'=g(x);\np lot(g(x),x=-2*Pi..4*Pi,color=coral,thickness=2,\n labelfont=[HELVET ICA,9],ytickmarks=3,\n font=[SYMBOL,9],xtickmarks=[evalf(-2*Pi)=`-2 p`,\n evalf(-Pi)=`-p`,0=`0`,evalf(Pi)=`p`,evalf(2*Pi)=`2p`,\n ev alf(3*Pi)=`3p`,evalf(4*Pi)=`4p`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"gG6#%\"xG-%$SumG6$*&-%$cosG6#*&%\"kG\"\"\"F'F1F1F0!\"\"/F0;F1\"$+ &" }}{PARA 13 "" 1 "" {GLPLOT2D 555 211 211 {PLOTDATA 2 "6)-%'CURVESG6 #7jq7$$!3)****>YH&=$G'!#<$\"3!yk!)*HM#Gz'F*7$$!3#)\\$35rX.F'F*$\"3;TK) f\"35bVF*7$$!3w)p'RFh]diF*$\"3!yuR<%4>yOF*7$$!3rZ]yVlmWiF*$\"3!e>U(o;t vKF*7$$!3m'Rt,'p#=B'F*$\"37!f6Kzy())HF*7$$!3W&4]HzZh?'F*$\"3o6JH#ouGe# F*7$$!3A%zEdio/='F*$\"3[-'[\"orw$H#F*7$$!3!=>!G\"H5\"HhF*$\"3i^@'eQoT) =F*7$$!3[)eLo&>vxgF*$\"3%)G=a_Lz\"f\"F*7$$!3u#QSzGN](fF*$\"3SmX900Iz6F *7$$!3+xr/>'=B(eF*$\"3yN-%\\a7o\"*)!#=7$$!393`Gq@d$p&F*$\"3#[D.??Q4U&F in7$$!3JRM_@d#[^&F*$\"3)fJ!R!ovx!HFin7$$!3e@ICpvy7^F*$!39^;DR#o1y*!#>7 $$!3T'z;rW%33ZF*$!39yoEuKP%[$Fin7$$!3*4y&G3[I0VF*$!3dDxG`hFin7$$!3ufc$[;8_a$F*$!387]Ux)30s'Fin7$$!3YxM'f OD`9$F*$!3.W;[&3#QMpFin7$$!3C9QL7+sYFF*$!3!3&pz0(f]u'Fin7$$!3sk(*H6aqO BF*$!3=UJ466n!4'Fin7$$!33uL`FVcv>F*$!3&*Rnn.[2<^Fin7$$!3KqaC.]+p:F*$!3 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KNESSG6#\"\"#-%+AXESLABELSG6%Q\"x6\"Q!Fd^m-%%FONTG6$%*HELVETICAG\"\"*- %'COLOURG6&%$RGBG$\"*++++\"!\")$\")AR!)\\Fa_m$\"\"!Fe_m-Fg^m6$%'SYMBOL GFj^m-%*AXESTICKSG6$7)/$!+3`=$G'!\"*%$-2pG/$!+aEfTJF``m%#-pG/Fe_m%\"0G /$\"+aEfTJF``m%\"pG/$\"+3`=$G'F``m%#2pG/$\"+izxC%*F``m%#3pG/$\"+iqjc7F a_m%#4pG\"\"$-%%VIEWG6$;F^`mFeam%(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 5 "When " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 20 ", the series gives: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "g(Pi) = cos*Pi+cos*2*Pi/2+cos*3*Pi/3+cos*4*Pi/4+` . . . `;" "6#/-%\"gG6#%#PiG,,*&%$cosG\"\"\"F'F+F+**F*F+\"\"#F+F'F+F-!\"\"F+ **F*F+\"\"$F+F'F+F0F.F+**F*F+\"\"%F+F'F+F2F.F+%(~.~.~.~GF+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-1+1/2-1/3 +1/4-1/5+` . . . `" "6#/%!G,.\"\"\"!\"\"*&F&F&\"\"#F'F&*&F&F&\"\"$F'F' *&F&F&\"\"%F'F&*&F&F&\"\"&F'F'%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -ln*2;" "6#/%!G,$*&%#l nG\"\"\"\"\"#F(!\"\"" }{TEXT -1 1 " " }{TEXT 274 1 "~" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``-0" "6#,&%!G\"\"\"\"\"!!\"\"" }{TEXT -1 13 ".69314 71806. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Sum(cos(k*Pi)/k,k=1..infinity);\nvalue(%);\nevalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%$cosG6#*&%\"kG\"\"\"% #PiGF,F,F+!\"\"/F+;F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ -%#lnG6#\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+1=ZJp!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 20 ", the series gives: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(0) = 1+1/2+1/3+ 1/4+1/5+` . . . `;" "6#/-%\"gG6#\"\"!,.\"\"\"F)*&F)F)\"\"#!\"\"F)*&F)F )\"\"$F,F)*&F)F)\"\"%F,F)*&F)F)\"\"&F,F)%(~.~.~.~GF)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = infinity;" "6# /%!G%)infinityG" }{TEXT -1 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" }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the f unction " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 12 " defin ed by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = cos* x+sin*x+cos*2*x/(2^2)+sin*2*x/(2^2)+cos*3*x/(3^2)+sin*3*x/(3^2)+cos*4* x/(4^2)+sin*4*x/(4^2)+` . . . `;" "6#/-%\"hG6#%\"xG,4*&%$cosG\"\"\"F'F +F+*&%$sinGF+F'F+F+**F*F+\"\"#F+F'F+*$F/F/!\"\"F+**F-F+F/F+F'F+*$F/F/F 1F+**F*F+\"\"$F+F'F+*$F5F/F1F+**F-F+F5F+F'F+*$F5F/F1F+**F*F+\"\"%F+F'F +*$F:F/F1F+**F-F+F:F+F'F+*$F:F/F1F+%(~.~.~.~GF+" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = Sum((cos*k*x+s in*k*x)/(k^2),k = 1 .. infinity);" "6#/%!G-%$SumG6$*&,&*(%$cosG\"\"\"% \"kGF,%\"xGF,F,*(%$sinGF,F-F,F.F,F,F,*$F-\"\"#!\"\"/F-;F,%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 61 "In order to get a pictu re, we take a finite sum of 250 terms." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "h := x -> Sum((sin(k*x) +cos(k*x))/k^2,k=1..250):\n'h(x)'=h(x);\nplot(h(x),x=-2*Pi..4*Pi,color =COLOR(RGB,0,.5,.7),\n thickness=2,labelfont=[HELVETICA,9],ytickmar ks=3,\n font=[SYMBOL,9],xtickmarks=[evalf(-2*Pi)=`-2p`,\n evalf( -Pi)=`-p`,0=`0`,evalf(Pi)=`p`,evalf(2*Pi)=`2p`,\n evalf(3*Pi)=`3p`, evalf(4*Pi)=`4p`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG- %$SumG6$*&,&-%$sinG6#*&%\"kG\"\"\"F'F2F2-%$cosGF/F2F2F1!\"#/F1;F2\"$]# " }}{PARA 13 "" 1 "" {GLPLOT2D 570 224 224 {PLOTDATA 2 "6)-%'CURVESG6# 7[t7$$!3)****>YH&=$G'!#<$\"3=LN!G8U4k\"F*7$$!3A%zEdio/='F*$\"3\"Q_SF5g E#=F*7$$!3[)eLo&>vxgF*$\"3!zAP$*=5N'=F*7$$!3<&)pQAOREgF*$\"37Zh+$)*RU' =F*7$$!3u#QSzGN](fF*$\"3KjxM6W*f&=F*7$$!3K!y$\\`pnBfF*$\"34iTpd#H0%=F* 7$$!3+xr/>'=B(eF*$\"3^`J'*Qa4>=F*7$$!393`Gq@d$p&F*$\"38GGJF.m47$$!3&yz1P_ w=$RF*$!3qV&eSF:XR\"F^o7$$!3-H7FW[aQPF*$!3o^?'y3n`G$F^o7$$!3ufc$[;8_a$ 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o7$$\"3;VFU(**ehB\"F``m$\"3!zv>4UD@/)F^o7$$\"3Tdqz7&y7C\"F``m$\"3?-PNG (zMo*F^o7$$\"3mr8%4k\"F*-%*THICKNESSG6#\"\"#-% +AXESLABELSG6%Q\"x6\"Q!Fhhm-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGB G$\"\"!Fdim$\"\"&!\"\"$\"\"(Fgim-F[im6$%'SYMBOLGF^im-%*AXESTICKSG6$7)/ $!+3`=$G'!\"*%$-2pG/$!+aEfTJFdjm%#-pG/Fdim%\"0G/$\"+aEfTJFdjm%\"pG/$\" +3`=$G'Fdjm%#2pG/$\"+izxC%*Fdjm%#3pG/$\"+iqjc7!\")%#4pG\"\"$-%%VIEWG6$ ;FbjmFi[n%(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 5 "When \+ " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 20 ", the series \+ gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(0)=1+1/4 +1/9+1/16+1/25+` . . . `" "6#/-%\"hG6#\"\"!,.\"\"\"F)*&F)F)\"\"%!\"\"F )*&F)F)\"\"*F,F)*&F)F)\"#;F,F)*&F)F)\"#DF,F)%(~.~.~.~GF)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=Sum(1/k^2,k=1..infinity)" "6#/%!G-%$SumG6$*&\" \"\"F)*$%\"kG\"\"#!\"\"/F+;F)%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi^2/6" "6#/%!G*&%#PiG\"\"# \"\"'!\"\"" }{TEXT -1 1 " " }{TEXT 275 1 "~" }{TEXT -1 14 " 1.64493406 7. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(evalf(Pi^2/6,13),10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nS$\\k\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the f unction " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 12 " defin ed by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s(x) = cos* x+cos*2*x/(2^2)+cos*3*x/(3^3)+cos*4*x/(4^4)+cos*5*x/(5^5)+` . . . `;" "6#/-%\"sG6#%\"xG,.*&%$cosG\"\"\"F'F+F+**F*F+\"\"#F+F'F+*$F-F-!\"\"F+* *F*F+\"\"$F+F'F+*$F1F1F/F+**F*F+\"\"%F+F'F+*$F4F4F/F+**F*F+\"\"&F+F'F+ *$F7F7F/F+%(~.~.~.~GF+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = Sum(cos*k*x/(k^k),k = 1 .. infinity);" "6 #/%!G-%$SumG6$**%$cosG\"\"\"%\"kGF*%\"xGF*)F+F+!\"\"/F+;F*%)infinityG " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "Since the series converges rapidly, Maple does not take \+ too long to plot a graph with the upper limit of the sum left as \"inf inity\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "s := x -> Sum(cos(k*x)/k^k,k=1..infinity):\n's(x)'=s (x);\nplot(s(x),x=-2*Pi..4*Pi,color=COLOR(RGB,.5,.1,.95),thickness=2, \n thickness=2,labelfont=[HELVETICA,9],ytickmarks=3,\n font=[SYM BOL,9],xtickmarks=[evalf(-2*Pi)=`-2p`,\n evalf(-Pi)=`-p`,0=`0`,eval f(Pi)=`p`,evalf(2*Pi)=`2p`,\n evalf(3*Pi)=`3p`,evalf(4*Pi)=`4p`]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"sG6#%\"xG-%$SumG6$*&-%$cosG6# *&%\"kG\"\"\"F'F1F1)F0F0!\"\"/F0;F1%)infinityG" }}{PARA 13 "" 1 "" {GLPLOT2D 628 209 209 {PLOTDATA 2 "6)-%'CURVESG6#7]t7$$!+&H&=$G'!\"*$ \"+(*fG\"H\"F*7$$!+g>vxgF*$\"+\\AAT7F*7$$!+?'=B(eF*$\"+^7&))4\"F*7$$!+ &RXHy&F*$\"+nu785F*7$$!+q@d$p&F*$\"+q'f<;*!#57$$!+X*)>/cF*$\"+H%z[5)FA 7$$!+Ad#[^&F*$\"+.>4')pFA7$$!+&=;VT&F*$\"+()Q#\\o&FA7$$!+Xm!QJ&F*$\"+: )*=qVFA7$$!+5rH8_F*$\"+cF^rIFA7$$!+qvy7^F*$\"+$*e,9=FA7$$!+!z6;,&F*$\" +6vd,h!#67$$!+4gV5\\F*$!+Gx.i^F_o7$$!+G-E4[F*$!+zSoa:FA7$$!+[W33ZF*$!+ F`R*\\#FA7$$!+GYp1XF*$!+9qz&4%FA7$$!+4[I0VF*$!+TY[K`FA7$$!+n1f=TF*$!+l Z<'>'FA7$$!+Dl(=$RF*$!+'38?$oFA7$$!+Y[aQPF*$!+yFA7$$!+n`KXJF*$!+KdGMyFA7$$!+GSnXIF*$!+j?[@yFA7$$!+!pAg%HF*$!+Fah!y (FA7$$!+_8PYGF*$!+$>&\\5xFA7$$!+9+sYFF*$!+Ji64wFA7$$!+9FrTDF*$!+q9P*G( FA7$$!+8aqOBF*$!+xY\\*y'FA7$$!+r[8c@F*$!+I=IihFA7$$!+HVcv>F*$!+xU:?`FA 7$$!+nYGsai1%FA7$$!+0]+p:F*$!+<[=XCFA7$$!+Jy%pY\"F*$!+uX.'[\"F A7$$!+d1*[O\"F*$!+&)HfFA7$$!+6`Ui5F*$\"+\"H0o;$FA7$$!+NJuS'*FA$\"+N[TRWFA7$$!+gJBd')FA$ \"+E$>gs&FA7$$!*=BPn(F*$\"+&[(p)*pFA7$$!++>g!y'FA$\"+lJ>;\")FA7$$!+:1[ ()eFA$\"+mOgr\"*FA7$$!+K$fV*\\FA$\"+1B&R,\"F*7$$!*0Q75%F*$\"+m+^*4\"F* 7$$!+qGAx>FA$\"+4U'[C\"F*7$$\")J#zY\"F*$\"+Op-\"H\"F*7$$\"+v<6Y>FA$\"+ HhHY7F*7$$\"*CJau$F*$\"+e<%*H6F*7$$\"+#Q\\>z%FA$\"+PMTM5F*7$$\"+DvYQeF A$\"+*f@sA*FA7$$\"+qc)\\)oFA$\"+HGI))zFA7$$\"*\"Q]JzF*$\"+(or\"omFA7$$ \"+!p\\y&))FA$\"+(4WPY&FA7$$\"+vb>%y*FA$\"+V#3BD%FA7$$\"+YT0r5F*$\"+)R Xm0$FA7$$\"+M()oj6F*$\"+g$Ri*=FA7$$\"+fNKl7F*$\"+zH5CoF_o7$$\"+%QepO\" F*$!+*=$HSXF_o7$$\"+4Kfo9F*$!+%yjA]\"FA7$$\"+M!G-d\"F*$!+*=*3cCFA7$$\" +5))yjF*$!+J(>*p]<'FA 7$$\"+:.FhBF*$!+w`OgoFA7$$\"+uHtYDF*$!+ME=*H(FA7$$\"+Lc>KFF*$!+q?d\"f( FA7$$\"++*=A$GF*$!+O%e!)p(FA7$$\"+m@CKHF*$!+K%)psxFA7$$\"+LaEKIF*$!+o] j&F*$\"+*o^:$GFA7$$\"+SD%**G&F *$\"+/EPfSFA7$$\"+SCY&Q&F*$\"+)f2vI&FA7$$\"+NB)4[&F*$\"+C_Z^lFA7$$\"++ 4/(e&F*$\"+j@6%*yFA7$$\"+g%*4$p&F*$\"+yqPc\"*FA7$$\"+?!e\"*z&F*$\"+LDe H5F*7$$\"+&e;_!fF*$\"+#Q4r7\"F*7$$\"+]K\"e4'F*$\"+NUa\\7F*7$$\"+:*4kG' F*$\"+&\\t7H\"F*7$$\"+SD\"**['F*$\"+=#)fS7F*7$$\"+l^T$p'F*$\"+sgT*4\"F *7$$\"+!)ph&y'F*$\"+DU(3,\"F*7$$\"+!z=y(oF*$\"+\"[qW5*FA7$$\"++1-qpF*$ \"+bit3!)FA7$$\"+:CAiqF*$\"+Z#z%\\oFA7$$\"+]L-jrF*$\"+b``TbFA7$$\"+&GC QE(F*$\"+'[#RBUFA7$$\"+?_iktF*$\"+%3mW#HFA7$$\"+bhUluF*$\"+U^ap;FA7$$ \"+!)>FgvF*$\"+;qpmaF_o7$$\"++y6bwF*$!+\"z:X2&F_o7$$\"+?O'*\\xF*$!+=6O %[\"FA7$$\"+X%4[%yF*$!+v+3zBFA7$$\"+v=4V!)F*$!+$)>RyRFA7$$\"+0VPT#)F*$ !+1NJE_FA7$$\"+?;BN%)F*$!+aCwXhFA7$$\"+N*)3H')F*$!+2)3k\"oFA7$$\"+!eR? $))F*$!+(>'H.tFA7$$\"+D-*\\.*F*$!+&3(3:wFA7$$\"+?IsK\"*F*$!+G8>8xFA7$$ \"+?eXI#*F*$!+(o/8y(FA7$$\"+?')=G$*F*$!+!p,8#yFA7$$\"+:9#fU*F*$!+*G.V$ yFA7$$\"+:t'e_*F*$!+/(e+#yFA7$$\"+:K\"ei*F*$!+1EaxxFA7$$\"+:\"fds*F*$! +=*4bq(FA7$$\"+:]qD)*F*$!+E9*=g(FA7$$\"+hTR-5!\")$!+yyx!H(FA7$$\"+?y@A 5Fhbm$!+dhV6oFA7$$\"+/PVS5Fhbm$!+z(pf='FA7$$\"+)e\\'e5Fhbm$!+iw1V`FA7$ $\"+Zq_z5Fhbm$!+Oc!z0%FA7$$\"+1XS+6Fhbm$!+&4%Q&Q#FA7$$\"+I5u46Fhbm$!+Y 8)e]\"FA7$$\"+bv2>6Fhbm$!+8j^maF_o7$$\"+!39%G6Fhbm$\"+Z,`w[F_o7$$\"+/1 vP6Fhbm$\"+k&G\"*e\"FA7$$\"+'G0x9\"Fhbm$\"+%H=X#GFA7$$\"+o*fw:\"Fhbm$ \"+/)>X5%FA7$$\"+]Yhn6Fhbm$\"+Nul0aFA7$$\"+K$pv<\"Fhbm$\"+3%****p'FA7$ $\"+Ls4(=\"Fhbm$\"+BCK.zFA7$$\"+M^i'>\"Fhbm$\"+r)f:/*FA7$$\"+NI:17Fhbm $\"+:.R35F*7$$\"+O4o:7Fhbm$\"+K$3+5\"F*7$$\"+)**ehB\"Fhbm$\"+Tc`T7F*7$ $\"+fqjc7FhbmF+-%*THICKNESSG6#\"\"#-%+AXESLABELSG6%Q\"x6\"Q!F`im-%%FON TG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$\"\"&!\"\"$\"\"\"F]jm$\"#&*!\"# -Fcim6$%'SYMBOLGFfim-%*AXESTICKSG6$7)/$!+3`=$G'F*%$-2pG/$!+aEfTJF*%#-p G/\"\"!%\"0G/$\"+aEfTJF*%\"pG/$\"+3`=$G'F*%#2pG/$\"+izxC%*F*%#3pG/$\"+ iqjc7Fhbm%#4pG\"\"$-%%VIEWG6$;F[[nFb\\n%(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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 49 "Dot product of functions and orthogonal functions" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 14 "We define the " }{TEXT 261 13 "inner product" }{TEXT -1 4 " or \+ " }{TEXT 261 11 "dot product" }{TEXT -1 20 " of two functions f(" } {TEXT 270 1 "x" }{TEXT -1 8 ") and g(" }{TEXT 271 1 "x" }{TEXT -1 13 " ) defined on " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 31 " to be the real number value of" }}{PARA 256 "" 0 "" {TEXT -1 4 " f \+ " }{TEXT 262 1 "." }{TEXT -1 6 " g = " }{XPPEDIT 18 0 "Int(f(x)*g(x), x = a .. b);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%\"gG6#F*F+/F*;%\"aG% \"bG" }{TEXT -1 15 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 28 "If we think of a function f(" }{TEXT 272 1 "x" }{TEXT -1 26 ") defined on the interval " }{XPPEDIT 18 0 "[a , b];" "6#7$%\"aG%\"bG" }{TEXT -1 7 " as an " }{TEXT 261 27 "infinite \+ dimensional vector" }{TEXT -1 63 " whose components are the infinite s et of values at the points " }{TEXT 282 1 "x" }{TEXT -1 249 " in the i nterval, then the definition just given is a natural extension of the \+ standard notion of the dot product of two finite dimensional vectors, \+ where the integral gives an infinite sum of products of values, or com ponents, of the two functions." }}{PARA 0 "" 0 "" {TEXT -1 91 "Once we have a notion of dot product, we can develop the notions of length an d projection. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The length or " }{TEXT 261 4 "norm" }{TEXT -1 15 " of a f unction " }{XPPEDIT 18 0 "phi(x);" "6#-%$phiG6#%\"xG" }{TEXT -1 41 " w ith respect to the dot product (i) is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(abs(phi)) = sqrt(phi*`.`*phi);" "6#/-%$absG6 #-F%6#%$phiG-%%sqrtG6#*(F)\"\"\"%\".GF.F)F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 6 " = " } {XPPEDIT 18 0 "sqrt(Int(phi(x)^2,x = a .. b));" "6#-%%sqrtG6#-%$IntG6$ *$-%$phiG6#%\"xG\"\"#/F-;%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Note that the " }{TEXT 261 22 "root mean square value" }{TEXT -1 35 " of the function over th e interval " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 5 " i s: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(``(1/(b-a ))*Int(phi(x)^2,x = a .. b));" "6#-%%sqrtG6#*&-%!G6#*&\"\"\"F+,&%\"bGF +%\"aG!\"\"F/F+-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F7;F.F-F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/sqrt(b-a) ;" "6#/%!G*&\"\"\"F&-%%sqrtG6#,&%\"bGF&%\"aG!\"\"F-" }{TEXT -1 1 " " } {XPPEDIT 18 0 "abs(abs(phi))" "6#-%$absG6#-F$6#%$phiG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "For e xample, the norm of the sine function on the interval " }{XPPEDIT 18 0 "[-pi,pi]" "6#7$,$%#piG!\"\"F%" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(abs(sin))^2 = Int(sin^2*x,x = -P i .. Pi);" "6#/*$-%$absG6#-F&6#%$sinG\"\"#-%$IntG6$*&F*F+%\"xG\"\"\"/F 0;,$%#PiG!\"\"F5" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "Int((1-cos*2*x)/2,x = -Pi .. Pi);" "6#-%$IntG6$*&,& \"\"\"F(*(%$cosGF(\"\"#F(%\"xGF(!\"\"F(F+F-/F,;,$%#PiGF-F1" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "x/2-sin*2 *x/4;" "6#,&*&%\"xG\"\"\"\"\"#!\"\"F&**%$sinGF&F'F&F%F&\"\"%F(F(" } {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([pi, ``],[-pi, ``]);" "6#-%* PIECEWISEG6$7$%#piG%!G7$,$F'!\"\"F(" }}{PARA 256 "" 0 "" {TEXT -1 4 " \+ = " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a bs(abs(sin)) = sqrt(Pi);" "6#/-%$absG6#-F%6#%$sinG-%%sqrtG6#%#PiG" } {TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sqrt(Int(sin (x)^2,x=-Pi..Pi));\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%% sqrtG6#-%$IntG6$*$)-%$sinG6#%\"xG\"\"#\"\"\"/F/;,$%#PiG!\"\"F5F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#%#PiG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We can find the length of the projectio n of a vector " }{TEXT 278 1 "v" }{TEXT -1 30 " in the direction of a \+ vector " }{TEXT 279 1 "u" }{TEXT -1 30 " by taking the dot product of \+ " }{TEXT 280 1 "v" }{TEXT -1 40 " with a unit vector in the direction \+ of " }{TEXT 281 1 "u" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "A unit vector (function) in the \"direc tion\" of " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 26 " is given by the function " }{XPPEDIT 18 0 "s(x)=sin*x/sqrt(Pi) " "6#/-%\"sG6#%\"xG*(%$sinG\"\"\"F'F*-%%sqrtG6#%#PiG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " For example, the length of the " }{TEXT 261 10 "projection" }{TEXT -1 5 " of " }{XPPEDIT 18 0 "g(x) = exp(x);" "6#/-%\"gG6#%\"xG-%$expG6#F' " }{TEXT -1 21 " in the direction of " }{XPPEDIT 18 0 "sin*x" "6#*&%$s inG\"\"\"%\"xGF%" }{TEXT -1 47 ", when considered as functions on the \+ interval " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 5 ", is:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g*`.`*s = Int(exp(x)*s(x),x = -Pi .. Pi)" "6#/*( %\"gG\"\"\"%\".GF&%\"sGF&-%$IntG6$*&-%$expG6#%\"xGF&-F(6#F0F&/F0;,$%#P iG!\"\"F6" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= 1/sqrt(Pi)" "6#/%!G*&\"\"\"F&-%%sqrtG6#%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)*sin*x,x = -Pi .. Pi);" "6#- %$IntG6$*(-%$expG6#%\"xG\"\"\"%$sinGF+F*F+/F*;,$%#PiG!\"\"F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "The integration by parts form ula " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)=u*v-Int(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 19 " used twice gives: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)*sin*x, x);" "6#-%$IntG6$*(-%$expG6#%\"xG\"\"\"%$sinGF+F*F+F*" }{TEXT -1 9 " \+ ... " }{XPPEDIT 18 0 "PIECEWISE([u=exp(x),v=-cos*x],[du/dx=exp(x),d v/dx=sin*x])" "6#-%*PIECEWISEG6$7$/%\"uG-%$expG6#%\"xG/%\"vG,$*&%$cosG \"\"\"F,F2!\"\"7$/*&%#duGF2%#dxGF3-F*6#F,/*&%#dvGF2F8F3*&%$sinGF2F,F2 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -exp(x)*cos*x-Int((-cos*x)*exp(x), x);" "6#/%!G,&*(-%$expG6#%\"xG\"\"\"%$cosGF+F*F+!\"\"-%$IntG6$*&,$*&F, F+F*F+F-F+-F(6#F*F+F*F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-exp(x)*cos*x+Int(exp(x)*cos*x,x)" "6#/%!G,&* (-%$expG6#%\"xG\"\"\"%$cosGF+F*F+!\"\"-%$IntG6$*(-F(6#F*F+F,F+F*F+F*F+ " }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = exp(x), v = sin *x],[du/dx = exp(x), dv/dx = cos*x])" "6#-%*PIECEWISEG6$7$/%\"uG-%$exp G6#%\"xG/%\"vG*&%$sinG\"\"\"F,F17$/*&%#duGF1%#dxG!\"\"-F*6#F,/*&%#dvGF 1F6F7*&%$cosGF1F,F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-exp(x)*cos*x+exp (x)*sin*x - Int(exp(x)*sin*x,x)" "6#/%!G,(*(-%$expG6#%\"xG\"\"\"%$cosG F+F*F+!\"\"*(-F(6#F*F+%$sinGF+F*F+F+-%$IntG6$*(-F(6#F*F+F1F+F*F+F*F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Int(exp(x)*sin*x,x) = -exp(x)*co s*x+exp(x)*sin*x+c;" "6#/*&\"\"#\"\"\"-%$IntG6$*(-%$expG6#%\"xGF&%$sin GF&F.F&F.F&,(*(-F,6#F.F&%$cosGF&F.F&!\"\"*(-F,6#F.F&F/F&F.F&F&%\"cGF& " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)*sin*x,x) = exp(x)*(s in*x-cos*x)/2+c;" "6#/-%$IntG6$*(-%$expG6#%\"xG\"\"\"%$sinGF,F+F,F+,&* (-F)6#F+F,,&*&F-F,F+F,F,*&%$cosGF,F+F,!\"\"F,\"\"#F6F,%\"cGF," }{TEXT -1 4 "'. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g*`. `*s = Int(exp(x)*s(x),x = -Pi .. Pi)" "6#/*(%\"gG\"\"\"%\".GF&%\"sGF&- %$IntG6$*&-%$expG6#%\"xGF&-F(6#F0F&/F0;,$%#PiG!\"\"F6" }{XPPEDIT 18 0 "``= 1/sqrt(Pi)" "6#/%!G*&\"\"\"F&-%%sqrtG6#%#PiG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(exp(x)*sin*x,x = -Pi .. Pi);" "6#-%$IntG6$*(-%$e xpG6#%\"xG\"\"\"%$sinGF+F*F+/F*;,$%#PiG!\"\"F0" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(x)*(sin*x-co s*x)/(2*sqrt(Pi));" "6#/%!G*(-%$expG6#%\"xG\"\"\",&*&%$sinGF*F)F*F**&% $cosGF*F)F*!\"\"F**&\"\"#F*-%%sqrtG6#%#PiGF*F0" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([Pi,``],[-Pi,``])" "6#-%*PIECEWISEG6$7$%#PiG% !G7$,$F'!\"\"F(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = exp(Pi)*(sin*Pi-cos*Pi)/(2*sqrt(Pi))-exp(Pi)*(sin( -Pi)-cos(-Pi))/(2*sqrt(Pi));" "6#/%!G,&*(-%$expG6#%#PiG\"\"\",&*&%$sin GF+F*F+F+*&%$cosGF+F*F+!\"\"F+*&\"\"#F+-%%sqrtG6#F*F+F1F+*(-F(6#F*F+,& -F.6#,$F*F1F+-F06#,$F*F1F1F+*&F3F+-F56#F*F+F1F1" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(exp(Pi)-exp(-Pi)) /(2*sqrt(Pi))" "6#/%!G*&,&-%$expG6#%#PiG\"\"\"-F(6#,$F*!\"\"F/F+*&\"\" #F+-%%sqrtG6#F*F+F/" }{TEXT -1 2 " " }{TEXT 287 1 "~" }{TEXT -1 15 " \+ 6.515678448. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Int(exp(x)*sin(x),x=-Pi..Pi)/sqrt(Pi);\nsimplify (expand(value(%)),power);\nevalf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$sinGF*F,/F+;,$%#PiG !\"\"F2*$F2#F3\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\" #F&*&,&-%$expG6#%#PiGF&-F+6#,$F-!\"\"F1F&F-#F1F'F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[%yc^'!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Two functions " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 24 " defined on an interval " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG " }{TEXT -1 12 " are called " }{TEXT 261 10 "orthogonal" }{TEXT -1 21 " on this interval if " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*g(x),x = a .. b) = 0;" "6#/-%$IntG6$*&-%\"fG6#%\"xG\"\" \"-%\"gG6#F+F,/F+;%\"aG%\"bG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "For example, every pair of functions " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)" "6#-% \"gG6#%\"xG" }{TEXT -1 12 " defined on " }{XPPEDIT 18 0 "[-a,a]" "6#7$ ,$%\"aG!\"\"F%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 13 " is even and " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6 #%\"xG" }{TEXT -1 27 " is odd, are orthogonal on " }{XPPEDIT 18 0 "[-a ,a]" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "sin*k*x;" "6 #*(%$sinG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 18 " is orthogonal to " } {XPPEDIT 18 0 "cos*m*x;" "6#*(%$cosG\"\"\"%\"mGF%%\"xGF%" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi, Pi];" "6#7$,$%#PiG!\"\"F% " }{TEXT -1 27 " for all positive integers " }{TEXT 276 1 "k" }{TEXT -1 5 " and " }{TEXT 277 1 "m" }{TEXT -1 10 ", because " }{XPPEDIT 18 0 "sin*k*x;" "6#*(%$sinG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 12 " is odd an d " }{XPPEDIT 18 0 "cos*m*x;" "6#*(%$cosG\"\"\"%\"mGF%%\"xGF%" }{TEXT -1 10 " is even. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Ort hogonal sine functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "sin*k*x;" "6#*(%$sinG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 18 " is orthogonal to " } {XPPEDIT 18 0 "sin*m*x;" "6#*(%$sinG\"\"\"%\"mGF%%\"xGF%" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 27 " for all positive integers " }{TEXT 285 1 "k" }{TEXT -1 5 " and " }{TEXT 286 1 "m" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "k<>m" "6#0%\"kG%\"mG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "This fo llows since" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(si n*k*x*sin*m*x,x = -Pi .. Pi);" "6#-%$IntG6$*.%$sinG\"\"\"%\"kGF(%\"xGF (F'F(%\"mGF(F*F(/F*;,$%#PiG!\"\"F/" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos((k-m)*x)-cos((k+m)*x),x = \+ -Pi .. Pi);" "6#-%$IntG6$,&-%$cosG6#*&,&%\"kG\"\"\"%\"mG!\"\"F-%\"xGF- F--F(6#*&,&F,F-F.F-F-F0F-F//F0;,$%#PiGF/F8" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(cos((k-m)*x)-cos(( k+m)*x),x = 0 .. Pi);" "6#/%!G-%$IntG6$,&-%$cosG6#*&,&%\"kG\"\"\"%\"mG !\"\"F/%\"xGF/F/-F*6#*&,&F.F/F0F/F/F2F/F1/F2;\"\"!%#PiG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "( because the integrand is an even \+ function ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = s in((k-m)*x)/(k-m)-sin((k+m)*x)/(k+m);" "6#/%!G,&*&-%$sinG6#*&,&%\"kG\" \"\"%\"mG!\"\"F-%\"xGF-F-,&F,F-F.F/F/F-*&-F(6#*&,&F,F-F.F-F-F0F-F-,&F, F-F.F-F/F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([ Pi, ``],[0 , ` `])" "6#-%*PIECEWISEG6$7$%#PiG%!G7$\"\"!F(" }{TEXT -1 5 ", if " } {XPPEDIT 18 0 "k<>m" "6#0%\"kG%\"mG" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 15 "Note also that \+ " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "Int(sin^2*k*x,x = \+ -Pi .. Pi);" "6#-%$IntG6$*(%$sinG\"\"#%\"kG\"\"\"%\"xGF*/F+;,$%#PiG!\" \"F/" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Int((1-cos*2*k*x)/2,x = -Pi .. Pi);" "6#/%!G-%$IntG6$*&,&\" \"\"F***%$cosGF*\"\"#F*%\"kGF*%\"xGF*!\"\"F*F-F0/F/;,$%#PiGF0F4" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = x/2-sin*2*k*x/4;" "6#/%!G,&*&%\"xG\"\"\"\"\"#!\"\"F(*,%$sinGF(F)F(% \"kGF(F'F(\"\"%F*F*" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([pi, ` `],[-pi, ``]);" "6#-%*PIECEWISEG6$7$%#piG%!G7$,$F'!\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Pi;" "6 #/%!G%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Orthogonal cosine functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " } {XPPEDIT 18 0 "sin*k*x;" "6#*(%$sinG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 18 " is orthogonal to " }{XPPEDIT 18 0 "sin*m*x;" "6#*(%$sinG\"\"\"%\" mGF%%\"xGF%" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi,Pi] " "6#7$,$%#PiG!\"\"F%" }{TEXT -1 27 " for all positive integers " } {TEXT 283 1 "k" }{TEXT -1 5 " and " }{TEXT 284 1 "m" }{TEXT -1 6 " wit h " }{XPPEDIT 18 0 "k<>m" "6#0%\"kG%\"mG" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 18 "This follows since" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*k*x*cos*m*x,x = -Pi .. Pi);" "6#-%$IntG6 $*.%$cosG\"\"\"%\"kGF(%\"xGF(F'F(%\"mGF(F*F(/F*;,$%#PiG!\"\"F/" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(cos((k+m)*x)+cos((k-m)*x),x = -Pi .. Pi);" "6#-%$IntG6$,&-%$cosG6#* &,&%\"kG\"\"\"%\"mGF-F-%\"xGF-F--F(6#*&,&F,F-F.!\"\"F-F/F-F-/F/;,$%#Pi GF4F8" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(cos((k+m)*x)+cos((k-m)*x),x = 0 .. Pi);" "6#/%!G-%$IntG 6$,&-%$cosG6#*&,&%\"kG\"\"\"%\"mGF/F/%\"xGF/F/-F*6#*&,&F.F/F0!\"\"F/F1 F/F//F1;\"\"!%#PiG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "( \+ because the integrand is an even function ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sin(k+m)/(k+m)+sin(k-m)/(k-m);" "6 #/%!G,&*&-%$sinG6#,&%\"kG\"\"\"%\"mGF,F,,&F+F,F-F,!\"\"F,*&-F(6#,&F+F, F-F/F,,&F+F,F-F/F/F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([ Pi, \+ ``],[0 , ``])" "6#-%*PIECEWISEG6$7$%#PiG%!G7$\"\"!F(" }{TEXT -1 5 ", i f " }{XPPEDIT 18 0 "k<>m" "6#0%\"kG%\"mG" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 6 " = 0. " }}{PARA 0 "" 0 "" {TEXT -1 15 "Note al so that " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "Int(cos^2* k*x,x = -Pi .. Pi);" "6#-%$IntG6$*(%$cosG\"\"#%\"kG\"\"\"%\"xGF*/F+;,$ %#PiG!\"\"F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = Int((1+cos*2*k*x)/2,x = -Pi .. Pi);" "6#/%!G-%$Int G6$*&,&\"\"\"F***%$cosGF*\"\"#F*%\"kGF*%\"xGF*F*F*F-!\"\"/F/;,$%#PiGF0 F4" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x/2+sin*2*k*x/4;" "6#/%!G,&*&%\"xG\"\"\"\"\"#!\"\"F(*,%$sinGF( F)F(%\"kGF(F'F(\"\"%F*F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([ pi, ``],[-pi, ``]);" "6#-%*PIECEWISEG6$7$%#piG%!G7$,$F'!\"\"F(" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi;" "6#/%!G%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 34 "The constant function with value 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 70 "The constan t function with value 1 is orthogonal to all the functions " } {XPPEDIT 18 0 "sin*k*x;" "6#*(%$sinG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos*k*x;" "6#*(%$cosG\"\"\"%\"kGF%%\"xGF%" } {TEXT -1 8 ", where " }{TEXT 288 1 "k" }{TEXT -1 23 " is a positive in teger." }}{PARA 0 "" 0 "" {TEXT -1 20 "This follows because" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(sin*k*x,x = -Pi .. Pi) = 0;" "6#/-%$IntG6$*(%$sinG\"\"\"%\"kGF)%\"xGF)/F+;,$%#PiG!\"\"F/\"\" !" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "since " }{XPPEDIT 18 0 "sin*k*x;" "6#*(%$sinG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 24 " is an \+ odd function, and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Int(cos*k*x,x = -Pi .. Pi);" "6#-%$IntG6$*(%$cosG\"\"\"%\"kGF(%\"xGF(/ F*;,$%#PiG!\"\"F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "`` = sin*k*x/k;" "6#/%!G**%$sinG\"\"\"%\"kGF'%\"xGF' F(!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``], [-Pi , `` ])" "6#-%*PIECEWISEG6$7$%#PiG%!G7$,$F'!\"\"F(" }{TEXT -1 2 " " }} {PARA 256 "" 0 "" {TEXT -1 6 " = 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 10 "Conclusion" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "The set of functions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{ 1, sin*x, sin*2*x, sin*3*x,` . . . `,cos*x, cos*2*x, cos*3*x,` . . . ` \}" "6#<+\"\"\"*&%$sinGF$%\"xGF$*(F&F$\"\"#F$F'F$*(F&F$\"\"$F$F'F$%(~. ~.~.~G*&%$cosGF$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 0 "" 0 "" {TEXT -1 4 "are " } {TEXT 261 19 "mutually orthogonal" }{TEXT -1 18 ", on the interval " } {XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 62 ", that is, any pair of functions from this set are orthogonal." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 63 "Derivation of the formulas for th e Fourier series coefficients " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 74 "We now show how to find the co efficients in the Fourier series expansion " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = c+a[1]*cos*x+b[1]*sin*x+a[2]*cos *2*x+b[2]*sin*2*x+a[3]*cos*3*x+b[3]*sin*3*x+` . . . `;" "6#/-%\"fG6#% \"xG,2%\"cG\"\"\"*(&%\"aG6#F*F*%$cosGF*F'F*F**(&%\"bG6#F*F*%$sinGF*F'F *F***&F-6#\"\"#F*F/F*F8F*F'F*F***&F26#F8F*F4F*F8F*F'F*F***&F-6#\"\"$F* F/F*F?F*F'F*F***&F26#F?F*F4F*F?F*F'F*F*%(~.~.~.~GF*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = c+``;" "6#/-%\"fG6#%\"xG,&%\"cG\"\"\"%!GF*" } {XPPEDIT 18 0 "Sum(a[k]*cos*k*x+b[k]*sin*k*x,k = 1 .. infinity)" "6#-% $SumG6$,&**&%\"aG6#%\"kG\"\"\"%$cosGF,F+F,%\"xGF,F,**&%\"bG6#F+F,%$sin GF,F+F,F.F,F,/F+;F,%)infinityG" }{TEXT -1 14 " -------- (i)," }}{PARA 0 "" 0 "" {TEXT -1 14 "of a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "[-Pi ,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 41 ", assuming that such an expan sion exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "To find the coefficient " }{TEXT 289 1 "c" }{TEXT -1 113 " take the dot product of each side of equation (i) with 1. This amounts to \+ simply integrating both sides between " }{XPPEDIT 18 0 "-Pi" "6#,$%#Pi G!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 80 "The integral of the whole of the in finite sum is 0, leaving us with the equation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x),x = -Pi .. Pi) = Int(c,x = -P i .. Pi);" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;,$%#PiG!\"\"F.-F%6$%\"cG/F*;, $F.F/F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = c*x;" "6#/%!G*&%\"cG\"\"\"%\"xGF'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([Pi ,`` ],[-Pi , ``])" "6#-%*PIECEWISEG6$7$%# PiG%!G7$,$F'!\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = 2*c*Pi;" "6#/%!G*(\"\"#\"\"\"%\"cGF'%#PiGF'" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1/(2*Pi);" "6#/%\"cG*&\"\"\"F&*& \"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x=-Pi..P i)" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$%#PiG!\"\"F-" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 11 "___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "This is just th e " }{TEXT 261 13 "average value" }{TEXT -1 17 " of the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interva l " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "To find t he coefficient " }{XPPEDIT 18 0 "a[k]" "6#&%\"aG6#%\"kG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "cos*k*x;" "6#*(%$cosG\"\"\"%\"kGF%%\"xGF%" } {TEXT -1 71 " in the Fourier series, take the dot product of both side s of (i) with " }{XPPEDIT 18 0 "cos*k*x;" "6#*(%$cosG\"\"\"%\"kGF%%\"x GF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "The dot product of " }{XPPEDIT 18 0 "cos*k*x;" "6#*(%$cosG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 22 " with the left side is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x)*cos*k*x,x = -Pi .. Pi);" "6#-%$IntG6$**-%\"fG6 #%\"xG\"\"\"%$cosGF+%\"kGF+F*F+/F*;,$%#PiG!\"\"F1" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "cos*k*x;" "6#*(%$c osG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 49 " is orthogonal to all terms exc ept for the term " }{XPPEDIT 18 0 "a[k]*cos*k*x;" "6#**&%\"aG6#%\"kG \"\"\"%$cosGF(F'F(%\"xGF(" }{TEXT -1 21 ", the dot product of " } {XPPEDIT 18 0 "cos*k*x;" "6#*(%$cosG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 23 " with the right side is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[k]*Int(cos^2*k*x,x = -Pi .. Pi);" "6#*&&%\"aG6#%\"kG \"\"\"-%$IntG6$*(%$cosG\"\"#F'F(%\"xGF(/F/;,$%#PiG!\"\"F3F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = a[k]* Int(``(1/2+cos*2*k*x/2),x = -Pi .. Pi);" "6#/%!G*&&%\"aG6#%\"kG\"\"\"- %$IntG6$-F$6#,&*&F*F*\"\"#!\"\"F**,%$cosGF*F2F*F)F*%\"xGF*F2F3F*/F6;,$ %#PiGF3F:F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = a[k]*``(x/2+sin*2*k*x/(4*k));" "6#/%!G*&&%\"aG6#% \"kG\"\"\"-F$6#,&*&%\"xGF*\"\"#!\"\"F**,%$sinGF*F0F*F)F*F/F**&\"\"%F*F )F*F1F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi , ``],[-Pi , \+ ``])" "6#-%*PIECEWISEG6$7$%#PiG%!G7$,$F'!\"\"F(" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = a[k]*Pi;" "6#/%! G*&&%\"aG6#%\"kG\"\"\"%#PiGF*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "Note that this is " }{XPPEDIT 18 0 "a[k]" "6#&%\"aG6#%\"k G" }{TEXT -1 33 " times the square of the norm of " }{XPPEDIT 18 0 "co s*k*x;" "6#*(%$cosG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " We now have" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(f(x)*cos*k*x,x \+ = -Pi .. Pi) = a[k]*Pi;" "6#/-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF,%\" kGF,F+F,/F+;,$%#PiG!\"\"F2*&&%\"aG6#F.F,F2F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[k] = 1/Pi" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*k*x,x = -Pi .. Pi);" "6#-% $IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF+%\"kGF+F*F+/F*;,$%#PiG!\"\"F1" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 14 "_____ _________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "By taking the dot product of both sides of (i) wi th " }{XPPEDIT 18 0 "sin*k*x;" "6#*(%$sinG\"\"\"%\"kGF%%\"xGF%" } {TEXT -1 30 " we find in a similar way that" }}{PARA 256 "" 0 "" {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\"\"\"%$sinGF+%\"kGF+F*F+/F*;,$%#PiG!\"\"F1" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 265 14 "_____ _________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 69 "Summary of the integral formulas for the Fourier series c oefficients " }}{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 25 " defined on the interval " }{XPPEDIT 18 0 " [-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 51 " or, equivalently, a peri odic function with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#Pi GF%" }{TEXT -1 6 ", the " }{TEXT 261 20 "Fourier coefficients" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{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 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k ] = 1/Pi" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x)*cos*k*x,x = -Pi .. Pi);" "6#-%$IntG6$**-%\"fG6 #%\"xG\"\"\"%$cosGF+%\"kGF+F*F+/F*;,$%#PiG!\"\"F1" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "k=1,2,` . . . `" "6%/%\"kG\"\"\"\"\"#%(~.~.~.~G" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{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\"\"\"%$sinGF+%\"kGF+F*F+/F*;,$ %#PiG!\"\"F1" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "k=1,2,` . . . `" "6%/%\"kG\"\"\"\"\"#%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 14 "Fourier series" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c+a[1]*cos*x+b[1]*sin*x+a[2 ]*cos*2*x+b[2]*sin*2*x+a[3]*cos*3*x+b[3]*sin*3*x+` . . . `;" "6#,2%\"c G\"\"\"*(&%\"aG6#F%F%%$cosGF%%\"xGF%F%*(&%\"bG6#F%F%%$sinGF%F+F%F%**&F (6#\"\"#F%F*F%F4F%F+F%F%**&F.6#F4F%F0F%F4F%F+F%F%**&F(6#\"\"$F%F*F%F;F %F+F%F%**&F.6#F;F%F0F%F;F%F+F%F%%(~.~.~.~GF%" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = c+``;" "6#/%!G,& %\"cG\"\"\"F$F'" }{XPPEDIT 18 0 "Sum(a[k]*cos*k*x+b[k]*sin*k*x,k = 1 . . infinity)" "6#-%$SumG6$,&**&%\"aG6#%\"kG\"\"\"%$cosGF,F+F,%\"xGF,F,* *&%\"bG6#F+F,%$sinGF,F+F,F.F,F,/F+;F,%)infinityG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The funct ion " }{XPPEDIT 18 0 "h[1](x) = c+a[1]*cos*x+b[1]*sin*x;" "6#/-&%\"hG 6#\"\"\"6#%\"xG,(%\"cGF(*(&%\"aG6#F(F(%$cosGF(F*F(F(*(&%\"bG6#F(F(%$si nGF(F*F(F(" }{TEXT -1 28 " is a sine wave with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 20 ", and is called the " }{TEXT 261 16 "fundamental wave" }{TEXT -1 43 " of the Fourier series. Each pair of terms " }{XPPEDIT 18 0 "h[k](x)=a[k]*cos*k*x+b[k]*sin*k* x" "6#/-&%\"hG6#%\"kG6#%\"xG,&**&%\"aG6#F(\"\"\"%$cosGF0F(F0F*F0F0**&% \"bG6#F(F0%$sinGF0F(F0F*F0F0" }{TEXT -1 8 ", where " }{TEXT 294 1 "k" }{TEXT -1 47 " is greater than 1, is a sine wave with period " } {XPPEDIT 18 0 "2*Pi/k" "6#*(\"\"#\"\"\"%#PiGF%%\"kG!\"\"" }{TEXT -1 12 " called the " }{TEXT 293 1 "k" }{TEXT -1 4 " th " }{TEXT 261 8 "ha rmonic" }{TEXT -1 24 " of the Fourier series. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([0, -Pi <= x and x < 0],[1, 0 <= x and x < Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"!31,$%#PiG!\" \"F'2F'F,7$\"\"\"31F,F'2F'F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 52 " can be extended to a periodic function with \+ period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 10 " so that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([0, (n-1)*pi <= x and x < n *Pi and ` n is even`],[1, (n-1)*pi <= x and x < n*Pi and ` n is odd`]) ;" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"!331*&,&%\"nG\"\"\"F3!\"\"F3% #piGF3F'2F'*&F2F3%#PiGF3%+~n~is~evenG7$F3331*&,&F2F3F3F4F3F5F3F'2F'*&F 2F3F8F3%*~n~is~oddG" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 16 " or, more simply," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f (x) = PIECEWISE([0, -Pi <= x and x < 0],[1, 0 <= x and x < Pi]);" "6#/ -%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"!31,$%#PiG!\"\"F'2F'F,7$\"\"\"31F,F' 2F'F0" }{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 "2*Pi" "6#*&\" \"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 89 "We can draw the graph of this function by first defining its restriction to the interval " }{XPPEDIT 18 0 "-Pi< =x" "6#1,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 706 "f := x -> piecewise(x<0,0,1):\n'f(x)'=f( x);\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n'f_(x)'='f(x-2*Pi*flo or((x+Pi)/(2*Pi)))';\npp := evalf(Pi):\np1 := plot(f_(x),x=-2*Pi-1.2.. 4*Pi+1.2,discont=true,\n thickness=2,color=COLOR( RGB,.4,0,1)):\np2 := plot([[-2*pp,0],[-pp,1],[0,0],[pp,1],[2*pp,0],[3* pp,1],[4*pp,0]],\n style=point,symbol=circle,color=bl ack):\np3 := plot([[[-2*pp,1],[-pp,0],[0,1],[pp,0],[2*pp,1],[3*pp,0],[ 4*pp,1]]$3],\n style=point,symbol=[circle,diamond,cross],color=b lack):\nplots[display]([p1,p2,p3],ytickmarks=2,labelfont=[HELVETICA,9] ,\n font=[SYMBOL,9],xtickmarks=[-3*pp=`-3p`,-2*pp=`-2p`,\n -pp=` -p`,0=`0`,pp=`p`,2*pp=`2p`,3*pp=`3p`,4*pp=`4p`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 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*F*FL7$$\"3#R$H*3#y@A5!#;FL7$$\"3q4mW*e\\'e5F_^lFL7$$\"3c%4Kv]//5\"F_^ lFL7$$\"3AW<./1vP6F_^lFL7$$\"3CdyDK$pv<\"F_^lFL7$$\"3]'[Df$4o:7F_^lFL7 $$\"3+++#*eqjc7F_^lFL-%*THICKNESSG6#\"\"#-%*AXESTICKSG6$%(DEFAULTGFe_l -%+AXESLABELSG6$Q\"x6\"Q!F^`l-%&COLORG6&%$RGBG$\"\"%!\"\"FLF+-%%VIEWG6 $;$!+3`=$G'!\"*$\"+iqjc7!\")Fi_l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 42.000000 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 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/(2*Pi)" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x=-Pi..Pi)" "6#-%$IntG6$-% \"fG6#%\"xG/F);,$%#PiG!\"\"F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\" \"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi=1/2" "6#/%#PiG*& \"\"\"F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = 1/Pi" "6#/&% \"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f (x)*cos*k*x,x = -Pi .. Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF+ %\"kGF+F*F+/F*;,$%#PiG!\"\"F1" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*k*x,x = 0 .. Pi);" "6#-%$Int G6$*(%$cosG\"\"\"%\"kGF(%\"xGF(/F*;\"\"!%#PiG" }{TEXT -1 3 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G* &\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*k*x/k;" "6#** %$sinG\"\"\"%\"kGF%%\"xGF%F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIE CEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F( 7$\"\"!F(" }{TEXT -1 6 "= 0, " }{XPPEDIT 18 0 "k=1,2,3,` . . . `" "6& /%\"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "f := x -> p iecewise(x<0,0,1):\n'f(x)'=f(x);\ninterface(showassumed=0): k := 'k': \+ assume(k,integer):\n1/Pi*Int('f(x)'*cos(k*x),x=-Pi..Pi);\n``=value(%); \nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEW ISEG6$7$\"\"!2F'F,7$\"\"\"%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#*&%#k|irGF,F+F,F,/F+;,$ %#PiG!\"\"F5*$F5F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 14 ": The comma nd " }{TEXT 260 24 "interface(showassumed=0)" }{TEXT -1 75 " prevents \+ Maple from displaying assumed variables with a trailing tilde ~. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {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\"\"\"%$sinGF+%\"kGF+F*F+/F*;,$%#PiG!\"\"F1" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(sin*k*x,x = 0 .. Pi);" "6#-%$IntG6$*(%$sinG\"\"\"%\"kGF(%\"xGF(/F*; \"\"!%#PiG" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``(-cos*k*x/k);" "6#-%!G6#,$**%$cosG\"\"\"%\"kGF)%\" xGF)F*!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = 1/( k*Pi);" "6#/%!G*&\"\"\"F&*&%\"kGF&%#PiGF&!\"\"" }{TEXT -1 2 " " } {XPPEDIT 18 0 "``(-cos(k*Pi)+1);" "6#-%!G6#,&-%$cosG6#*&%\"kG\"\"\"%#P iGF,!\"\"F,F," }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "`` = (1-(-1)^k)/(k*Pi);" "6#/%!G*&,&\"\"\"F'),$F'!\"\"% \"kGF*F'*&F+F'%#PiGF'F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = PIECEWI SE([0, `if k is even`],[2/(k*Pi), `if k is odd`]);" "6#/%!G-%*PIECEWIS EG6$7$\"\"!%-if~k~is~evenG7$*&\"\"#\"\"\"*&%\"kGF.%#PiGF.!\"\"%,if~k~i s~oddG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "These Fourier coefficients can be found using Maple \+ as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 194 "f := x -> piecewise(x<0,0,1):\n'f(x)'=f(x);\ninter face(showassumed=0): k := 'k': assume(k,integer):\n1/Pi*Int('f(x)'*sin (k*x),x=-Pi..Pi);\n``=value(%);\nb := unapply(rhs(%),k):\n'b(k)'=b(k); \nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEW ISEG6$7$\"\"!2F'F,7$\"\"\"%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#*&%#k|irGF,F+F,F,/F+;,$ %#PiG!\"\"F5*$F5F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(%#PiG!\" \",&)F(%#k|irG\"\"\"F,F(F,F+F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"bG6#%\"kG,$*(%#PiG!\"\",&)F+%#k|irG\"\"\"F/F+F/F.F+F+" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The first few sine terms are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seq(b(k)*sin(k*x),k=1..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.,$*(\"\"#\"\"\"%#PiG!\"\"-%$sinG6#%\"xGF&F&\"\"!,$*& #F%\"\"$F&*&F'F(-F*6#,$*&F1F&F,F&F&F&F&F&F-,$*&#F%\"\"&F&*&F'F(-F*6#,$ *&F:F&F,F&F&F&F&F&F-,$*&#F%\"\"(F&*&F'F(-F*6#,$*&FCF&F,F&F&F&F&F&F-,$* &#F%\"\"*F&*&F'F(-F*6#,$*&FLF&F,F&F&F&F&F&F-,$*&#F%\"#6F&*&F'F(-F*6#,$ *&FUF&F,F&F&F&F&F&F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 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 4 " is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2+2/Pi*sin*x+2/(3*Pi)*sin*3*x+2/ (5*Pi)*sin*5*x+2/(7*Pi)*sin*7*x+` . . . `;" "6#,.*&\"\"\"F%\"\"#!\"\"F %**F&F%%#PiGF'%$sinGF%%\"xGF%F%*,F&F%*&\"\"$F%F)F%F'F*F%F.F%F+F%F%*,F& F%*&\"\"&F%F)F%F'F*F%F1F%F+F%F%*,F&F%*&\"\"(F%F)F%F'F*F%F4F%F+F%F%%(~. ~.~.~GF%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum((1-(-1)^k)/(k*Pi)*sin*k*Pi,k = 1 .. infinity);" "6 #-%$SumG6$*,,&\"\"\"F(),$F(!\"\"%\"kGF+F(*&F,F(%#PiGF(F+%$sinGF(F,F(F. F(/F,;F(%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 98 "We can define a function of two variable s to construct finite truncated Fourier series as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "FS : = (x,n) -> 1/2+sum((1-(-1)^k)/(k*Pi)*sin(k*x),k=1..n); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),& #\"\"\"\"\"#F/-%$sumG6$**,&F/F/)!\"\"%\"kGF7F/F8F7%#PiGF7-%$sinG6#*&F8 F/9$F/F//F8;F/9%F/F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 33 "The terms as far as the term in " }{XPPEDIT 18 0 "sin*7*x;" "6#*(%$sinG\"\"\"\"\"(F%%\"xGF%" }{TEXT -1 10 " are . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "FS(x,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,#\"\"\"\"\"#F%*(F&F %%#PiG!\"\"-%$sinG6#%\"xGF%F%*&#F&\"\"$F%*&F(F)-F+6#,$*&F0F%F-F%F%F%F% F%*&#F&\"\"&F%*&F(F)-F+6#,$*&F8F%F-F%F%F%F%F%*&#F&\"\"(F%*&F(F)-F+6#,$ *&F@F%F-F%F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 106 "The following picture compares the graphs of some trun cated Fourier series with the graph of the function " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "f := x -> piecewise(x<0 ,0,1):\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nFS := (x,n) -> 1/2 +sum((1-(-1)^k)/(k*Pi)*sin(k*x),k=1..n);\nplot([f_(x),FS(x,1),FS(x,3), FS(x,5),FS(x,7),FS(x,9)],x=-4..10,\n color=[black,red,blue,green,mag enta,coral],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&#\"\"\"\"\"#F/-%$sumG6$** ,&F/F/)!\"\"%\"kGF7F/F8F7%#PiGF7-%$sinG6#*&F8F/9$F/F//F8;F/9%F/F)F)F) " }}{PARA 13 "" 1 "" {GLPLOT2D 763 201 201 {PLOTDATA 2 "6*-%'CURVESG6% 7dq7$$!\"%\"\"!$\"\"\"F*7$$!3cmmmT)R[p$!#;KHMF0F+7$$!3sm\"zp'3naLF0F+7$$!33+]P9,-!G$F0F+7$$!3v;H2QZpUKF0F+ 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x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nFS := (x,n) -> \+ 1/2+sum((1-(-1)^k)/(k*Pi)*sin(k*x),k=1..n):\nplot([f_(x),FS(x,40)],x=- 4..4,numpoints=80,\n color=[black,COLOR(RGB,.8,0,1)],linestyle=[ 3,1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 491 251 251 {PLOTDATA 2 "6&-%'CUR VESG6%7]r7$$!\"%\"\"!$\"\"\"F*7$$!3An$\\fF\\S*Q!#$F0F+7 $$!3+&f2'ot)z<$F0F+7$$!3W75[e(Q^;$F0F+7$$!3%4s[Vy#F 0F[o7$$!3QpbkJx2zEF0F[o7$$!3gF#eE8Iwd#F0F[o7$$!3[]S#R*f]&[#F0F[o7$$!3_ X;`#[hfP#F0F[o7$$!3Uj]S7H;$G#F0F[o7$$!3kz.V9h@v@F0F[o7$$!3csF0F[o7$$!3qvg)3b-](=F0F[o7$$!3#>xTt)H%3x\"F0F[o7$$!3 3#R6H*>>v;F0F[o7$$!3_@xTt%>?d\"F0F[o7$$!3!=:sF[o7$$!3R$\\;`-\"4GPFcuF[o7$$!3W=&f2'o'zY#FcuF[o 7$$!3'4.\"[yZ!z$=FcuF[o7$$!3[VD?'pUy?\"FcuF[o7$$!3P(*Hj]l6G*)!#?F[o7$$ !3)*f0CRh!yx&FcvF[o7$$!3eA\"[ys&\\FEFcvF[o7$$\"3=[JWNo9G_!#@F+7$$\"3S \"QouzHDd#FcuF+7$$\"3KJBeE\"yF4&FcuF+7$$\"3Ne$ouRTZ-\"FdsF+7$$\"3d$[yA )\\?S:FdsF+7$$\"3Y#QIWb&fiDFdsF+7$$\"3Ve2')3j1-NFdsF+7$$\"3GC#eE\"e!)y 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h%=>zo(QTZ\"Fcu7$Fb`l$!3[RvOh%\\GX\"Fcu7$$\"3!f@xT4WBs$F0$!3Q![@% Qw'fA&Fcv7$$\"3g#R6H:)*fu$F0$\"3FYPOZRbNwFcv7$$\"3s![yA=Dyv$F0$\"3#G\\ S!eB)H>\"Fcu7$$\"3Ipbk6AlpPF0$\"3=!\\!fQ4!)\\8Fcu7$$\"3'yl75Cz9y$F0$\" 3^l.WDn)*37Fcu7$Fe`l$\"3L2*)o?v\"R5)Fcv7$$\"3lnuzV\")f?QF0$!3-L(>HQ!RC _Fcv7$$\"3I*=:s,!*y%QF0$!3NWB!)4g5Fcu-%&COLORG6&F`al$\"\")!\"\"F[oF+-Fbal6#F,-%+AXES LABELSG6$Q\"x6\"Q!F\\it-%%VIEWG6$;F(F[al%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "x=P i/2" "6#/%\"xG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 22 ", the series beco mes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2+2/Pi" "6 #,&*&\"\"\"F%\"\"#!\"\"F%*&F&F%%#PiGF'F%" }{XPPEDIT 18 0 "``(1-1/3+1/5 -1/7+1/9-1/11+` . . . `)" "6#-%!G6#,0\"\"\"F'*&F'F'\"\"$!\"\"F**&F'F' \"\"&F*F'*&F'F'\"\"(F*F**&F'F'\"\"*F*F'*&F'F'\"#6F*F*%(~.~.~.~GF'" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1 /2+2/Pi" "6#/%!G,&*&\"\"\"F'\"\"#!\"\"F'*&F(F'%#PiGF)F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*1=1/2+2/Pi" "6#/*&%'arctanG\"\"\"F&F&,&*&F&F &\"\"#!\"\"F&*&F)F&%#PiGF*F&" }{TEXT -1 1 " " }{TEXT 292 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi/4=1" "6#/*&%#PiG\"\"\"\"\"%!\"\"F&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 7 ": Wh en " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 20 ", or more gene rally " }{XPPEDIT 18 0 "x=k*Pi" "6#/%\"xG*&%\"kG\"\"\"%#PiGF'" }{TEXT -1 7 " where " }{TEXT 291 1 "k" }{TEXT -1 58 " is an integer, the Four ier series converges to the value " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$ \"\"#!\"\"" }{TEXT -1 79 ". Consequently it is cannot be regarded as a genuine expansion of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "In fact the Fou rier series converges to " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" } {TEXT -1 8 " where: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = PIECEWISE([1/2, x = -Pi],[-1, -Pi < x and x < 0],[1/2, x = 0 ],[1, 0 < x and x < Pi]);" "6#/-%\"FG6#%\"xG-%*PIECEWISEG6&7$*&\"\"\"F -\"\"#!\"\"/F',$%#PiGF/7$,$F-F/32,$F2F/F'2F'\"\"!7$*&F-F-F.F//F'F97$F- 32F9F'2F'F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "F(x);" "6#-%\"FG6#%\"xG " }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*& \"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 72 "Note that the integrals needed to calcul ate the Fourier coefficients of " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\" xG" }{TEXT -1 77 " are exactly the same as those used to calculate the Fourier coefficients of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 79 "The Fourier series obtai ned in this example is the Fourier series expansion of " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" }{TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x)=1/2+2/Pi*sin*x+2/(3*Pi)*sin*3 *x+2/(5*Pi)*sin*5*x+2/(7*Pi)*sin*7*x+` . . . `" "6#/-%\"FG6#%\"xG,.*& \"\"\"F*\"\"#!\"\"F***F+F*%#PiGF,%$sinGF*F'F*F**,F+F**&\"\"$F*F.F*F,F/ F*F2F*F'F*F**,F+F**&\"\"&F*F.F*F,F/F*F5F*F'F*F**,F+F**&\"\"(F*F.F*F,F/ F*F8F*F'F*F*%(~.~.~.~GF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "We can construct an " }{TEXT 295 9 "animation" }{TEXT -1 73 " to show successive truncated Fourier seri es approximations in sequence.\n" }{TEXT 261 4 "Note" }{TEXT -1 15 ": \+ The function " }{TEXT 0 2 "FS" }{TEXT -1 87 " has been changed to avoi d duplication of frames caused by the zero sine coefficients. " }} {PARA 0 "" 0 "" {TEXT -1 254 "To play the animation, click on the grap hic and use the controls in the context bar, use the animation menu, o r right click on the graphic to obtain a context-sensitive menu. You c an 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 0 300 "f := x -> piecewise(x<0,0,1):\nf_ := x -> f(x-2*Pi*f loor((x+Pi)/(2*Pi))):\nFS := (x,n) -> 1/2+sum(2/((2*k-1)*Pi)*sin((2*k- 1)*x),k=1..n): \nframes := [seq(plot([f_(x),FS(x,n)],x=-4..4,numpoints =80,\n color=[black,COLOR(RGB,.8,0,1)],linestyle=[3,1]),n=1..15) ]:\nplots[display](frames,insequence=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 649 245 245 {PLOTDATA 2 "6#-%(ANIMATEG617&-%'CURVESG6%7]r7$$ !\"%\"\"!$\"\"\"F.7$$!3An$\\fF\\S*Q!#$F4F/7$$!3+&f2'ot) z<$F4F/7$$!3W75[e(Q^;$F4F/7$$!3%4s[Vy#F4F_o7$$!3Qpb kJx2zEF4F_o7$$!3gF#eE8Iwd#F4F_o7$$!3[]S#R*f]&[#F4F_o7$$!3_X;`#[hfP#F4F _o7$$!3Uj]S7H;$G#F4F_o7$$!3kz.V9h@v@F4F_o7$$!3csF4F_o7$$!3qvg)3b-](=F4F_o7$$!3#>xTt)H%3x\"F4F_o7$$!33#R6H*>>v;F4 F_o7$$!3_@xTt%>?d\"F4F_o7$$!3!=:sF_o7$$!3R$\\;`-\"4GPFguF_o7$$!3W=&f2'o'zY#FguF_o7$$!3'4.\"[y Z!z$=FguF_o7$$!3[VD?'pUy?\"FguF_o7$$!3P(*Hj]l6G*)!#?F_o7$$!3)*f0CRh!yx &FgvF_o7$$!3eA\"[ys&\\FEFgvF_o7$$\"3=[JWNo9G_!#@F/7$$\"3S\"QouzHDd#Fgu F/7$$\"3KJBeE\"yF4&FguF/7$$\"3Ne$ouRTZ-\"FhsF/7$$\"3d$[yA)\\?S:FhsF/7$ $\"3Y#QIWb&fiDFhsF/7$$\"3Ve2')3j1-NFhsF/7$$\"3GC#eE\"e!)yXFhsF/7$$\"3& **[f2Ya=a&FhsF/7$$\"3SF,\"[=l'olFhsF/7$$\"3)oyAe1a9b(FhsF/7$$\"31y)*=: [rU')FhsF/7$$\"3m@j]Skm(e*FhsF/7$$\"3MxTt)\\>g1\"F4F/7$$\"3Cj]SK)oP;\" F4F/7$$\"3G()*=:C\"pq7F4F/7$$\"3+$ouz>ZHO\"F4F/7$$\"3?/Vajf,o9F4F/7$$ \"3/L10/s]p:F4F/7$$\"3G5[yU@$4n\"F4F/7$$\"3#\\Nou&R)>x\"F4F/7$$\"39n$ \\f6j!p=F4F/7$$\"3-!>:sD3S(>F4F/7$$\"3O9\"HjeGS2#F4F/7$$\"3!GR6HF&Hz@F 4F/7$$\"3;dX;tpeuAF4F/7$$\"3O$\\f29-*zBF4F/7$$\"3ErO\\>dx![#F4F/7$$\"3 cBR6pPS\"e#F4F/7$$\"3wD?'p:5lo#F4F/7$$\"3y*=:s(pJ$y#F4F/7$$\"3cuz../Y# )GF4F/7$$\"39?'pb_E>*HF4F/7$$\"39)3n$HG[TIF4F/7$$\"39cX;L\"R54$F4F/7$$ \"3))\\S#RG4P5$F4F/7$$\"3iVNoM%zj6$F4F/7$$\"3\\!Hj+^9F7$F4F/7$$\"3NPIW &e\\!HJF4F/7$$\"3z5H8Br@KJF4F/7$$\"3A%yA3m%QNJF4F/7$$\"3ldE^)>_&QJF4F/ 7$$\"3`JD?O(><9$F4F_o7$$\"3W>:sP+1nJF4F_o7$$\"3#p]S#R.S#>$F4F_o7$$\"3, 3')327%RC$F4F_o7$$\"334n$\\2#[&H$F4F_o7$$\"3cCR6\\UB!R$F4F_o7$$\"3?85[ )*pB\"\\$F4F_o7$$\"31IWN34Z\"f$F4F_o7$$\"3uQIWN+p)p$F4F_o7$$\"3WY(z.F1 Lz$F4F_o7$$\"3=K10kPZ-RF4F_o7$$\"\"%F.F_o-%'COLOURG6&%$RGBGF.F.F.-%*LI NESTYLEG6#\"\"$-F(6%7dp7$F,$\"3g2:!HKaz\")*Fhs7$F2$\"3w;9M]r(3N*Fhs7$F 6$\"3+,0*f;yX!*)Fhs7$F9$\"3Uh[7*)GDj$)Fhs7$F<$\"355id\\h(=y(Fhs7$F?$\" 3mcvx$)p>trFhs7$FB$\"3UOdNX;!ye'Fhs7$FE$\"3)e&z!*zQAmfFhs7$FK$\"3eeMWh gO8`Fhs7$Fdo$\"3eRwas7GfYFhs7$Fjo$\"3eVxyKeH!*RFhs7$F]p$\"3D/BS[0:5MFh s7$F`p$\"3oqG1$))yPx#Fhs7$Fcp$\"3I^%o]G,%f@Fhs7$Ffp$\"3_&4q/A=Fgu7$Fb q$!3Cj4xJ'4JQ#Fgu7$Feq$!3_skjN!odf&Fgu7$Fhq$!3-,/&zo&3O&)Fgu7$F[r$!3'H R(zmQ*Q2\"Fhs7$F^r$!3[TPL&fQ#R7Fhs7$$!3,#eE,\\m8Fhs7$$!3mO\\fZeV=:F4$!3kmP\\qGZd8Fhs7$Fgr$!3dg+'*QN]I8Fhs7$$! 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$!3-6!*>vW>3KFgu7$Fjgx$!3%y\"QQjKWBEFgu7$F``l$!3!eu¥?8#Fgu7$Fc`l$\"3'zy9n)R#\\d\"Fgu7$Febr$!3 q&R)f%*e#[H#Fgv7$Fjbr$!3k^%yUf9Fs\"Fgu7$Fg^hl$!3a\"yi**)G6j?Fgu7$F_cr$ !3IMZ-[8'R1#Fgu7$F__hl$!3t=^=t$Q0u\"Fgu7$Ff`l$!3G=$zDbFv:\"Fgu7$F^ix$ \"3.m*H'GvP,=Fgv7$Fjfo$\"3N0h`E\">4M\"Fgu7$Fb^jm$\"37zbGd*[rn\"Fgu7$Ff ix$\"3I]=(z?vdz\"Fgu7$Fj^jm$\"3-+\"pV\"fL*o\"Fgu7$Fi`l$\"3?MNEg`*zP\"F gu7$Fbgo$\"3N4_Q*)=EI:Fgv7$Fggo$!3#*Q/Iyd['3\"Fgu7$F`_am$!3ei#4e;SxX\" Fgu7$F\\ho$!3]7R**\\Y]z:Fgu7$Fh_am$!3,aOA_J_R9Fgu7$F\\al$!3+zB,9-Fo5Fg u7$Fdho$!3s(ye-A!o_bFaw7$Fiho$\"3%z`TuM:.P*Fgv7$Fj`jm$\"3IwQb9:Ii7Fgu7 $F^io$\"3LXeq)y9ZT\"Fgu7$Fbajm$\"3w\")4.=I=y8Fgu7$F_al$\"3+<_m_`ch6Fgu FdcmFjcmF\\dmFbdm" 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 2" }}{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 4 " by:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "-Pi<=x" "6#1,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 "``< Pi" "6#2%!G%#PiG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "In order to draw the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 ", we start by def ining " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the \+ interval " }{XPPEDIT 18 0 "-Pi<=x" "6#1,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 173 "f : = x -> x:\n'f(x)'=f(x);\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n' f_(x)'='f(x-2*Pi*floor((x+Pi)/(2*Pi)))';\nplot(f_(x),x=-2*Pi..4*Pi,col or=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6#%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\" fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floorG6#,$*(F.!\"\",&F'F,F/F,F,F/F5F, F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 521 194 194 {PLOTDATA 2 "6'-%'CURVE SG6#7^p7$$!3)****>YH&=$G'!#<$\"3)fq*)fiefD\"!#D7$$!3+xr/>'=B(eF*$\"3I# 4C8)om3T!#=7$$!3JRM_@d#[^&F*$\"3Ep9ccef$o(F37$$!3e@ICpvy7^F*$\"3kkl$zt (Rq6F*7$$!3T'z;rW%33ZF*$\"3#)*yi+'35v:F*7$$!3*4y&G3[I0VF*$\"3C0Q*))\\! )y(>F*7$$!3&yz1P_w=$RF*$\"3Q)ysMy38N#F*7$$!3ufc$[;8_a$F*$\"3]ERMU@(zt# F*7$$!3go&*Rl#p_M$F*$\"3k<+yTg\"z$HF*7$$!3YxM'fOD`9$F*$\"3y3h@T*fy8$F* 7$$!3]]Dm[*oG8$F*Fgn7$$!36B;OJDT?JF*Fjn7$$!3;'pgS6cz5$F*F]o7$$!3?p(fnp *\\&4$F*F`o7$$!3K:z:ioeqIF*Fco7$$!3UhgbFSnXIF*Ffo7$$!3k`BNe$[e*HF*Fio7 $$!3%ek[\"*oAg%HF*F\\p7$$!3EI7u]8PYGF*F_p7$$!3C9QL7+sYFF*Fbp7$$!3[*y;= r7F*F[q7$$!3KqaC.]+p:F*F^q7$ $!3?D1J2jxg6F*Faq7$$!3#\\1&ReJstwF3Fdq7$$!3Fjq7$$\"3K]VSe7VXPF3F^r7$$\"3/7#zJ$Q]JzF3Far7$$\"3m.cXO()oj6F *Fdr7$$\"3Yst,P!G-d\"F*Fgr7$$\"3aP@K*e\\t&>F*Fjr7$$\"3K2XV<.FhBF*F]s7$ $\"32AlCOc>KFF*F`s7$$\"3W/4bp@CKHF*Fcs7$$\"3#oGbGq)GKJF*Ffs7$$\"3KpLt5 dFXJF*$!3#p@Ykf4z8$F*7$$\"3!=X6'=FEeJF*$!3VM\"o&)eA\\7$F*7$$\"3GM&*[E( \\7<$F*$!3%>0!p!eN>6$F*7$$\"3y;wOMnB%=$F*$!3Wp>\"Gd[*)4$F*7$$\"3w\"yB, v5-@$F*$!3Y/e0dX(H2$F*7$$\"3IY*zew%=OKF*$!3%*R'*HT0+ZIF*7$$\"3#eF#R(zK \")G$F*$!3U5ty4D0&*HF*7$$\"3y0Y!*G33SLF*$!3X!)\\FyW5VHF*7$$\"3El#H>*o( RW$F*$!3(4K]_T3#RGF*7$$\"3uCR&\\&H(ya$F*$!3\\hcA_BJNFF*7$$\"3w%HA:hc(G PF*$!3[\"HdcpGWb#F*7$$\"3yk14o-k4RF*$!3W@*)3R]atBF*7$$\"3B\\x@x)e.I%F* $!3+P='*Hk#G)>F*7$$\"3\"*R%*GK%4Sq%F*$!3KY,*[(e*4kG'F*$\"3,0mraBhCK!#?7$$\"35\\lLp^T$p'F*$ \"3))G'p:i)H-TF37$$\"3eo<`=CAiqF*$\"3UB=_86P!z(F37$$\"3W3AmchUluF*$\"3 ?AE[\\3C#=\"F*7$$\"3`n3%=X4[%yF*$\"3I\"GhY9C;c\"F*7$$\"3Z\"=I'4VPT#)F* $\"3C&f]C+*=e>F*7$$\"3UwedV*)3H')F*$\"3=!H'ROO!fM#F*7$$\"3Us`o)eR?$))F *$\"3?'y0:Ga)[DF*7$$\"3Vo[zL-*\\.*F*$\"3?#G:m#\\!=v#F*7$$\"3#*eK/IIsK \"*F*$\"3osO'GsP&\\GF*7$$\"3=^;HEeXI#*F*$\"3%\\17\">0FZHF*7$$\"3UYeTCA Kz#*F*$\"3=giB9-;#=i _$*F*$\"3sbDU9lVpIF*7$$\"3oQUm?]0x$*F*$\"3W_Y[8(pQ4$F*7$$\"3%zG&>?;F*Q *F*$\"3r,d,8j31JF*7$$\"3UNjs>#)[,%*F*$\"3>\\na7HI=JF*7$$\"3n%Qd#>[q8%* F*$\"3W)zx?^>08$F*7$$\"3#RV)y=9#fU*F*$!3aQ2d&>\\/9$F*7$$\"3Afu$)=K\"ei *F*$!3C8<_&Rd0%HF*7$$\"3^%['))=]qD)*F*$!3'zosafl1u#F*7$$\"3#R$H*3#y@A5 !#;$!3JL)HaS#>WBF*7$$\"3q4mW*e\\'e5Fj_l$!3_vI*)>Z()z>F*7$$\"3c%4Kv]//5 \"Fj_l$!3!pAQ!RbKi:F*7$$\"3AW<./1vP6Fj_l$!3GI " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The Fourier series coefficients of " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 30 " can be determined as follows." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "c = 1/(2*Pi)" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x=-Pi..Pi)" "6#-%$IntG6$-% \"fG6#%\"xG/F);,$%#PiG!\"\"F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\" \"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x,x = -Pi .. Pi )" "6#-%$IntG6$%\"xG/F&;,$%#PiG!\"\"F*" }{TEXT -1 2 " " }}{PARA 256 " " 0 "" {TEXT -1 5 " = 0." }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "a[k] = 1/Pi" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*cos*k*x,x = -Pi .. Pi);" "6#-%$In tG6$**%\"xG\"\"\"%$cosGF(%\"kGF(F'F(/F';,$%#PiG!\"\"F." }{TEXT -1 2 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 7 " = 0, " }{XPPEDIT 18 0 "k=1,2,3,` \+ . . . `" "6&/%\"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "because " }{XPPEDIT 18 0 "x*cos*k*x;" "6#**%\"xG\"\"\"%$cosGF%%\"kGF%F$F%" } {TEXT -1 7 " is an " }{TEXT 261 3 "odd" }{TEXT -1 10 " function." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "interface(showassumed=0): k := 'k': assume(k,integer):\n1/Pi*Int( x*cos(k*x),x=-Pi..Pi);\nvalue(%);\nk := 'k':\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&%\"xG\"\"\"-%$cosG6#*&%#k|irGF)F(F)F)/F(;, $%#PiG!\"\"F2*$F2F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 14 ": The comma nd " }{TEXT 260 24 "interface(showassumed=0)" }{TEXT -1 75 " prevents \+ Maple from displaying assumed variables with a trailing tilde ~. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {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\"\"\"%$sinGF+%\"kGF+F*F+/F*;,$%#PiG!\"\"F1" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Int(x*sin*k*x,x = -Pi .. Pi);" "6#-%$IntG6$**%\"xG\"\"\"%$sinGF(%\"kGF (F'F(/F';,$%#PiG!\"\"F." }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sin*k*x,x = 0 .. Pi);" "6#-%$IntG 6$**%\"xG\"\"\"%$sinGF(%\"kGF(F'F(/F';\"\"!%#PiG" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 8 "because " }{XPPEDIT 18 0 "x*sin*k*x;" "6#* *%\"xG\"\"\"%$sinGF%%\"kGF%F$F%" }{TEXT -1 7 " is an " }{TEXT 261 4 "e ven" }{TEXT -1 10 " function." }}{PARA 0 "" 0 "" {TEXT -1 3 "Now" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sin*k*x,x) = Int( u*``(dv/dx),x);" "6#/-%$IntG6$**%\"xG\"\"\"%$sinGF)%\"kGF)F(F)F(-F%6$* &%\"uGF)-%!G6#*&%#dvGF)%#dxG!\"\"F)F(" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "u = x" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*k*x/k;" "6#/%\"vG,$**%$cosG\"\"\"%\"kGF(%\"xGF(F)!\"\"F +" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the int egration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v- Int(v*``(du/dx),x);" "6#/-%$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/Pi" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``(-x*cos*k*x/k);" "6#-%!G6#,$*,%\"xG\"\"\"%$cosGF)% \"kGF)F(F)F+!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi , ` `],[0 , ``]" "6#-%*PIECEWISEG6$7$%#PiG%!G7$\"\"!F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "- 2/Pi" "6#,$*&\"\"#\"\"\"%#PiG!\"\"F(" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(-cos*k*x/k,x = 0 .. Pi);" "6#-%$IntG6$,$**%$cosG \"\"\"%\"kGF)%\"xGF)F*!\"\"F,/F+;\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = -2*x*cos*k*x/(Pi*k)+2*sin*k*x/(Pi*k^2);" "6#/%!G,& *.\"\"#\"\"\"%\"xGF(%$cosGF(%\"kGF(F)F(*&%#PiGF(F+F(!\"\"F.*,F'F(%$sin GF(F+F(F)F(*&F-F(*$F+F'F(F.F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWI SE([Pi, ``],[``,``],[0, ``])" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\" !F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -2*Pi*cos*k*Pi/(k*Pi)+0;" "6#/% !G,&*.\"\"#\"\"\"%#PiGF(%$cosGF(%\"kGF(F)F(*&F+F(F)F(!\"\"F-\"\"!F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -2*cos*k*Pi/k;" "6#/%!G,$*,\"\"#\" \"\"%$cosGF(%\"kGF(%#PiGF(F*!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = ``(-2/k)*(-1)^k;" "6#/%!G*&-F$6#,$*&\"\"#\"\"\"%\"kG!\"\"F-F+),$F+F- F,F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (-1)^(k+1)*``(2/k);" "6#/%! G*&),$\"\"\"!\"\",&%\"kGF(F(F(F(-F$6#*&\"\"#F(F+F)F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = PIECEWISE([-2/k, `if k is even`],[2/k, `if k is od d`]);" "6#/%!G-%*PIECEWISEG6$7$,$*&\"\"#\"\"\"%\"kG!\"\"F.%-if~k~is~ev enG7$*&F+F,F-F.%,if~k~is~oddG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 64 "These Fourier coefficien ts can be found using Maple as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "interface(showassume d=0): k := 'k': assume(k,integer):\n1/Pi*Int(x*sin(k*x),x=-Pi..Pi);\nv alue(%);\nb := unapply(%,k):\n'b(k)'=b(k);\nk := 'k':\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&%\"xG\"\"\"-%$sinG6#*&%#k|irGF)F(F) F)/F(;,$%#PiG!\"\"F2*$F2F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\" #\"\"\")!\"\",&F&F&%#k|irGF&F&F*F(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"bG6#%\"kG,$*(\"\"#\"\"\")!\"\",&F+F+%#k|irGF+F+F/F-F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The first few sine terms are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "seq(b(k)*sin(k*x),k=1..6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(,$*&\"\"#\"\"\"-%$sinG6#%\"xGF&F&,$-F( 6#,$*&F%F&F*F&F&!\"\",$*&#F%\"\"$F&-F(6#,$*&F4F&F*F&F&F&F&,$*&#F&F%F&- F(6#,$*&\"\"%F&F*F&F&F&F0,$*&#F%\"\"&F&-F(6#,$*&FDF&F*F&F&F&F&,$*&#F&F 4F&-F(6#,$*&\"\"'F&F*F&F&F&F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "It follows that the Fourier series of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*sin*x-sin*2*x+2*sin*3*x/3-sin*4*x/2+2*sin*5*x/5-sin*6*x/3+` . . . `;" "6#,0*(\"\"#\"\"\"%$sinGF&%\"xGF&F&*(F'F&F%F&F(F&!\"\"*,F%F&F'F& \"\"$F&F(F&F,F*F&**F'F&\"\"%F&F(F&F%F*F**,F%F&F'F&\"\"&F&F(F&F0F*F&**F 'F&\"\"'F&F(F&F,F*F*%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum((-1)^(k+1)*``(2*sin(k*x)/k),k \+ = 1 .. infinity);" "6#/%!G-%$SumG6$*&),$\"\"\"!\"\",&%\"kGF+F+F+F+-F$6 #*(\"\"#F+-%$sinG6#*&F.F+%\"xGF+F+F.F,F+/F.;F+%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 " We can define a function of two variables to construct finite truncate d Fourier series as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "FS := (x,n) -> sum((-1)^(k+1 )*2*sin(k*x)/k,k=1..n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$ %\"xG%\"nG6\"6$%)operatorG%&arrowGF)-%$sumG6$,$**\"\"#\"\"\")!\"\",&% \"kGF3F3F3F3-%$sinG6#*&F7F39$F3F3F7F5F3/F7;F39%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(9*x);" "6#-%$sinG6#*&\"\"*\"\"\"%\" xGF(" }{TEXT -1 10 " are . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "FS(x,9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*&\"\"#\"\"\"-%$sinG6#%\"xGF&F&-F(6#,$*&F%F&F*F&F&!\" \"*&#F%\"\"$F&-F(6#,$*&F2F&F*F&F&F&F&*&#F&F%F&-F(6#,$*&\"\"%F&F*F&F&F& F/*&#F%\"\"&F&-F(6#,$*&F@F&F*F&F&F&F&*&#F&F2F&-F(6#,$*&\"\"'F&F*F&F&F& F/*&#F%\"\"(F&-F(6#,$*&FNF&F*F&F&F&F&*&#F&F=F&-F(6#,$*&\"\")F&F*F&F&F& F/*&#F%\"\"*F&-F(6#,$*&FfnF&F*F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture compares the \+ graphs of some truncated Fourier 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 0 221 "f_ \+ := x -> x-2*Pi*floor((x+Pi)/(2*Pi)):\nFS := (x,n) -> sum((-1)^(k+1)*2* sin(k*x)/k,k=1..n): \nplot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x ,5)],x=-4..10,\n color=[black,red,blue,green,magenta,coral],linestyl e=[3,1$5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 701 302 302 {PLOTDATA 2 "6*- %'CURVESG6%7^p7$$!\"%\"\"!$\"3C'ezrI&=$G#!#<7$$!3cmmmT)R[p$F-$\"3m>H^l aM)e#F-7$$!3'***\\iI23iNF-$\"3E'eald/6s#F-7$$!3PLLe>;KHMF-$\"3'GD'f(oj Q&GF-7$$!3sm\"zp'3naLF-$\"3^>/?SW^GHF-7$$!33+]P9,-!G$F-$\"3;'e/G>lJ+$F -7$$!3v;H2QZpUKF-$\"3[pm5p0\\SIF-7$$!3VL3xh$p`?$F-$\"3!Gv3a%f\"y2$F-7$ $!3)>z>On1n=$F-$\"3C%zfNjyk4$F-7$$!35](oa)R/oJF-$\"38O3r@89:JF-7$$!3;H KRTEreJF-$\"33djylEZCJF-7$$!3A3xJ(H\"Q\\JF-$\"3-y=')4S!Q8$F-7$$!3s(=UK &*\\+9$F-F`o7$$!3ymm;4'=28$F-Fco7$$!3)pmmmVF/)HF-Ffo7$$!3vmm;ki8IGF-Fi o7$$!3+LLeMD)4`#F-F\\p7$$!3)om;HtGOD#F-F_p7$$!3%)***\\i$\\Wm>F-Fbp7$$! 3Mmm\"H/R%p;F-Fep7$$!3+++D^cQt8F-Fhp7$$!3%HLL$[$e)o5F-F[q7$$!33kmmT`I1 !)!#=F^q7$$!3#Q*****\\ap')\\F`qFbq7$$!3q)*****\\noa>F`qFeq7$$\"3uH+++X :s'*!#>Fhq7$$\"3,NL$3d#e?OF`qF\\r7$$\"3Gqmm;FpvnF`qF_r7$$\"3?tmmm[[[%* F`qFbr7$$\"3?++v`xvb7F-Fer7$$\"3cmmmO^'4`\"F-Fhr7$$\"3u++v`7\"H$=F-F[s 7$$\"35,+Dh`V?@F-F^s7$$\"3Rnm;/mV?CF-Fas7$$\"3EnmT&RJfp#F-Fds7$$\"3B+] iq0^WGF-Fgs7$$\"3?LL$eu*3$*HF-Fjs7$$\"3Z$eRsus;.$F-F]t7$$\"3GLek[dDqIF -F`t7$$\"3'z&*[$\\sa*3$F-Fct7$$\"35$3_+vQ)3JF-Fft7$$\"3nXOS+X[=JF-Fit7 $$\"3B3_v]-8GJF-F\\u7$$\"3!3x16+wx8$F-F_u7$$\"3OL$e9v@u9$F-$!3(GD@dbjd 8$F-7$$\"3WL3FaxeCKF-$!3!Gv3Hb(feIF-7$$\"3_LL3dPv,LF-$!3s_i4]:V\")HF-7 $$\"3&pmm;A+hV$F-$!3G>H^&3&3ZGF-7$$\"3Q++D'oY/d$F-$!3%eeH4iQFr#F-7$$\" 3#RLL3TU1'QF-$!3J_iM'*GaACF-7$$\"3'********)HWgTF-$!3G'ezrJUF7#F-7$$\" 3w++]n$RPX%F-$!3Z&ez'RfWH=F-7$$\"37,+v$p=vt%F-$!37&eHMhmca\"F-7$$\"3k* ***\\_sg_]F-$!3f'ezY0y0B\"F-7$$\"3olmmO$GdL&F-$!3]0#H^qpXZ*F`q7$$\"3t, ++D0-QcF-$!31Xez@yk^kF`q7$$\"3qKL3x@%>\"fF-$!3QND'4IJCr$F`q7$$\"3*>++] *3T6iF-$!3iC%ez@Tu<(Fjq7$$\"3ImmT?w=$\\'F-$\"3u+3PKJ-+@F`q7$$\"3[++v)[ Dxy'F-$\"3[UTq:=SX]F`q7$$\"3'pmm;4!pvqF-$\"3F23([%y/DzF`q7$$\"3M***\\P MirP(F-$\"378/dOq(R4\"F-7$$\"3oNLL`f^nwF-$\"3Y\\P:Y1L%Q\"F-7$$\"31ML$e XWW'zF-$\"3!yu`'[\"f7o\"F-7$$\"3momTSU\"*e#)F-$\"3W#3PK$*Gd(>F-7$$\"3& =+++R,&H&)F-$\"3i:/#G3;jC#F-7$$\"3lnm;*zC'R))F-$\"3U\"3()>\\Rkb#F-7$$ \"3#RLLL(G+<\"*F-$\"3pZP:mv\"Q$GF-7$$\"3-o;a8V([E*F-$\"3y\"3ij+*o\")HF -7$$\"3L++v`du7%*F-$\"359/dY/cHJF-7$$\"3>Cc^$\\\"f@%*F-$\"3'z.Oj=1%QJF -7$$\"3$)\\7GLsVI%*F-$!3jAz2\"QLf8$F-7$$\"3Zvo/tHGR%*F-$!3+(H78k(3FJF- 7$$\"35,D\"GrG\"[%*F-$!3Orma,>C=JF-7$$\"3i]PM#>?eY*F-$!3'=U:?U]05$F-7$ $\"35+](=n6N[*F-$!3OsT[U*eG3$F-7$$\"3)3]P4j%*)=&*F-$!3er;U$)fZZIF-7$$ \"3m,++!fxUb*F-$!3\"3u%QF-$\"3OQ+,GtK(H\"F-7$F/$\"3%3DGq-d]vcF`q7$FC$\"3=iXG&y:(fFF `q7$Fco$!3WC)*)[-mZ<#Fjq7$Ffo$!3&fR=Fko$4KF`q7$Fio$!3YObK>c!*GhF`q7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" } {TEXT -1 7 ": When " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 20 ", or more generally " }{XPPEDIT 18 0 "x=k*Pi" "6#/%\"xG*&%\"kG\"\" \"%#PiGF'" }{TEXT -1 7 " where " }{TEXT 290 1 "k" }{TEXT -1 58 " is an integer, the Fourier series converges to the value " }{XPPEDIT 18 0 " 0;" "6#\"\"!" }{TEXT -1 79 ". Consequently it is cannot be regarded as a genuine expansion of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 87 ", but rather, it is an expansion of a function \+ obtained by modifying the definition of " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 44 " so that it has the value 0 at these points ." }}{PARA 0 "" 0 "" {TEXT -1 9 "That is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x)=PIECEWISE([0,x=-Pi],[x,-Pi " 0 "" {MPLTEXT 1 0 193 "f_ := x -> x-2*Pi*floor(1/2+x/(2*Pi)):\nFS : = (x,n) -> sum((-1)^(k+1)*2*sin(k*x)/k,k=1..n): \nplot([f_(x),FS(x,20) ],x=-4..4,numpoints=80,\n color=[black,COLOR(RGB,.8,0,1)],lines tyle=[3,1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 408 278 278 {PLOTDATA 2 "6& -%'CURVESG6%7bq7$$!\"%\"\"!$\"3C'ezrI&=$G#!#<7$$!3An$\\fF\\S*QF-$\"3+> -BJg8*Q#F-7$$!3Xa$ou6i=!QF-$\"3yJ7r*=B8[#F-7$$!3yJD?;!)=)p$F-$\"3Waq(4 H(*\\e#F-7$$!3Q-ip&pEQf$F-$\"3%QQ$[6'e$*o#F-7$$!379\"HjQh**[$F-$\"35s/ 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$pHOCF17$Fdcp$!3[*)>%)H1#*zBF17$F]ar$!39f*QLSv7O#F17$Fgal$!3#*=L(*QJaw BF17$F\\dp$!3MDv5x7o@CF17$Fadp$!3gJz**\\=htCF17$Fhc^l$!37[Qd5c]%\\#F17 $Ffdp$!3:J:*[*zc3DF17$F`d^l$!3ZD$[1pCR^#F17$F\\bl$!3H$R&yHaU4DF17$Fcep $!3Dt;_F0w9CF17$Fabl$!3IDZU0=^_AF1FhdmFaemFcemFiem" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 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 fu nction " }{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, -Pi <= x and x < 0],[0, 0 <= x and x < Pi/2],[1, Pi/2 <= x and x < \+ Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$,$\"\"\"!\"\"31,$%#PiGF.F'2F' \"\"!7$F431F4F'2F'*&F2F-\"\"#F.7$F-31*&F2F-F:F.F'2F'F2" }{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 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{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 defining " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "-Pi<=x " "6#1,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 200 "f := x -> piecewise(x<0,-1, x f(x-2*Pi*floor((x+Pi)/(2*Pi))): \n'f_(x)'='f(x-2*Pi*floor((x+Pi)/(2*Pi)))';\nplot(f_(x),x=-2*Pi..6*Pi, color=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$!\"\"2F'\"\"!7$F.2F',$*&\"\"#F,%#PiG \"\"\"F57$F5%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#% \"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floorG6#,$*(F.!\"\",&F'F,F/F, F,F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 591 173 173 {PLOTDATA 2 "6' 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F*F<7$$\"3wR4!3nwQN#F*F<7$$\"3%oa$pKfy$f#F*F<7$$\"3\\gl=&o@Lt#F*F<7$$ \"37u&zwVdG(GF*F<7$$\"3<\"3ERJDE%HF*F<7$$\"3y(es,>$R7IF*F<7$$\"3'3%eHG rFZIF*F<7$$\"3Q%4>k1h@3$F*F<7$$\"37@2[NIg*4$F*F<7$$\"3!zMUX+Xq6$F*F<7$ $\"3muRgtp[MJF*F<7$$\"3U,cmU*G>:$F*Fbo7$$\"3Gt\\U6X&*)R$F*Fbo7$$\"3;XV =!3!)fk$F*Fbo7$$\"3#GQN4[K!)=%F*Fbo7$$\"3*o)pn]X>/ZF*Fbo7$$\"3_za\\@bv U_F*Fbo7$$\"3EpmC8EKPdF*Fbo7$$\"3Tp,KC80/gF*Fbo7$$\"3epORN+yqiF*Fbo7$$ \"3AVy*e/'4)G'F*F+7$$\"3u1p1GFK'F*F+7$$\"3zm.TxS/ SjF*F+7$$\"3'\\r=%)4wYP'F*F+7$$\"38jqU>\"3$4kF*F+7$$\"3[fPWh@dykF*F+7$ $\"3qc/Y.i$ya'F*F+7$$\"3C]Q\\(Gkjo'F*F+7$$\"3#QCF:P#*[#oF*F+7$$\"3omfG !eqg1(F*F+7$$\"3b*oW!*y[sI(F*F+7$$\"3qYqzhysnvF*F+7$$\"3'QS\\X$p?GyF*F +7$$\"35aun2d-XyF*F+7$$\"3Y.b!3[W='yF*F<7$$\"3#GbLRDj'yyF*F<7$$\"3=-;1 F?[&*yF*F<7$$\"3!4q\"HzF*F<7$$\"3i*zt&>rvizF*F<7$$\"3#))*f37A.I!)F *F<7$$\"3C'>)f/tI(4)F*F<7$$\"3!HfA'*[d=B)F*F<7$$\"3a*)pkuwSm$)F*F<7$$ \"3\\(fVp))p'H')F*F<7$$\"3W0-C*4KH*))F*F<7$$\"3S@zKE=lZ\"*F*F<7$$\"3NP cT`:P-%*F*F<7$$\"3iK5>!)z/?%*F*F<7$$\"36Ek'pSCxV*F*Fbo7$$\"3i>=uL3Sb%* F*Fbo7$$\"3)[@<0ExIZ*F*Fbo7$$\"3l.!oS6I%3&*F*Fbo7$$\"3T#z=w'HyV&*F*Fbo 7$$\"3%*p.su')[9'*F*Fbo7$$\"3qX>#=Q%>&o*F*Fbo7$$\"3)*)4Dgz0m#)*F*Fbo7$ $\"3/a#G-@$Hjo7F`blF+7$$\"3)R)fHYBCv7F`blF+7$$ \"31(\\#fg<&=G\"F`blF+7$$\"3CBb=*eq]H\"F`blF+7$$\"3W\\&yxT*G38F`blF+7$ $\"3v\"pT+1PTL\"F`blF+7$$\"31M[I-Z)*f8F`blF+7$$\"3-%=7`u9NP\"F`blF+7$$ \"3'R`>$)yWqQ\"F`blF+7$$\"3#*3K#)4)4QR\"F`blF+7$$\"3!R)oKJ[d+9F`blF+7$ $\"3Q@(y?MdRS\"F`blF+7$$\"3()e0$G&)RtS\"F`blF+7$$\"3hxk?36.49F`blF+7$$ \"3O'R#ejBs59F`blF+7$$\"35:$e*=OT79F`blF+7$$\"3%QBMV([599F`blF<7$$\"3v ,RL`p;S9F`blF<7$$\"3mpNLK!HiY\"F`blF<7$$\"3O0-Ms7)G\\\"F`blF<7$$\"32To M7N`>:F`blF<7$$\"3e=$[`G\\F`\"F`blF<7$$\"3G'z\\$e]'fa\"F`blF<7$$\"3AN0 &[%Hd_:F`blF<7$$\"3)RF^8$3=f:F`blF<7$$\"3OV;guZ[i:F`blF<7$$\"3s7?&yr)y l:F`blF<7$$\"3M(>x%*oSuc\"F`blF<7$$\"35#Q-6m#4p:F`blF<7$$\"3)ocFFjW2d \"F`blF<7$$\"3[^FN/mRs:F`blFbo7$$\"3=#Hb+X%o'f\"F`blFbo7$$\"3)G$yv&Hs4 i\"F`blFbo7$$\"37>?()>bkw;F`blFbo7$$\"3Uq]?:.WE`z " 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/(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 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\" #F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-Pi/2);" "6#-%!G6# ,$*&%#PiG\"\"\"\"\"#!\"\"F+" }{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 94 "f := x -> piecewise(x<0,-1,x " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "a[k] = 1/Pi" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*k*x,x = -Pi .. Pi);" "6#-% $IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF+%\"kGF+F*F+/F*;,$%#PiG!\"\"F1" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G *&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((-1)*`.`*cos *k*x,x = -Pi .. 0)+1/Pi;" "6#,&-%$IntG6$*,,$\"\"\"!\"\"F)%\".GF)%$cosG F)%\"kGF)%\"xGF)/F.;,$%#PiGF*\"\"!F)*&F)F)F2F*F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(1*`.`*cos*k*x,x = Pi/2 .. Pi);" "6#-%$IntG6$*,\"\" \"F'%\".GF'%$cosGF'%\"kGF'%\"xGF'/F+;*&%#PiGF'\"\"#!\"\"F/" }{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 "``(-sin*k*x/k);" "6#-%!G6#,$**%$sinG\"\"\"%\"kGF)%\"xGF)F*!\"\"F ," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([0, ``],[``, ``],[-Pi, `` ]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$%#PiG!\"\"F(" }{XPPEDIT 18 0 " ``+1/Pi" "6#,&%!G\"\"\"*&F%F%%#PiG!\"\"F%" }{XPPEDIT 18 0 "``(sin* k*x/k);" "6#-%!G6#**%$sinG\"\"\"%\"kGF(%\"xGF(F)!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "PIECEWISE([Pi,``],[Pi/2,``])" "6#-%*PIECEWISEG6$7$%# PiG%!G7$*&F'\"\"\"\"\"#!\"\"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)/k)" "6#-%!G6#,&\"\"!\"\"\"*& -%$sinG6#*(%\"kGF(%#PiGF(\"\"#!\"\"F(F.F1F1" }{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 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "interface(showassume d=0): k := 'k': assume(k,integer):\nf := x -> piecewise(x<0,-1,x " 0 "" {MPLTEXT 1 0 1 ";" }}}{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 3 " , " }{XPPEDIT 18 0 "sin(k*Pi/2)" "6#-%$si nG6#*(%\"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'*&FIF'F*F'F+%(~.~.~.~G" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(a(k),k=1..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.,$*&\"\"\"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%*&\"#6F%F&F%F'F%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b[k] = 1/Pi;" "6#/&%\"bG6#%\"kG*&\"\"\"F)%#P iG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin*k*x,x = -Pi .. P i);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$sinGF+%\"kGF+F*F+/F*;,$%#PiG! \"\"F1" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = 1/Pi; " "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((-1 )*`.`*sin*k*x,x = -Pi .. 0)+1/Pi;" "6#,&-%$IntG6$*,,$\"\"\"!\"\"F)%\". GF)%$sinGF)%\"kGF)%\"xGF)/F.;,$%#PiGF*\"\"!F)*&F)F)F2F*F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1*`.`*sin*k*x,x = Pi/2 .. Pi);" "6#-%$IntG6$ *,\"\"\"F'%\".GF'%$sinGF'%\"kGF'%\"xGF'/F+;*&%#PiGF'\"\"#!\"\"F/" } {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 "``(cos*k*x/k);" "6#-%!G6#**%$cosG\"\"\"%\"kGF(%\"xGF(F )!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([0, ``],[``, ``],[-P i, ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$%#PiG!\"\"F(" } {XPPEDIT 18 0 " ``+1/Pi" "6#,&%!G\"\"\"*&F%F%%#PiG!\"\"F%" }{XPPEDIT 18 0 "``(-cos*k*x/k);" "6#-%!G6#,$**%$cosG\"\"\"%\"kGF)%\"xGF)F*!\"\"F ," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi,``],[Pi/2,``])" "6#-% *PIECEWISEG6$7$%#PiG%!G7$*&F'\"\"\"\"\"#!\"\"F(" }{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 "` `(1/k-cos*k*Pi/k)+1/Pi" "6#,&-%!G6#,&*&\"\"\"F)%\"kG!\"\"F)**%$cosGF)F *F)%#PiGF)F*F+F+F)*&F)F)F.F+F)" }{XPPEDIT 18 0 "``(-cos*k*Pi/k+cos(k*P i/2)/k)" "6#-%!G6#,&**%$cosG\"\"\"%\"kGF)%#PiGF)F*!\"\"F,*&-F(6#*(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 "``= 1/Pi" "6#/%!G*& \"\"\"F&%#PiG!\"\"" }{XPPEDIT 18 0 "``(1/k-2*cos*k*Pi/k+cos(k*Pi/2)/k) ;" "6#-%!G6#,(*&\"\"\"F(%\"kG!\"\"F(*,\"\"#F(%$cosGF(F)F(%#PiGF(F)F*F* *&-F-6#*(F)F(F.F(F,F*F(F)F*F(" }{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'%$cosGF'%\"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 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "interface(showassumed=0): k := 'k': assume(k,integer):\nf := x \+ -> piecewise(x<0,-1,x " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 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)-%$c osG6#*(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 coefficients " }{XPPEDIT 18 0 "b[k];" "6#&%\"bG6#%\"kG" } {TEXT -1 6 " are: " }}{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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(b(k),k=1..12); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6.,$*&\"\"$\"\"\"%#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&*&F.F&F'F&F(F(,$*(F%F&\"#6F(F' F(F&F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{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 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = -1/4-c os*x/Pi+3*sin*x/Pi-sin*2*x/Pi+cos*3*x/(3*Pi)+sin*3*x/Pi-cos*5*x/(5*Pi) +3*sin*5*x/(5*Pi)-sin*6*x/(3*Pi)+cos*7*x/(7*Pi)+3*sin*7*x/(7*Pi)+`. . \+ . `;" "6#/-%\"FG6#%\"xG,:*&\"\"\"F*\"\"%!\"\"F,*(%$cosGF*F'F*%#PiGF,F, **\"\"$F*%$sinGF*F'F*F/F,F***F2F*\"\"#F*F'F*F/F,F,**F.F*F1F*F'F**&F1F* F/F*F,F***F2F*F1F*F'F*F/F,F***F.F*\"\"&F*F'F**&F9F*F/F*F,F,*,F1F*F2F*F 9F*F'F**&F9F*F/F*F,F***F2F*\"\"'F*F'F**&F1F*F/F*F,F,**F.F*\"\"(F*F'F** &FAF*F/F*F,F**,F1F*F2F*FAF*F'F**&FAF*F/F*F,F*%'.~.~.~GF*" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/4;" "6#/%!G,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "Sum(-sin(Pi*k/2)*cos*k*x/(k*Pi)+(1-2*(-1)^k+cos(k*Pi/2) )*sin*k*x/(k*Pi),k = 1 .. infinity);" "6#-%$SumG6$,&*,-%$sinG6#*(%#PiG \"\"\"%\"kGF-\"\"#!\"\"F-%$cosGF-F.F-%\"xGF-*&F.F-F,F-F0F0*,,(F-F-*&F/ F-),$F-F0F.F-F0-F16#*(F.F-F,F-F/F0F-F-F)F-F.F-F2F-*&F.F-F,F-F0F-/F.;F- %)infinityG" }{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 0 106 "FS := (x, n) -> -1/4+\nsum(-sin(1/2*Pi*k)/(k*Pi)*cos(k*x)+(1-2*(-1)^k+cos(k*Pi/2 ))/(k*Pi)*sin(k*x),k=1..n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSG f*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&#\"\"\"\"\"%!\"\"-%$sumG6$,& **-%$sinG6#,$*(\"\"#F1%\"kGF/%#PiGF/F/F/F=F1F>F1-%$cosG6#*&F=F/9$F/F/F 1**,(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 term s as far as the terms in " }{XPPEDIT 18 0 "sin(7*x);" "6#-%$sinG6#*&\" \"(\"\"\"%\"xGF(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(7*x);" "6#-% $cosG6#*&\"\"(\"\"\"%\"xGF(" }{TEXT -1 10 " are . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "FS(x,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8#\"\"\"\"\"%!\"\"*&%#PiGF'-%$cosG6 #%\"xGF%F'*(\"\"$F%F)F'-%$sinGF,F%F%*&F)F'-F16#,$*&\"\"#F%F-F%F%F%F'*& #F%F/F%*&F)F'-F+6#,$*&F/F%F-F%F%F%F%F%*&F)F'-F1F " 0 "" {MPLTEXT 1 0 307 "f := x -> piecewi se(x<0,-1,x f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nFS := (x,n) -> -1/4+\nsum(-sin(1/2*Pi*k)/(k*Pi)*cos(k*x)+(1-2*(-1)^k+cos (k*Pi/2))/(k*Pi)*sin(k*x),k=1..n):\nplot([f_(x),seq(FS(x,i),i=1..7)],x =-4..4,\n color=[black,red,blue,green,magenta,coral,brown,navy],line style=[3,1$5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 617 360 360 {PLOTDATA 2 "6,-%'CURVESG6%7ip7$$!\"%\"\"!$\"\"\"F*7$$!3ommmmFiDQ!#1DBF0FL7$$!3kmmmw))yr@F0FL7$$!3;+++S(R#**>F0FL7$$!30 ++++@)f#=F0FL7$$!3-+++gi,f;F0FL7$$!3qmmm\"G&R2:F0FL7$$!3XLLLtK5F8F0FL7 $$!3eLLL$HsV<\"F0FL7$$!3+-++]&)4n**!#=FL7$$!37PLLL\\[%R)FaqFL7$$!3G)** ***\\&y!pmFaqFL7$$!3Y******\\O3E]FaqFL7$$!3NKLLL3z6LFaqFL7$$!3sLLL$)[` PFL7$$!3gSnmmmr[R!#?FL7$$\"3ATK3x19j:Fhr$F*F* 7$$\"31BK$3-)*\\2(FhrF\\s7$$\"3[?$ek`&oe7FdrF\\s7$$\"3n=L$3Fr)4=FdrF\\ s7$$\"3.:LeRFC7HFdrF\\s7$$\"3S6LL3Uh9SFdrF\\s7$$\"38/L$e9d$>iFdrF\\s7$ $\"3'oHLL3+TU)FdrF\\s7$$\"3AGL$efeLG\"FaqF\\s7$$\"3yELL$=2Vs\"FaqF\\s7 $$\"3Khmmm7+#\\#FaqF\\s7$$\"3)e*****\\`pfKFaqF\\s7$$\"36HLLLm&z\"\\Faq F\\s7$$\"3>(******z-6j'FaqF\\s7$$\"3q\"******4#32$)FaqF\\s7$$\"3r$**** *\\#y'G**FaqF\\s7$$\"3G******H%=H<\"F0F\\s7$$\"35mmm1>qM8F0F\\s7$$\"3( HLLL5r5U\"F0F\\s7$$\"3%)*******HSu]\"F0F\\s7$$\"3gm;HKh+F:F0F\\s7$$\"3 7LLek>dY:F0F\\s7$$\"3Gm\"H2)[Nc:F0F\\s7$$\"3m**\\(ozPhc\"F0F\\s7$$\"3M ;z%\\DH5d\"F0F+7$$\"3/L3-82#fd\"F0F+7$$\"3s\\P4r@\"3e\"F0F+7$$\"3Smm;H Oq&e\"F0F+7$$\"3p***\\PHN[i\"F0F+7$$\"3'HLL$ep'Rm\"F0F+7$$\"3Umm;\\%H& \\4N=F0F+7$$\"3#emm;@2h*>F0F+7$$\"3]*****\\c9W;#F 0F+7$$\"3Lmmmmd'*GBF0F+7$$\"3j*****\\iN7]#F0F+7$$\"3aLLLt>:nEF0F+7$$\" 35LLL.a#o$GF0F+7$$\"3#)****\\F'f4#HF0F+7$$\"3ammm^Q40IF0F+7$$\"3&)*** \\(e\"\\P/$F0F+7$$\"3;LL$eY/C3$F0F+7$$\"3#)**\\P>@t,JF0F+7$$\"3[mm\"Hx f57$F0F+7$$\"3!)*\\(o*fB28$F0F+7$$\"38L$ekU(QSJF0F+7$$\"3e\\P%)R$>_9$F 0FL7$$\"3Ym\"HKD^+:$F0FL7$$\"3M$e9m;$)[:$F0FL7$$\"3y******z]rfJF0FL7$$ \"3?LLLo$\"31rg0(**R lj)Fdr7$Fhn$!3cKG#HwB#enFdr7$F^o$!3srm(\\9_vJ#Faq7$Fao$!3;@'**R(e0?SFa q7$Fdo$!3#pVhU*>otcFaq7$Fgo$!3WsVg*fY'zsFaq7$Fjo$!3u\"RN\"R%*4w&)Faq7$ F]p$!3y58!H&Qtj)*Faq7$F`p$!3IuN)*)=cO4\"F07$Fcp$!3=d7J47ND\"F07$$!3_LLL$yP2D\"F0$!3olX@,Gec7F07$F\\ q$!35C-SC\"zPD\"F07$$!3!pmmT2Tb3\"F0$!3p[DS'GfJC\"F07$F_q$!3vRu*\\f2ZA \"F07$$!3cpmmT?\"F07$Fcq$!3cAHvy3Kt6F07$Ffq$!3aaP+ b))y!4\"F07$Fiq$!3AC`Xo>Y*))*Faq7$F\\r$!3.N(49vg^h)Faq7$F_r$!3Y0`@hx0' G(Faq7$Ffr$!3n6SW99y?dFaq7$Fft$!3sUTn+3X(*RFaq7$F\\u$!3wHa0KD`dCFaq7$F _u$!3%y,9b@'*e'zFdr7$Fbu$\"3ClPD4pG(p)Fdr7$Feu$\"3G0QYscv/CFaq7$Fhu$\" 31D]$yDe&fPFaq7$F[v$\"3o677XN-q]Faq7$F^v$\"3p'f&[%>R)RgFaq7$Fdv$\"3;SS K:SgGoFaq7$F\\x$\"3E3g&f?&o&4(Faq7$Fbx$\"3G%4SAR@SI(Faq7$Fex$\"3KkAvoW 2juFaq7$Fhx$\"3!y.oT'HB\\vFaq7$$\"3Omm\"zvX`(=F0$\"3=J8sBLNkvFaq7$$\"3 &GLLed*f:>F0$\"39&G![5z;jvFaq7$$\"3M***\\PR`e&>F0$\"3+JM*\\kyca(Faq7$F [y$\"3\"*QH^iQ\">^(Faq7$$\"3mKLL))3E!3#F0$\"3+kY-$y>!*Q(Faq7$F^y$\"3_? @+bX8'>(Faq7$Fay$\"3aow([0$QAmFaq7$Fdy$\"3kvRZ7>+edFaq7$Fgy$\"3w$QIxTI Sp%Faq7$Fjy$\"3#o+q>/!)=S$Faq7$F`z$\"3M!)ysw[\"H&>Faq7$F^\\l$\"3o!*4gN AH&4&Fdr7$Fd\\l$!3u/'\\q[:4B\"Faq7$Fg\\l$!3eG>*)>t#H#GFaq7$Fj\\l$!3Z9q a)Qc/^%Faq7$F]]l$!3e+@Q**ohsgFaq7$F`]l$!3O!pL(=#>jk(Faq-Fc]l6&Fe]l$\"* ++++\"!\")F\\sF\\s-Fg]l6#F,-F$6%7[p7$F($\"3#)owue)on&**Faq7$F.$\"336Bd L#=$>\"*Faq7$F2$\"3)3q>>5#4tyFaq7$$!3m****\\AHe)e$F0$\"3QJg7)GM&zpFaq7 $F5$\"3OoLB3!**H'fFaq7$$!3Mmmm1rQ$\"3jFU.C8%Q%)*Fdr7$FX$!3f$[@LIKr!GFdr7$F hn$!3!>\\:`*pa_:Faq7$F[o$!3u<(p2>`\"eGFaq7$F^o$!38i\"eUgfQ8%Faq7$$!3eL LL3(4Jv#F0$!3c2a!f+^HS&Faq7$Fao$!3%osoM,$3.mFaq7$$!3]LLL)>kOe#F0$!3#H- #=]#GYr(Faq7$Fdo$!3d&QZZV$pF()Faq7$Fgo$!3)3%Q\")4xlX5F07$Fjo$!3Ex]2!e: X:\"F07$$!3SLLL3V^&3#F0$!3Kk@0LYH(>\"F07$F]p$!3>sa#=YapA\"F07$$!37+++I `#f&>F0$!3N*)=(oZWqB\"F07$$!35+++?4h7>F0$!3CV9*3TOSC\"F07$$!33+++5lHp= F0$!3]L?$**)H.[7F07$F`p$!3'H)=$*Q5:\\7F07$$!3;+++S1C%y\"F0$!3'H2e&egiZ 7F07$$!3/+++!=*\\UzwsuOC\"F07$$!3#*******>xv+R=W#H=\"F07$Fip$! 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Consequently it is cannot be regarded as a genuine expansion of the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 87 ", but rather, it \+ is an expansion of a function obtained by modifying the definition of \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 35 " in an obvious way at these points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "f := x -> piecewise(x<0,-1,x f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nFS := (x,n) -> -1/4+\nsu m(-sin(1/2*Pi*k)/(k*Pi)*cos(k*x)+(1-2*(-1)^k+\n cos(k*Pi/ 2))/(k*Pi)*sin(k*x),k=1..n):\nplot([f_(x),FS(x,20)],x=-Pi..5*Pi,color= [black,magenta],linestyle=[3,1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 669 245 245 {PLOTDATA 2 "6&-%'CURVESG6%7es7$$!3*****4tk#fTJ!#<$!\"\"\"\"!7 $$!3i%RP<(fsIFF*F+7$$!3WZQ@uIBtBF*F+7$$!35VO$>#\\>r>F*F+7$$!3wJw!)*z\" \\m:F*F+7$$!3;Io(4;7P;\"F*F+7$$!3:`/)RwQG!z!#=F+7$$!3mB6Fv^?OSF@F+7$$! 3gt7\"4=mn.#F@F+7$$!3-`B9b'=Ft$!#?F+7$$\"3Z-!ydk)pB()FJ$F-F-7$$\"3!e$) 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45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can also constr uct an " }{TEXT 295 9 "animation" }{TEXT -1 72 " to show successive tr uncated Fourier series approximations in sequence." }}{PARA 0 "" 0 "" {TEXT -1 254 "To play the animation, click on the graphic and use the \+ controls in the context bar, use the animation menu, or right click on the graphic to obtain a context-sensitive menu. 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cco$\"3$\\p_9,;T!**FM7$Fhco$\"3MFk:E\"4Y%**FM7$F]do$\"3mkjNZRK05F.7$Ff [n$\"3Pq%>KzPB,\"F.7$$\"3wNd@haFB&)F.$\"3$ek/Eij:,\"F.7$Fjhw$\"3)>nU?2 pw+\"F.7$$\"3kW))H3sRa&)F.$\"3TC'R4'QX,5F.7$Fedo$\"37+#3=c)fV**FM7$Fed bl$\"3qqcGP3R\"))*FM7$Fbiw$\"3@us\">$>IW)*FM7$F]ebl$\"3>aXomkVV)*FM7$F \\_l$\"3O3>a9?d\"))*FM7$F_]r$\"3c2kC-'[^+\"F.7$Fd]r$\"3a')GrqVY>5F.7$F b^jm$\"3)[A;j`J,-\"F.7$Fi]r$\"3zq$RMZB`,\"F.7$Fj^jm$\"3hL7_A/%f+\"F.7$ F]eo$\"3K%\\iegA>%**FM7$F[fbl$\"3[&*>kbx!4$)*FM7$F_]u$\"3)f3\\5uQpv*FM 7$Fcfbl$\"3$zC&R(=vGu*FM7$Fa^r$\"3Y>%pJI%=(z*FM7$Fg]u$\"3qwi;%HYd+\"F. 7$F__l$\"3'HzK-(>CI5F.7$Fi^r$\"3_BKVw6#\\)**FM7$Fa\\n$\"35@NeqOM*e*FM7 $Fd__l$\"3/:u=yol>**FM7$Fa_r$\"3:9r[dB4S5F.7$$\"3)ze]**pdh4*F.$\"3GYtU LLI\\5F.7$F\\`_l$\"3Y4$QCb!)f0\"F.7$$\"3>L@\"[`P?6*F.$\"3![-?J.w&f5F.7 $Fa`_l$\"33CqI,Upf5F.7$Ff`_l$\"3;u5(fie)[5F.7$Fb_l$\"3'Rl]t+BV-\"F.7$F c][l$\"3a1#3WP:**[*FM7$Fi_r$\"3SM1>8vT'3*FM7$F[^[l$\"3id;)*)H)Qa'*FM7$ Fi\\n$\"3Q9ee')y^&4\"F.7$Fjh^m$\"3[i$*p0%z8:\"F.7$F[_u$\"3UMaaZ1Wt6F.7 $Fbi^m$\"3Mu$Q:%*f=9\"F.7$Fa`r$\"3[?n6,YEU5F.7$Fja]o$\"3Sm;f%Riyk*FM7$ Fjidl$\"3['eIZ)*p(*o)FM7$Fbb]o$\"3s>(\\Z:a'yo,V[FM7$Fbjdl$\"3'en#*zv2-J$FM7$Fb_dn$\"3@:I^Agz, " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT 266 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "Find the Fourier series for the funct ion " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 95 " defined a s follows, and compare the graphs of some truncated Fourier series wit h the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 " ." }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)=min(abs(x) ,Pi/2)" "6#/-%\"fG6#%\"xG-%$minG6$-%$absG6#F'*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-Pi<=x" "6#1,$%#PiG!\"\"%\"xG" } {XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 133 "f := x -> min(abs(x),Pi/2);\nf_ := x -> f(x- 2*Pi*floor((x+Pi)/(2*Pi)));\nplot(f_(x),x=-2*Pi..4*Pi,color=COLOR(RGB, .4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\" xG6\"6$%)operatorG%&arrowGF(-%$minG6$-%$absG6#9$,$%#PiG#\"\"\"\"\"#F(F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f_Gf*6#%\"xG6\"6$%)operatorG %&arrowGF(-%\"fG6#,&9$\"\"\"*(\"\"#F1%#PiGF1-%&floorG6#,$*&,&F0F1F4F1F 1F4!\"\"#F1F3F1F;F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 463 133 133 {PLOTDATA 2 "6'-%'CURVESG6#7[t7$$!3)****>YH&=$G'!#<$\"3)fq*)fiefD\"!#D 7$$!3A%zEdio/='F*$\"3.?z_9o;F5!#=7$$!3[)eLo&>vxgF*$\"3Yx*fM]LV0#F37$$! 3u#QSzGN](fF*$\"3)[.#R#>+:3$F37$$!3+xr/>'=B(eF*$\"3I#4C8)om3TF37$$!3eU im%RXHy&F*$\"3aOM8D\"*R-]F37$$!393`Gq@d$p&F*$\"3y!yU*o88'*eF37$$!3ttV! f%*)>/cF*$\"3-D@v7O')*y'F37$$!3JRM_@d#[^&F*$\"3Ep9ccef$o(F37$$!3aNLX$= ;VT&F*$\"3!p]isB\"p)o)F37$$!3*3B$QXm!QJ&F*$\"3Y`N'zh'y$p*F37$$!3BEJJ2r H8_F*$\"3+gk')*>)))p5F*7$$!3e@ICpvy7^F*$\"3kkl$zt(Rq6F*7$$!3Hl9r)y6;,& F*$\"3%47o%=Ndr7F*7$$!3+4*z\"3gV5\\F*$\"3Cx'***)H\\FP\"F*7$$!3q_$[wAg# 4[F*$\"3aL7`z]#RZ\"F*7$$!3T'z;rW%33ZF*$\"3c'*[zEjzq:F*7$$!3q)G,xi%p1XF 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3UwedV*)3H')F*Fgp7$$\"3Vo[zL-*\\.*F*Fgp7$$\"3#RV)y=9#fU*F*Fgp7$$\"3^%[ '))=]qD)*F*Fgp7$$\"3#R$H*3#y@A5!#;Fgp7$$\"3q4mW*e\\'e5FdilFgp7$$\"3#4) z'*=$)3p5FdilFgp7$$\"39_$*[[q_z5FdilFgp7$$\"3uP+D8ku%3\"FdilFgp7$$\"3M B2,yd'**3\"FdilFgp7$$\"39m5Rgad#4\"FdilFgp7$$\"3'*39xU^=&4\"FdilFgp7$$ \"3O!ehR)**['4\"FdilFgp7$$\"3w^<:D[z(4\"FdilFgp7$$\"3;B>Mm'*4*4\"FdilF gp7$$\"3c%4Kv]//5\"Fdil$\"3!pAQ!RbKi:F*7$$\"3(p+d;.T(46Fdil$\"3w-\"*y( Hg*o9F*7$$\"3R>>ybv2>6Fdil$\"3gy*Rl0&fv8F*7$$\"3!=$o!*zSTG6Fdil$\"3Wa3 H:)HAG\"F*7$$\"3AW<./1vP6Fdil$\"3GI\"Fdil$\"3gQXUMF>,gF37$$\" 3Fz&3].`h?\"Fdil$\"3-)zLFk-%[]F37$$\"3]'[Df$4o:7Fdil$\"3?vI/^Dh&4%F37$ $\"3$[6um'*>fA\"Fdil$\"39U-=w%4<2$F37$$\"3;VFU(**ehB\"Fdil$\"354uJ,k!y /#F37$$\"3mr8 " 0 "" {MPLTEXT 1 0 85 "f := x -> min(abs(x),Pi/2) :\n'f(x)'=f(x);\n1/(2*Pi)*Int('f(x)',x=-Pi..Pi);\n``=value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$minG6$,$*&\"\"#!\"\"% #PiG\"\"\"F0-%$absGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\" \"#F&,$-%$IntG6$-%\"fG6#%\"xG/F/;,$%#PiG!\"\"F3*$F3F4F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"$\"\"\"\"\")!\"\"%#PiGF(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The cosin e coefficients can be calculated as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "interface(s howassumed=0): k := 'k': assume(k,integer):\nf := x -> min(abs(x),Pi/2 ):\n'f(x)'=f(x);\n1/Pi*Int('f(x)'*cos(k*x),x=-Pi..Pi);\n``=value(%);\n a := unapply(rhs(%),k):\n'a(k)'=a(k);\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$minG6$,$*&\"\"#!\"\"%#PiG\"\"\"F0-%$ab sGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%\"fG6#%\"xG\"\" \"-%$cosG6#*&%#k|irGF,F+F,F,/F+;,$%#PiG!\"\"F5*$F5F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\"%#PiG!\"\",&F(F*-%$cosG6#,$*(F'F* F)F(%#k|irGF(F(F(F(F1!\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG 6#%\"kG,$**\"\"#\"\"\"%#PiG!\"\",&F+F--%$cosG6#,$*(F*F-F,F+%#k|irGF+F+ F+F+F4!\"#F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The first few cosine terms are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seq(a(k)*cos (k*x),k=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,,$*(\"\"#\"\"\"%#Pi G!\"\"-%$cosG6#%\"xGF&F(,$*&F'F(-F*6#,$*&F%F&F,F&F&F&F(,$*&#F%\"\"*F&* &F'F(-F*6#,$*&\"\"$F&F,F&F&F&F&F(\"\"!,$*&#F%\"#DF&*&F'F(-F*6#,$*&\"\" &F&F,F&F&F&F&F(,$*&#F&F6F&*&F'F(-F*6#,$*&\"\"'F&F,F&F&F&F&F(,$*&#F%\"# \\F&*&F'F(-F*6#,$*&\"\"(F&F,F&F&F&F&F(F=,$*&#F%\"#\")F&*&F'F(-F*6#,$*& F6F&F,F&F&F&F&F(,$*&#F&FAF&*&F'F(-F*6#,$*&\"#5F&F,F&F&F&F&F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The sine \+ coefficients are all 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "interface(showassumed=0): k := 'k' : assume(k,integer):\n1/Pi*Int('f(x)'*sin(k*x),x=-Pi..Pi);\n``=value(% );\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%\"fG6# %\"xG\"\"\"-%$sinG6#*&%#k|irGF,F+F,F,/F+;,$%#PiG!\"\"F5*$F5F6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"!" }}}{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 268 1 "c" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(a[k]*cos(k*x),k = 1 .. inf inity);" "6#-%$SumG6$*&&%\"aG6#%\"kG\"\"\"-%$cosG6#*&F*F+%\"xGF+F+/F*; F+%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= 3/8*Pi" "6#/%!G*(\"\"$\"\"\"\"\")!\"\"%#PiGF'" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(2*[(cos(Pi*k/2)-1)/(Pi*k^2)]*cos( k*x),k = 1 .. infinity);" "6#-%$SumG6$*(\"\"#\"\"\"7#*&,&-%$cosG6#*(%# PiGF(%\"kGF(F'!\"\"F(F(F2F(*&F0F(*$F1F'F(F2F(-F-6#*&F1F(%\"xGF(F(/F1;F (%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 " We can d efine a function of two variables to construct finite truncated Fourie r series as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "FS := (x,n)->3*Pi/8+\n sum(2*(cos(1/2*P i*k)-1)/(Pi*k^2)*cos(k*x),k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&*(\"\"$\"\"\"\"\")!\" \"%#PiGF0F0-%$sumG6$,$*,\"\"#F0,&-%$cosG6#,$*(F9F2%\"kGF0F3F0F0F0F0F2F 0F3F2F@!\"#-F<6#*&F@F09$F0F0F0/F@;F09%F0F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The terms as far as the t erm in " }{XPPEDIT 18 0 "cos(10*x)" "6#-%$cosG6#*&\"#5\"\"\"%\"xGF(" } {TEXT -1 10 " are . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "FS(x,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*(\"\"$\"\"\"\"\")!\"\"%#PiGF&F&*(\"\"#F&F)F(-%$cosG6 #%\"xGF&F(*&F)F(-F-6#,$*&F+F&F/F&F&F&F(*&#F+\"\"*F&*&F)F(-F-6#,$*&F%F& F/F&F&F&F&F(*&#F+\"#DF&*&F)F(-F-6#,$*&\"\"&F&F/F&F&F&F&F(*&#F&F7F&*&F) F(-F-6#,$*&\"\"'F&F/F&F&F&F&F(*&#F+\"#\\F&*&F)F(-F-6#,$*&\"\"(F&F/F&F& F&F&F(*&#F+\"#\")F&*&F)F(-F-6#,$*&F7F&F/F&F&F&F&F(*&#F&F?F&*&F)F(-F-6# ,$*&\"#5F&F/F&F&F&F&F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Here are a few graphs of truncated Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 274 "f := x -> min(abs(x),Pi/2):\nf_ := x -> f(x-2*Pi*floor((x+Pi)/( 2*Pi))):\nFS := (x,n)->3*Pi/8+sum(2*(cos(1/2*Pi*k)-1)/(Pi*k^2)*cos(k*x ),k=1..n):\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,green,magenta,coral],linestyle=[3,1 $5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 570 199 199 {PLOTDATA 2 "6*-%'CURV ESG6%7[t7$$!3)****>YH&=$G'!#<$\"3)fq*)fiefD\"!#D7$$!3A%zEdio/='F*$\"3. ?z_9o;F5!#=7$$!3[)eLo&>vxgF*$\"3Yx*fM]LV0#F37$$!3u#QSzGN](fF*$\"3)[.#R #>+:3$F37$$!3+xr/>'=B(eF*$\"3I#4C8)om3TF37$$!3eUim%RXHy&F*$\"3aOM8D\"* R-]F37$$!393`Gq@d$p&F*$\"3y!yU*o88'*eF37$$!3ttV!f%*)>/cF*$\"3-D@v7O')* y'F37$$!3JRM_@d#[^&F*$\"3Ep9ccef$o(F37$$!3aNLX$=;VT&F*$\"3!p]isB\"p)o) F37$$!3*3B$QXm!QJ&F*$\"3Y`N'zh'y$p*F37$$!3BEJJ2rH8_F*$\"3+gk')*>)))p5F *7$$!3e@ICpvy7^F*$\"3kkl$zt(Rq6F*7$$!3Hl9r)y6;,&F*$\"3%47o%=Ndr7F*7$$! 3+4*z\"3gV5\\F*$\"3Cx'***)H\\FP\"F*7$$!3q_$[wAg#4[F*$\"3aL7`z]#RZ\"F*7 $$!3T'z;rW%33ZF*$\"3c'*[zEjzq:F*7$$!3q)G,xi%p1XF*Fgp7$$!3*4y&G3[I0VF*F gp7$$!3&yz1P_w=$RF*Fgp7$$!3ufc$[;8_a$F*Fgp7$$!3YxM'fOD`9$F*Fgp7$$!3C9Q L7+sYFF*Fgp7$$!3sk(*H6aqOBF*Fgp7$$!33uL`FVcv>F*Fgp7$$!3?A%*QlYGsF3$\"3C6Z-\\GAx>F37$$!3qA0)\\f_@:*!#>$\"3qA0)\\f_@:*Fbw7$$\"3+ngG+L #zY\"FbwFfw7$$\"3gUD7i?XY5F3Fiw7$$\"3_yk@%z6h%>F3F\\x7$$\"3U9/JE:xXGF3 F_x7$$\"3K]VSe7VXPF3Fbx7$$\"3ul!)4-%\\>z%F3Fex7$$\"3=\"y\"zXvYQeF3Fhx7 $$\"3g'\\&[*o&)\\)oF3F[y7$$\"3/7#zJ$Q]JzF3F^y7$$\"3?=M-;(\\y&))F3Fay7$ $\"3OCw'))f&>%y*F3Fdy7$$\"31$=r\"[T0r5F*Fgy7$$\"3m.cXO()oj6F*Fjy7$$\"3 'e/'fhNKl7F*F]z7$$\"31)[OnQepO\"F*F`z7$$\"3EIp(=@$fo9F*Fcz7$$\"3Yst,P! 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HX!Qc:F*7$Fhhq$\"3m&H+(Q3%\\c\"F*7$Ffil$\"3!o+adg9wd\"F*7$Fe\\w$\"39+t y9(G[e\"F*7$Fiil$\"3%fqx3140f\"F*7$F]]w$\"3%f=_5Y^Lf\"F*7$F\\jl$\"3!*) pn3-#3#f\"F*7$Fbjl$\"35->1()H)Gd\"F*7$Fd[m$\"3)))enA45q_\"F*7$Fi[m$\"3 KcGUf+%HY\"F*7$F^\\m$\"3EO)fTWQ2Q\"F*7$Fc\\m$\"3iEX!)G'QmG\"F*7$Fh\\m$ \"3;X\">Z%yx(=\"F*7$F]]m$\"3_`G4yg$R3\"F*7$Fb]m$\"3E^]Z#4th&)*F37$Fg]m $\"3%o+j^N`c#*)F37$F\\^m$\"3HCy*[$o!*3!)F37$Fa^m$\"3ye3I%4'3%4(F37$Ff^ m$\"3)o%Q+B+d/hF37$F[_m$\"3]yhGgATM]F37$F`_m$\"3$y;luM\\8#RF37$Fe_m$\" 3U!RChV--x#F37$Fj_m$\"3Sg]$GHb))z\"F37$F_`m$\"3#*Ri!GLsi9\"F37$Fd`m$\" 3-i,bD+&*f\"*Fbw-Fi`m6&F[amF[go$\")AR!)\\F]goF^goF_go-%+AXESLABELSG6$Q \"x6\"Q!Fj[z-%%VIEWG6$;$!+3`=$G'!\"*$\"+iqjc7F]go%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 137 "We don't need very many terms to obt ain a Fourier series with a graph which is indistinguishable from the \+ graph of the original function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 15 ": The function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 79 " in this example is continuous everywhere, and the Fourier series \+ converges to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 15 " \+ at all points." }}{PARA 0 "" 0 "" {TEXT -1 72 "Consequently, the Fouri er series is a genuine expansion of the function " }{XPPEDIT 18 0 "f(x )" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(FS(x,30),x=-2*Pi..4*Pi ,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 567 207 207 {PLOTDATA 2 " 6&-%'CURVESG6$7hu7$$!3)****>YH&=$G'!#<$\"3u)z\\NxqS0#!#>7$$!3A%zEdio/= 'F*$\"3+MvB\"*[h05!#=7$$!3[)eLo&>vxgF*$\"31.A'[Fa+1#F37$$!3u#QSzGN](fF *$\"3SQi%*p%o,3$F37$$!3+xr/>'=B(eF*$\"3r$)o;$po\"3TF37$$!3eUim%RXHy&F* $\"3GN+BJ[q)*\\F37$$!393`Gq@d$p&F*$\"3,v&**oz?B!fF37$$!3ttV!f%*)>/cF*$ \"3eN([y\")QDy'F37$$!3JRM_@d#[^&F*$\"3wbj\"GIL5p(F37$$!3aNLX$=;VT&F*$ \"3O$4%z>2W#o)F37$$!3*3B$QXm!QJ&F*$\"3+0QZS%z*)p*F37$$!3BEJJ2rH8_F*$\" 3=U'f<#=Zp5F*7$$!3e@ICpvy7^F*$\"3ejx-j,qq6F*7$$!3Hl9r)y6;,&F*$\"3gtDwo ?Wr7F*7$$!3+4*z\"3gV5\\F*$\"3g,rcbmbs8F*7$$!3q_$[wAg#4[F*$\"351PF?16v9 F*7$$!3T'z;rW%33ZF*$\"3-#)G&)oa'Hc\"F*7$$!3+FLS$e(\\&p%F*$\"3[B*[ftP!o :F*7$$!3gd)*o>2\"Ho%F*$\"3Od:%*>ebr:F*7$$!3A)Qwf&QKqYF*$\"3K[eB+^lt:F* 7$$!3q>HE#*ptdYF*$\"3%[Qel]SXd\"F*7$$!3!=)f$[EjDj%F*$\"3I^/zb5st:F*7$$ !3+V!4u`*Q2YF*$\"3'>W6-^H8d\"F*7$$!3Sl^b#3Uqb%F*$\"3%=4/l'Hio:F*7$$!3q )G,xi%p1XF*$\"35*Q*pcp#3d\"F*7$$!3SMN*zr**fS%F*$\"3U6(fWG`1d\"F*7$$!3* 4y&G3[I0VF*$\"3QA2%=@e4d\"F*7$$!3&yz1P_w=$RF*$\"3I)\\2-d\"F*7$$!3uf c$[;8_a$F*$\"3k%*G<#[*4q:F*7$$!3YxM'fOD`9$F*$\"3+5U'HlG,d\"F*7$$!3C9QL 7+sYFF*$\"3e-T!GW^,d\"F*7$$!3sk(*H6aqOBF*$\"3#>!Q!oL7+d\"F*7$$!3iplTp[ 8c@F*$\"3Y$e5Hp$Gq:F*7$$!33uL`FVcv>F*$\"3a=YUqX(3d\"F*7$$!39)Rhk\\CR(= F*$\"3u'[Xid'zq:F*7$$!3?A%*QlYGs;F*$\"3Yx\\j+&zUd\"F*7$$!3OkM,h\\T%f\"F*$\" 3YU&\\+c/,d\"F*7$$!3KqaC.]+p:F*$\"3M0!yd/J*f:F*7$$!3/f1\"F*7$$!3Xe1v:JuS'*F3$\"3EGA\"QEBkk*F37$$!3ohG2PJBd')F3$\"39G *[[6<2l)F37$$!3#\\1&ReJstwF3$\"3/kriQiC\"o(F37$$!3)f!e\"e(=g!y'F3$\"3) ReC4o8Jx'F37$$!3/ZlB$f![()eF3$\"3-7^uK8\"R*eF37$$!35)Gd1JfV*\\F3$\"3e& [7A>b.*\\F37$$!3F3$\"3y\\Y[C@Q!*>F37$$!3qA0)\\f_@:*F-$\"3XF$R.) *4(R()F-7$$\"3+ngG+L#zY\"F-$\"31'*)G)o#zaE#F-7$$\"3gUD7i?XY5F3$\"3y6mj wqLG5F37$$\"3_yk@%z6h%>F3$\"3Spwe:h@i>F37$$\"3U9/JE:xXGF3$\"3`a\"RiZp( HGF37$$\"3K]VSe7VXPF3$\"3]QH\"Q:z0w$F37$$\"3ul!)4-%\\>z%F3$\"3iP!)Qqr; \"y%F37$$\"3=\"y\"zXvYQeF3$\"3#3p)*e)R=YeF37$$\"3g'\\&[*o&)\\)oF3$\"3O wha5UkzoF37$$\"3/7#zJ$Q]JzF3$\"366t=Y1%[$zF37$$\"3?=M-;(\\y&))F3$\"3K; Y#HA&)R&))F37$$\"3OCw'))f&>%y*F3$\"3Eq7TA!=#)y*F37$$\"31$=r\"[T0r5F*$ \"3+@5)zE`12\"F*7$$\"3m.cXO()oj6F*$\"3KZkEph4k6F*7$$\"3'e/'fhNKl7F*$\" 3'H5B6.X]E\"F*7$$\"31)[OnQepO\"F*$\"3=TmXF0*pO\"F*7$$\"3EIp(=@$fo9F*$ \"3An7ma`Np9F*7$$\"3Yst,P!G-d\"F*$\"31E3\\7-dg:F*7$$\"3K?k.MJU%f\"F*$ \"3]%\\co!o5q:F*7$$\"3@oa0J#='=;F*$\"3oSjT'>'>u:F*7$$\"3'e^u!GL\"Gk\"F *$\"31$\\,=2[Ud\"F*7$$\"3sjN4D%3qm\"F*$\"3c2Q#>\")3Ad\"F*7$$\"3Df;8>') R:zN[')Qr:F*7$$\"3aP@K*e\\t&>F*$\"3c(oJa\"o@q:F*7$$\"3K2XV<.F hBF*$\"3'4s\"*[W4/d\"F*7$$\"32AlCOc>KFF*$\"35!oO6f#4q:F*7$$\"3#oGbGq)G KJF*$\"3SBrY*e^,d\"F*7$$\"3uCR&\\&H(ya$F*$\"31X+!ya$4q:F*7$$\"3yk14o-k 4RF*$\"3YT)\\jd-2d\"F*7$$\"3B\\x@x)e.I%F*$\"3?=s'fYJ6d\"F*7$$\"3Yrc)f^ r7S%F*$\"3_M,z(=_/d\"F*7$$\"3d%f`Z:%=-XF*$\"3k(H&f(G&3r:F*7$$\"37cv8u/ k_XF*$\"31?L&eUm'o:F*7$$\"3o<:_$z'4.YF*$\"3!4RqW994d\"F*7$$\"3Y)\\8K& \\KGYF*$\"3mvbVk%oLd\"F*7$$\"3Cza!H6`Nl%F*$\"3MwZHC!4Yd\"F*7$$\"3ip9v# >nhm%F*$\"31qXqF+2u:F*7$$\"3-gufs7yyYF*$\"3')zpv\">#Rs:F*7$$\"3S]MW_`R \"p%F*$\"3cL![J0]$p:F*7$$\"3\"*R%*GK%4Sq%F*$\"3egL4wFyk:F*7$$\"3&3I+>w KF![F*$\"39\"HuEA_@[\"F*7$$\"3zh6^\"4c9!\\F*$\"3-HSa2\"y6Q\"F*7$$\"3uA ?7@%z,+&F*$\"3'yF0D>qJG\"F*7$$\"3o$)Gt]F!*)4&F*$\"3ab;@xxJ%=\"F*7$$\"3 aA/R[EU%>&F*$\"3glM]>ol)3\"F*7$$\"3Whz/YD%**G&F*$\"3Yu%f9!*pQ$**F37$$ \"3I+bqVCY&Q&F*$\"3CW[\")RVrv*)F37$$\"3?RIOTB)4[&F*$\"3FlIkdcFB!)F37$$ \"396],-4/(e&F*$\"3SXaWWu+epF37$$\"31$)pmi%*4$p&F*$\"3S\\I7zV\"p!fF37$ $\"3-b*=L-e\"*z&F*$\"3e!p1i(o&3$[F37$$\"3(p#4(Re;_!fF*$\"33\\Vlk:\"Qz$ F37$$\"3Qr<'y\"\\^+gF*$\"3*Qe71t0(4GF37$$\"3!\\h_'F*$\"3_e(QJG$G/))F-7$$\"3%GIM&>*4kG'F*$\"3klu:&QGV1 #F-7$$\"3Sk[)>Bh\")Q'F*$\"3.@`0$pJA.\"F37$$\"3)fUNWa7**['F*$\"3w\"z^/r [<2#F37$$\"3a()f)o&Qm\"f'F*$\"3]&QYa+nO3$F37$$\"35\\lLp^T$p'F*$\"3NQ-) R#)3@5%F37$$\"3a``j\")ph&y'F*$\"3@siw,KZ@]F37$$\"3%)eT$Rz=y(oF*$\"3o9. 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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 0 291 "f := x -> m in(abs(x),Pi/2):\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nFS := (x ,n)->3*Pi/8+sum(2*(cos(1/2*Pi*k)-1)/(Pi*k^2)*cos(k*x),k=1..n):\nframes := [seq(plot([f_(x),FS(x,i)],x=-2*Pi..4*Pi,\n color=[black,red],line style=[3,1],thickness=1),i=1..8)]:\nplots[display](frames,insequence=t rue);" }}{PARA 13 "" 1 "" {GLPLOT2D 657 306 306 {PLOTDATA 2 "6#-%(ANIM ATEG6*7&-%'CURVESG6&7[t7$$!3)****>YH&=$G'!#<$\"3)fq*)fiefD\"!#D7$$!3A% zEdio/='F.$\"3.?z_9o;F5!#=7$$!3[)eLo&>vxgF.$\"3Yx*fM]LV0#F77$$!3u#QSzG N](fF.$\"3)[.#R#>+:3$F77$$!3+xr/>'=B(eF.$\"3I#4C8)om3TF77$$!3eUim%RXHy &F.$\"3aOM8D\"*R-]F77$$!393`Gq@d$p&F.$\"3y!yU*o88'*eF77$$!3ttV!f%*)>/c F.$\"3-D@v7O')*y'F77$$!3JRM_@d#[^&F.$\"3Ep9ccef$o(F77$$!3aNLX$=;VT&F.$ \"3!p]isB\"p)o)F77$$!3*3B$QXm!QJ&F.$\"3Y`N'zh'y$p*F77$$!3BEJJ2rH8_F.$ \"3+gk')*>)))p5F.7$$!3e@ICpvy7^F.$\"3kkl$zt(Rq6F.7$$!3Hl9r)y6;,&F.$\"3 %47o%=Ndr7F.7$$!3+4*z\"3gV5\\F.$\"3Cx'***)H\\FP\"F.7$$!3q_$[wAg#4[F.$ \"3aL7`z]#RZ\"F.7$$!3T'z;rW%33ZF.$\"3c'*[zEjzq:F.7$$!3q)G,xi%p1XF.F[q7 $$!3*4y&G3[I0VF.F[q7$$!3&yz1P_w=$RF.F[q7$$!3ufc$[;8_a$F.F[q7$$!3YxM'fO D`9$F.F[q7$$!3C9QL7+sYFF.F[q7$$!3sk(*H6aqOBF.F[q7$$!33uL`FVcv>F.F[q7$$ !3?A%*QlYGsF7$\"3C6Z-\\GAx>F77$$!3qA0)\\f_@:*!#>$\"3qA0)\\f_@ :*Ffw7$$\"3+ngG+L#zY\"FfwFjw7$$\"3gUD7i?XY5F7F]x7$$\"3_yk@%z6h%>F7F`x7 $$\"3U9/JE:xXGF7Fcx7$$\"3K]VSe7VXPF7Ffx7$$\"3ul!)4-%\\>z%F7Fix7$$\"3= \"y\"zXvYQeF7F\\y7$$\"3g'\\&[*o&)\\)oF7F_y7$$\"3/7#zJ$Q]JzF7Fby7$$\"3? =M-;(\\y&))F7Fey7$$\"3OCw'))f&>%y*F7Fhy7$$\"31$=r\"[T0r5F.F[z7$$\"3m.c XO()oj6F.F^z7$$\"3'e/'fhNKl7F.Faz7$$\"31)[OnQepO\"F.Fdz7$$\"3EIp(=@$fo 9F.Fgz7$$\"3Yst,P!G-d\"F.Fjz7$$\"3+b(pJ\"))yjF. 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following cases show that the given functions are orthogonal on the indicated interval." }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "f(x)=x" "6#/-%\"fG6# %\"xGF'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "g(x)=x^2" "6#/-%\"gG6#%\"xG *$F'\"\"#" }{TEXT -1 3 "; " }{XPPEDIT 18 0 "[-2,2]" "6#7$,$\"\"#!\"\" F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "f(x)=x^3" "6#/-%\"fG6#%\"xG*$F'\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "g(x)=x^2+1" "6#/-%\"gG6#%\"xG,&*$F'\"\"#\"\"\"F+F+" }{TEXT -1 2 "; " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "f(x)=exp(x)" "6#/-% \"fG6#%\"xG-%$expG6#F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "g(x)=x*exp(- x)-exp(-x)" "6#/-%\"gG6#%\"xG,&*&F'\"\"\"-%$expG6#,$F'!\"\"F*F*-F,6#,$ F'F/F/" }{TEXT -1 3 "; " }{XPPEDIT 18 0 "[0,2]" "6#7$\"\"!\"\"#" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "f(x )=cos*x" "6#/-%\"fG6#%\"xG*&%$cosG\"\"\"F'F*" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "g(x)=sin^2*x" "6#/-%\"gG6#%\"xG*&%$sinG\"\"#F'\"\"\"" } {TEXT -1 3 "; " }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(e) " }{XPPEDIT 18 0 "f(x)=x" "6#/-% \"fG6#%\"xGF'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "g(x)=cos*2*x" "6#/-% \"gG6#%\"xG*(%$cosG\"\"\"\"\"#F*F'F*" }{TEXT -1 3 "; " }{XPPEDIT 18 0 "[-Pi/2,Pi/2]" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(f) " }{XPPEDIT 18 0 "f(x)=exp( x)" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "g( x)=sin*x" "6#/-%\"gG6#%\"xG*&%$sinG\"\"\"F'F*" }{TEXT -1 3 "; " } {XPPEDIT 18 0 "[Pi/4,5*Pi/4]" "6#7$*&%#PiG\"\"\"\"\"%!\"\"*(\"\"&F&F%F &F'F(" }{TEXT -1 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 41 "Find the Fourier se ries for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 95 " defined as follows, and compare the graphs of some trunc ated Fourier series with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x) = PIECEWISE([-1, -Pi <= x and x < 0],[1, 0 <= x an d x < Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,$\"\"\"!\"\"31,$%#PiGF .F'2F'\"\"!7$F-31F4F'2F'F2" }{TEXT -1 3 ", " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " } {XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "f := x -> piecewise(x<0,-1,1):\n'f(x)'=f(x);\nf_ := x -> f(x-2*Pi *floor((x+Pi)/(2*Pi))):\n'f_(x)'='f(x-2*Pi*floor((x+Pi)/(2*Pi)))';\npl ot(f_(x),x=-2*Pi..5*Pi,thickness=2,\n color=COLOR(RGB,.4, 0,1),ytickmarks=3);" }}{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,%#PiGF,-%&floo rG6#,$*(F.!\"\",&F'F,F/F,F,F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 522 182 182 {PLOTDATA 2 "6(-%'CURVESG6#7_r7$$!3)****>YH&=$G'!#<$\"\"\" \"\"!7$$!3y6^yR3%Q!eF*F+7$$!3+\\Tn#zlnQ&F*F+7$$!3C*R!o\"G@x\"\\F*F+7$$ !3'R()*>RwcXWF*F+7$$!3QgPcg!ec(RF*F+7$$!3%oQfX*[#yv$F*F+7$$!3K8]bG<**R NF*F+7$$!3:[4b:\\@FMF*F+7$$!3V$)oa-\"QWJ$F*F+7$$!3%3&[/'p\\!eKF*F+7$$! 3q=Ga*Gh;?$F*F+7$$!3i-=H'3nM<$F*F+7$$!37'ySI)GFXJF*F+7$$!33y_T\"yv68$F *$!\"\"F-7$$!3gp(*yz'yq6$F*FX7$$!3dhU;y:)H5$F*FX7$$!3a`(QlZ%)))3$F*FX7 $$!35\\L'Qf;c&GF*FX7$$!35Xz=6([Bi#F*FX7$$!3y3M&>84t:#F*FX7$$!3aB/3k(e* y;F*FX7$$!3Yljo*\\FwD\"F*FX7$$!3#>;s,)G3Ly!#=FX7$$!3!))e_0#)[F_pFX7$$!31A]=S9ebx!#>FX7$$!3?!)=/p 1,()[F\\qFX7$$!3NQ()*y*)R%=?F\\qFX7$$!3>u;FB^aTe!#?FX7$$\"32NSWK(38])F fqF+7$$\"3Wuf\")eiT%G#F\\qF+7$$\"3OXvQW;q=PF\\qF+7$$\"31HQn'=VeX*F\\qF +7$$\"3G6g*GZ)H>:F_pF+7$$\"3X'HZA8[Kg$F_pF+7$$\"3k\"e)f\"z(>(o&F_pF+7$ $\"3[!*)Q4KBV1\"F*F+7$$\"3SI$\\#HX;%[\"F*F+7$$\"3c!*H(HmRD(>F*F+7$$\"3 W?-I3W$[S#F*F+7$$\"3:$)yiLB)>k#F*F+7$$\"3'eab*e-8zGF*F+7$$\"3W&36JF*F+7$$\"3z4blx(yJ8$F*F+7$$ \"3=_*\\Ue#HZJF*FX7$$\"3c%RW3R19;$F*FX7$$\"3MzK./Sj*=$F*FX7$$\"3dk@A<; 'y@$F*FX7$$\"3eM**fVoJuKF*FX7$$\"39/x(*p?xILF*FX7$$\"3G$))4Z\\#RmNF*FX 7$$\"3Vi?W>H,-QF*FX7$$\"3ozEsC\"fZB%F*FX7$$\"3i-&*4pV`,ZF*FX7$$\"3Y1X@ jEQ'=&F*FX7$$\"38S1aGXW3cF*FX7$$\"37r!)>* )HclnF*F+7$$\"3E_j&\\U=c))F*F+7$$\"3/BvNv%>82*F*F+7$$\"3= O!)QPqX'G*F*F+7$$\"3U\\?`#ed_M*F*F+7$$\"3kignF\"eSS*F*F+7$$\"31l?'REe( =%*F*F+7$$\"3Dp![-SeMV*F*FX7$$\"3WtS`O&e\"[%*F*FX7$$\"3)e2?GneGY*F*FX7 $$\"3E%3#RX*eA\\*F*FX7$$\"3(34kz@f;_*F*FX7$$\"35>@D3.ER'*F*FX7$$\"3cX, a)Rhov*F*FX7$$\"3Cjws?$o\"y**F*FX7$$\"34=:HCv%*>5!#;FX7$$\"31JpOCM@m5F ^]lFX7$$\"33bsK)zY96\"F^]lFX7$$\"3%pz_)[>!)e6F^]lFX7$$\"3>R?5(e5W?\"F^ ]lFX7$$\"3Ybx5(HJxA\"F^]lFX7$$\"3urM62?0^7F^]lFX7$$\"3s-bG#[(\\_7F^]lF X7$$\"3oLvXdH%RD\"F^]lFX7$$\"3%[cHEV)Qb7F^]lFX7$$\"3#ef,y!R$oD\"F^]lF+ 7$$\"3%zlX\"e[sf7F^]lF+7$$\"3))>(*[3ehi7F^]lF+7$$\"3'R%y<4xRo7F^]lF+7$ $\"3/of')4'zTF\"F^]lF+7$$\"3=;AC6Mu&G\"F^]lF+7$$\"3Mk%=E@2tH\"F^]lF+7$ $\"3J0\"3w2f&=8F^]lF+7$$\"3HYxfU4\")R8F^]lF+7$$\"3=rup8]_)Q\"F^]lF+7$$ \"3OAPhfa4K9F^]lF+7$$\"3'4=WDk]&y9F^]lF+7$$\"313k:!=9I_\"F^]lF+7$$\"3' ****\\OK'zq:F^]lF+-%*THICKNESSG6#\"\"#-%*AXESTICKSG6$%(DEFAULTG\"\"$-% +AXESLABELSG6$Q\"x6\"Q!Fabl-%&COLORG6&%$RGBG$\"\"%FY$F-F-F+-%%VIEWG6$; $!+3`=$G'!\"*$\"+Fjzq:!\")F[bl" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________ ______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Find the Fourier \+ series for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 95 " defined as follows, and compare the graphs of some trunc ated Fourier series with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x)=abs(x)" "6#/-%\"fG6#%\"xG-%$absG6#F'" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "-Pi<=x" "6#1,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 " `` " 0 "" {MPLTEXT 1 0 178 "f := x -> abs(x):\n' f(x)'=f(x);\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n'f_(x)'='f(x- 2*Pi*floor((x+Pi)/(2*Pi)))';\nplot(f_(x),x=-2*Pi..6*Pi,color=COLOR(RGB ,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\" xG-%$absGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#, &F'\"\"\"*(\"\"#F,%#PiGF,-%&floorG6#,$*(F.!\"\",&F'F,F/F,F,F/F5F,F,F5 " }}{PARA 13 "" 1 "" {GLPLOT2D 598 136 136 {PLOTDATA 2 "6'-%'CURVESG6# 7]q7$$!3)****>YH&=$G'!#<$\"3)fq*)fiefD\"!#D7$$!3OM/dxTF4gF*$\"3r=:4'H6 \"RF!#=7$$!3'y'3_gIONdF*$\"3q$=(emCAyaF37$$!3[C3$))yt$y]8$F*7$$!3SQ!43OJ.*GF*$\"3SQ!43OJ.*GF*7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 34 "______________ ____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "f := x -> piecewise(x<0,0,x):\n'f(x)'=f( x);\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n'f_(x)'='f(x-2*Pi*flo or((x+Pi)/(2*Pi)))';\nplot(f_(x),x=-2*Pi..6*Pi,color=COLOR(RGB,.4,0,1) ,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIE CEWISEG6$7$\"\"!2F'F,7$F'%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floorG6#,$*(F.! \"\",&F'F,F/F,F,F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 624 135 135 {PLOTDATA 2 "6'-%'CURVESG6#7^r7$$!3)****>YH&=$G'!#<$\"3)fq*)fiefD\"!#D 7$$!3OM/dxTF4gF*$\"3r=:4'H6\"RF!#=7$$!3'y'3_gIONdF*$\"3q$=(emCAyaF37$$ !3[C3$))yt$y]8$F*7$$!3=(R/, 'f*>8$F*$\"\"!Fio7$$!30S!=o)\\)e6$F*Fho7$$!3Y#oJN,u(*4$F*Fho7$$!3MD`CS Im$3$F*Fho7$$!3i5En$4T90$F*Fho7$$!3O'*)*4Z\">#>IF*Fho7$$!3QnW&RDvZ&HF* Fho7$$!3SQ!43OJ.*GF*Fho7$$!3Y!==XdV9w#F*Fho7$$!31AtA)ybDj#F*Fho7$$!3GW :JAR'fO#F*Fho7$$!31mdRc?P*4#F*Fho7$$!3+TUb^#)*yc\"F*Fho7$$!3!\\9So677- \"F*Fho7$$!3'4.l\"=n!pR&F3Fho7$$!3suA!HA6lo#F3Fho7$$\"3y^\"[gBF%)Q#!#? 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" }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "f := x -> x ^2:\n'f(x)'=f(x);\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n'f_(x)' ='f(x-2*Pi*floor((x+Pi)/(2*Pi)))';\nplot(f_(x),x=-2*Pi..6*Pi,color=COL OR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"f G6#%\"xG*$)F'\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#% \"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floorG6#,$*(F.!\"\",&F'F,F/F, F,F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 529 174 174 {PLOTDATA 2 "6' -%'CURVESG6#7[v7$$!3)****>YH&=$G'!#<$\"39]z>q?Vx:!#L7$$!3!zgd`^2Z@'F*$ \"3Q-U\"G(z?*o%!#?7$$!3s;_4O(Hi9'F*$\"33zX5XGov=!#>7$$!3aDG$o&>vxgF*$ \"3-#\\pM9'G?UF97$$!3OM/dxTF4gF*$\"3pfhP#pIF](F97$$!35^c/>'=B(eF*$\"3/ $[\"=aV6)o\"!#=7$$!3'y'3_gIONdF*$\"3Vc?MR@4,IFI7$$!3[C3oT+e[DNCF*7$$!3GDBS`Rv(e%F*$\"3o &*4Pvu[uGF*7$$!3/#3$p7H&GX%F*$\"37+nIum6]LF*7$$!3zQQ)>(=&zJ%F*$\"3%zF6 [XU@'QF*7$$!3a&fu7$30$=%F*$\"3dHZ)o\"[c5WF*7$$!3#zvj;&4z[SF*$\"3-%*4'f 8=D*\\F*7$$!3<@H0s5`9RF*$\"3+([$e&)G_5cF*7$$!3U%3UC>r-y$F*$\"3-7Avl!zX E'F*7$$!3CZ7$GJ6gk$F*$\"3ksrYwmoapF*7$$!3!R7rz@N:_$F*$\"3?3^A:7rEwF*7$ $!3+,56B\"fqR$F*$\"3+$Hm.NC(H$)F*7$$!3mx3DGIesKF*$\"3EL2*=3EP1*F*7$$!3 Ka2RLp5[JF*$\"3JE%)z4krG)*F*7$$!3O'*)*4Z\">#>IF*$\"3ZEY@eUo:\"*F*7$$!3 SQ!43OJ.*GF*$\"3GGiFv`,a$)F*7$$!3Y!==XdV9w#F*$\"3#euZ_hqbi(F*7$$!31AtA )ybDj#F*$\"3Ix\"H\"y*\\.$pF*7$$!3SL%p_&)f#*\\#F*$\"3S*QhT#)*HYiF*7$$!3 GW:JAR'fO#F*$\"3-F)z;G&y(f&F*7$$!3=bON*)zmKAF*$\"3u$\\%o]j![)\\F*7$$!3 1mdRc?P*4#F*$\"3v(Qv6.jtS%F*7$$!3#[Q&=0O]m>F*$\"3)f62#Hk8nQF*7$$!3a.]( R:NO$=F*$\"3-NH*y(y@iLF*7$$!3FAYw-nw+(>%=bXH\"F*$\"3;G=$\\7ten\"F*7$$!3!\\9So 677-\"F*$\"3#)H6e(=uG/\"F*7$$!3+SKGVR^/yFI$\"3M^-4*yV54'FI7$$!3'4.l\"= n!pR&FI$\"3j%[dC@gE\"HFI7$$!3'GlL0(*3F97$$\"3;#HU?vF]1#FI$\"3QU[q;'RVE%F97$$\"3/Hi$>\"* 3au#FI$\"3`p)QP4qs`(F97$$\"3\"G5C^^N $pY&FI$\"3QNZmXit))HFI7$$\"3mTe*z#oi*3)FI$\"3\"o+B;A1Ua'FI7$$\"3oqzW5? 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