{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 261 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 263 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 269 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Hea ding 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Time s" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "More period " }{XPPEDIT 18 0 "2*P i" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 24 " Fourier series examples" }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 52 "load Fourier series and Fourier transform procedures" } }{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 269 9 "fourier .m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 13 " FourierSeries" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command si milar to the one that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/procd rs/fourier.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 31 "Convergence of a Fourier series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 37 " be an periodic f unction with period " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 14 "Fou rier series" }{TEXT 258 1 " " }{TEXT -1 16 "of the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 " is" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = c+``;" "6#/-%\"FG6#%\"xG, &%\"cG\"\"\"%!GF*" }{XPPEDIT 18 0 "Sum(a[k]*cos*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 12 " --- ---- (i)" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{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%\"cG\"\"\"*(&%\"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#\"\"$F%F*F%F;F%F+F%F%**&F.6#F;F%F0F%F;F%F+F%F%%(~.~.~.~GF%" } {TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 10 " where " }{XPPEDIT 18 0 "c = 1/(2*Pi);" "6#/%\"cG*&\" \"\"F&*&\"\"#F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "Int(f(x),x = -Pi .. Pi); " "6#-%$IntG6$-%\"fG6#%\"xG/F);,$%#PiG!\"\"F-" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "a[k] = 1/Pi;" "6#/&%\"aG6#%\"kG*&\"\" \"F)%#PiG!\"\"" }{TEXT -1 2 " " }{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 5 "and " }{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 0 "" 0 "" {TEXT -1 83 "assuming that the integr als involved in the definition of the Fourier coefficients " } {XPPEDIT 18 0 "c, a[k]" "6$%\"cG&%\"aG6#%\"kG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" }{TEXT -1 47 " all exist, and g ive finite real number values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 58 "Under suitable conditions the Fourier ser ies converges to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 53 ", in which case the Fourier series (i) is called the " }{TEXT 259 24 "Fourier series expansion" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "In \+ fact, if " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 24 " and \+ the derivative f '(" }{TEXT 265 1 "x" }{TEXT -1 6 ") are " }{TEXT 259 20 "piecewise continuous" }{TEXT -1 14 ", that is, if " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 9 " and f '(" }{TEXT 264 1 "x" } {TEXT -1 165 ") are continuous except at only a finite number points i n any period, and do not have infinite discontinuities at these points , then the Fourier series converges to " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 22 " at every point where " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 15 " is continuous." }}{PARA 0 "" 0 "" {TEXT -1 30 "At any point of discontinuity " }{XPPEDIT 18 0 "x = d;" " 6#/%\"xG%\"dG" }{TEXT -1 76 ", the Fourier series converges to the ave rage of the left and right limits: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(Limit(f(x),x = d,le ft)+Limit(f(x),x = d,right))/2 = (f(d-``)+f(d+``))/2;" "6#/*&,&-%&Limi tG6%-%\"fG6#%\"xG/F,%\"dG%%leftG\"\"\"-F'6%-F*6#F,/F,F.%&rightGF0F0\" \"#!\"\"*&,&-F*6#,&F.F0%!GF8F0-F*6#,&F.F0F>F0F0F0F7F8" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 51 "that is, to the average of the two va lues to which " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 15 " approaches as " }{XPPEDIT 18 0 "x->d" "6#f*6#%\"xG7\"6$%)operatorG%&a rrowG6\"%\"dGF*F*F*" }{TEXT -1 30 " from the left and the right. 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" }}{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 0 190 "f := x -> piecewise(x<0,0,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=-6.28..10,color=COLOR(RGB,. 4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG -%*PIECEWISEG6$7$\"\"!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 539 193 193 {PLOTDATA 2 "6'-%'CURVESG6#7^r7$$!3E++++++!G'!#< $\"3C,A$G==Y,\"!#A7$$!3/n;a#e&G\">'F*$\"3#*p`_)QfbW)!#?7$$!3(HL$3l6d-h F*$\"3=cl^Ko9iK!#>7$$!3y**\\iZn&Q,'F*$\"3GU[<*H(y`sF97$$!3emm;IB9DfF*$ \"3=TF)QtZ>G\"!#=7$$!3i*\\7'\\CwqdF*$\"37(=s&\\=xDEFD7$$!3bL$e!pDQ;cF* $\"35***fc\"*eiW%FD7$$!3]mm^D*\\\"p_F*$\"3RjUEbxEG5F*7$$!31nm1iFQ%4&F* $\"3EapE\"\\^KT\"F*7$$!3vmmh)f:'>\\F*$\"31l.6]BKf=F*7$$!3e*\\()H2zcu%F *$\"3mJ&z,UDRO#F*7$$!3IL$etaU#\\UF*$\"3C#=f$*4;q8%F*7$$!3EM$eL$GC#3%F* $\"3b7/2tx9W[F*7$$!3V+]sofE:RF*$\"3L.sU$*>/2cF*7$$!3kL$ekDyDu$F*$\"3)4 auh=%oakF*7$$!3%om\">W0*)pNF*$\"3Opw&3Zo>O(F*7$$!3S++!RfBQ[$F*$\"3@=cx ,gUOyF*7$$!3]L$3OkcxR$F*$\"3O'eeDd)pD$)F*7$$!3jmmJ$p*o6LF*$\"3Fql?$='y H))F*7$$!3=+]-VFiDKF*$\"3En&>P$))o[$*F*7$$!30@!)oL,\\.KF*$\"31UsihB_%[ *F*7$$!3!>/^V_d8=$F*$\"3)G/FVfN8i*F*7$$!3M_Do>7HqJF*$\"3!>*R(**f4,p*F* 7$$!3yiS,:\\AfJF*$\"3cr*==`G\"f(*F*7$$!3?tbM5'e\"[JF*$\"3$=)>')*Q#RG)* F*7$$!3k$3xcI#4PJF*$\"\"!Fat7$$!3ODJ+(3FG4$F*F`t7$$!3km\"H$o=c[IF*F`t7 $$!3m\\7)4VJ+'HF*F`t7$$!35LLj$*4]rGF*F`t7$$!3')**\\#)yca:FF*F`t7$$!3im m,k.ffDF*F`t7$$!3`*****3(GX3AF*F`t7$$!3E++]tN(e&=F*F`t7$$!3x****4*3)4; :F*F`t7$$!3^m;>)R\\v?\"F*F`t7$$!3\"GLLBh^lS)FDF`t7$$!3UILLphZ)H&FDF`t7 $$!3#=+]#\\b/$o\"FDF`t7$$!3uqs;H%>6H)F3F`t7$$\"3TcmmgJA<:FD$\"3v=LE>h' >I#F97$$\"33F$3d:CGF$FD$\"3U#z=azP62\"FD7$$\"3u(**\\2:D%G]FD$\"3#R^p' \\f]GDFD7$$\"3B++]hC<+nFD$\"3E(H%Q,6B*[%FD7$$\"3s-+Ds(>>P)FD$\"3C`!*Hn S!*3qFD7$$\"33nmTl00'=\"F*$\"3r&>DP%fr19F*7$$\"3#p;H$36BY8F*$\"36xJ.(> QB\"=F*7$$\"3wm;C^;T1:F*$\"3'[k'*H1w#pAF*7$$\"3?+DT1!)=z;F*$\"3%oez3Os '>GF*7$$\"3mLLehV'>&=F*$\"3hkZd'*>xHMF*7$$\"3cLepO/VJ?F*$\"3uCU8>'4n7% F*7$$\"3YL$3=^'*3@#F*$\"35ci#fQj!))[F*7$$\"3%ommxDArO#F*$\"3ZiV@$ynKg& F*7$$\"3B+]s.![L_#F*$\"3)R%H!z9&GnjF*7$$\"3b$e9l/7xg#F*$\"3&zFPv6i,!oF *7$$\"3'o;/$*3w?p#F*$\"3Q;KgqOFZsF*7$$\"3=]P4K,WwFF*$\"3+h252)>'3xF*7$ $\"3]LL)[@*F*7$$\"3S]i&H&\\pyIF*$\"3A*GeLhi$y%*F*7$ $\"3^L3d3JFAJF*$\"3qwX]k$*e[(*F*7$$\"3izWZZw;LJF*$\"30l2=2&Rn\")*F*7$$ \"3HD\"yj=iS9$F*F`t7$$\"3'4x\"GDn&\\:$F*F`t7$$\"3i;a=k7&e;$F*F`t7$$\"3 S3F*>MSw=$F*F`t7$$\"3=++!)>%H%4KF*F`t7$$\"3Q**\\d8.'*zLF*F`t7$$\"3c)** \\t?\"\\]NF*F`t7$$\"33+](Q-'[!)QF*F`t7$$\"3!****\\]+*)oC%F*F`t7$$\"3Em m1!Q=hd%F*F`t7$$\"3Z****\\55kF\\F*F`t7$$\"3MK$3-wshC&F*F`t7$$\"3E++!z4 7Wf&F*F`t7$$\"3tk;uv\"y?#fF*F`t7$$\"3SK$ex!4L$4'F*F`t7$$\"35+]xROekiF* F`t7$$\"3i;Hij!*H[jF*$\"3P]8;y7!)RUF37$$\"3:L3Z([9?V'F*$\"30[kh!\\7]@# F97$$\"3o\\(=8\"*Hd^'F*$\"3kBSU\"H*p2aF97$$\"3?mm;N`W*f'F*$\"32')Q-`S? +5FD7$$\"3IK3-;*HZx'F*$\"3-O*3[-hhT#FD7$$\"3S)*\\(o\\9+&pF*$\"3w*R)H#o 6mW%FD7$$\"3uMLtlUl(G(F*$\"39_!)p))y&*35F*7$$\"3qL$e2)pHguF*$\"3#)f`2# *=f&Q\"F*7$$\"3lKLy&pRHj(F*$\"3Ft]b4p$=#=F*7$$\"3#**\\i>%G:/yF*$\"3f8O '=CULJ#F*7$$\"3?n;9))fOvzF*$\"39#H@qXvM'GF*7$$\"3!R$3(z3$pK\")F*$\"3$e O27.z1U$F*7$$\"3s****z(=?+H)F*$\"3;\\Z2QiQFSF*7$$\"3]L$e&)zM.Z)F*$\"3Q 7$Qw*Gi$y%F*7$$\"3EnmJ4%\\1l)F*$\"3GNx!\\F')[g&F*7$$\"3/+]sY\\#>\"))F* $\"3wp#3Y;CXR'F*7$$\"3#GLLT[+K(*)F*$\"3d'\\TAl\"=OsF*7$$\"3w\\(oV;x\"f !*F*$\"3XT0=\"zIhq(F*7$$\"3qmTgWQ:X\"*F*$\"3i\">3#4R'3>)F*7$$\"3j$eR[_ I6B*F*$\"3')[WK15Q!p)F*7$$\"3d+]20s5<$*F*$\"3>8$HD3#o/#*F*7$$\"3%*\\iv JDDe$*F*$\"3uDcmN%QgX*F*7$$\"3I*\\P%eyR*R*F*$\"37^thk1y5(*F*7$$\"3w5y5 !>%o4%*F*$\"3cN9&\\E&*\\x*F*7$$\"3*R7y<_q*>%*F*$\"3%Q*R_#[@%R)*F*7$$\" 3BP%[M&oDI%*F*F`t7$$\"3o[(=^=V0W*F*F`t7$$\"39v$f%[e6h%*F*F`t7$$\"3%)** **z6&)o\"[*F*F`t7$$\"3O+D;l\"zRc*F*F`t7$$\"35**\\_=)pik*F*F`t7$$\"3b* \\i#4\\8B)*F*F`t7$$\"#5FatF`t-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6 \"Q!F\\[m-%&COLORG6&%$RGBG$\"\"%!\"\"F`t$\"\"\"Fat-%%VIEWG6$;$!$G'!\"# Fbjl%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 43.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The constant co efficient in the Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c = 1/(2*Pi);" "6#/%\"cG*&\"\" \"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x \+ = -Pi .. Pi);" "6#-%$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 "Int(x^2,x = 0 .. Pi);" "6#-%$IntG6$*$%\"xG\"\"#/F';\"\"!%#PiG" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` =``(1/(2*Pi))*``(Pi^3/3)" "6#/%!G*&-F$6#*&\"\"\"F)*&\"\"#F)%#PiGF)!\" \"F)-F$6#*&F,\"\"$F1F-F)" }{XPPEDIT 18 0 "`` = Pi^2/6" "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 36 "The coefficients of the cosine terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 51 "The coeff icients of the cosine terms are given by: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = 1/Pi;" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#P iG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*k*x,x = -Pi .. P i);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF+%\"kGF+F*F+/F*;,$%#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 "Int(x^2*cos*k*x,x = 0 .. Pi);" "6#-%$IntG6$**%\"xG\"\"# %$cosG\"\"\"%\"kGF*F'F*/F';\"\"!%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "Int(x^2*cos*k*x,x) = Int(u*``( dv/dx),x);" "6#/-%$IntG6$**%\"xG\"\"#%$cosG\"\"\"%\"kGF+F(F+F(-F%6$*&% \"uGF+-%!G6#*&%#dvGF+%#dxG!\"\"F+F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x^2;" "6#/%\"uG*$%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = sin*k*x/k;" "6#/%\"vG**%$sinG\"\"\"%\"kGF'%\"xGF'F(!\"\"" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the integ ration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-In t(v*``(du/dx),x);" "6#/-%$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 0 "" 0 "" {TEXT -1 0 "" }}{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(x^2*cos*k*x,x = 0 \+ .. Pi);" "6#-%$IntG6$**%\"xG\"\"#%$cosG\"\"\"%\"kGF*F'F*/F';\"\"!%#PiG " }{TEXT -1 10 " ... " }{XPPEDIT 18 0 "PIECEWISE([u=x^2,v=sin*k*x /k],[du/dx=2*x,dv/dx=cos*k*x])" "6#-%*PIECEWISEG6$7$/%\"uG*$%\"xG\"\"# /%\"vG**%$sinG\"\"\"%\"kGF0F*F0F1!\"\"7$/*&%#duGF0%#dxGF2*&F+F0F*F0/*& %#dvGF0F7F2*(%$cosGF0F1F0F*F0" }{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 "``(x^2*sin*k*x/k); " "6#-%!G6#*,%\"xG\"\"#%$sinG\"\"\"%\"kGF*F'F*F+!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWI SEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-1/Pi; " "6#,$*&\"\"\"F%%#PiG!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*x *sin*k*x/k,x = 0 .. Pi);" "6#-%$IntG6$*.\"\"#\"\"\"%\"xGF(%$sinGF(%\"k GF(F)F(F+!\"\"/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0-2/(k *Pi);" "6#/%!G,&\"\"!\"\"\"*&\"\"#F'*&%\"kGF'%#PiGF'!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sin*k*x,x = 0 .. Pi);" "6#-%$IntG6$**% \"xG\"\"\"%$sinGF(%\"kGF(F'F(/F';\"\"!%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "Int(x*sin*k*x,x) = Int(u* ``(dv/dx),x);" "6#/-%$IntG6$**%\"xG\"\"\"%$sinGF)%\"kGF)F(F)F(-F%6$*&% \"uGF)-%!G6#*&%#dvGF)%#dxG!\"\"F)F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = \+ -cos*k*x/k;" "6#/%\"vG,$**%$cosG\"\"\"%\"kGF(%\"xGF(F)!\"\"F+" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*`` (du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"x G,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 16 ", again we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[ k]=-2/(k*Pi)" "6#/&%\"aG6#%\"kG,$*&\"\"#\"\"\"*&F'F+%#PiGF+!\"\"F." } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sin*k*x,x = 0 .. Pi)" "6#-%$IntG6 $**%\"xG\"\"\"%$sinGF(%\"kGF(F'F(/F';\"\"!%#PiG" }{TEXT -1 7 " ... \+ " }{XPPEDIT 18 0 "PIECEWISE([u=x,v=-cos*k*x/k],[du/dx=1,dv/dx=sin*k*x] )" "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%\"vG,$**%$cosG\"\"\"%\"kGF/F)F/F0! \"\"F17$/*&%#duGF/%#dxGF1F//*&%#dvGF/F6F1*(%$sinGF/F0F/F)F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = -2/ (k*Pi);" "6#/&%\"aG6#%\"kG,$*&\"\"#\"\"\"*&F'F+%#PiGF+!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-x*cos*k*x/k);" "6#-%!G6#,$*,%\"xG\"\"\"%$ cosGF)%\"kGF)F(F)F+!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE( [Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"! F(" }{TEXT -1 2 "+ " }{XPPEDIT 18 0 "2/(k*Pi);" "6#*&\"\"#\"\"\"*&%\"k GF%%#PiGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(-cos*k*x/k,x = 0 \+ .. Pi);" "6#-%$IntG6$,$**%$cosG\"\"\"%\"kGF)%\"xGF)F*!\"\"F,/F+;\"\"!% #PiG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``=2*Pi*cos*k*Pi/(Pi*k^2)-``(2*si n*k*x/(Pi*k^2))" "6#/%!G,&*.\"\"#\"\"\"%#PiGF(%$cosGF(%\"kGF(F)F(*&F)F (*$F+F'F(!\"\"F(-F$6#*,F'F(%$sinGF(F+F(%\"xGF(*&F)F(*$F+F'F(F.F." } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 2*cos*k*Pi/(k^2)+0;" "6#/%!G,&*,\"\"#\"\"\"%$cosGF(%\"kGF(% #PiGF(*$F*F'!\"\"F(\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "(-1)^k*` `(2/(k^2));" "6#*&),$\"\"\"!\"\"%\"kGF&-%!G6#*&\"\"#F&*$F(F-F'F&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "PIECEWISE([2/(k^2), `if k is even`], [-2/(k^2), `if k is odd`]);" "6#-%*PIECEWISEG6$7$*&\"\"#\"\"\"*$%\"kGF (!\"\"%-if~k~is~evenG7$,$*&F(F)*$F+F(F,F,%,if~k~is~oddG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "matrix([[k,`|`,1, 2,3,4,5,6,7,8],[a[k],`|`,-2, 1/2, -2/9, 1/8, -2/25, 1/18, -2/49, 1/32] ])" "6#-%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\" (\"\")7,&%\"aG6#F(F),$F+!\"\"*&F*F*F+F7,$*&F+F*\"\"*F7F7*&F*F*F1F7,$*& F+F*\"#DF7F7*&F*F*\"#=F7,$*&F+F*\"#\\F7F7*&F*F*\"#KF7" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "interface(showassumed=0): k := 'k': assume(k,integer ):\na[k]=1/Pi*Int(x^2*cos(k*x),x=0..Pi);\n``=value(rhs(%));\naa := una pply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..8)],['a'[k],`|`,s eq(aa(k),k=1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%#k|i rG,$-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$cosG6#*&F'F0F.F0F0/F.;\"\"!%#PiG*$F 8!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#k|irG! \"#)!\"\"F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,%\" kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&%\"aG6#F(F)!\"##F *F+#F6\"\"*#F*F1#F6\"#D#F*\"#=#F6\"#\\#F*\"#KQ)pprint456\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "The coefficients of the sine \+ terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 48 "The coefficients of the sine terms are given by:" }} {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 "Int(x^2*sin*k*x,x = 0 .. Pi);" "6#-% $IntG6$**%\"xG\"\"#%$sinG\"\"\"%\"kGF*F'F*/F';\"\"!%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "Int(x^2*sin* k*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$**%\"xG\"\"#%$sinG\"\"\"%\" kGF+F(F+F(-F%6$*&%\"uGF+-%!G6#*&%#dvGF+%#dxG!\"\"F+F(" }{TEXT -1 8 ", \+ where " }{XPPEDIT 18 0 "u = x^2;" "6#/%\"uG*$%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*k*x/k;" "6#/%\"vG,$**%$cosG\"\"\"%\"kG F(%\"xGF(F)!\"\"F+" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "H ence, using the integration by parts formula: " }{XPPEDIT 18 0 "Int(u* ``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G 6#*&%#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] = 1/Pi;" "6#/&%\"bG6#%\"kG*&\"\"\"F)%#PiG!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2*cos*k*x,x = 0 .. Pi);" "6#-% $IntG6$**%\"xG\"\"#%$cosG\"\"\"%\"kGF*F'F*/F';\"\"!%#PiG" }{TEXT -1 10 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = x^2, v = -cos*k*x/k],[d u/dx = 2*x, dv/dx = sin*k*x]);" "6#-%*PIECEWISEG6$7$/%\"uG*$%\"xG\"\"# /%\"vG,$**%$cosG\"\"\"%\"kGF1F*F1F2!\"\"F37$/*&%#duGF1%#dxGF3*&F+F1F*F 1/*&%#dvGF1F8F3*(%$sinGF1F2F1F*F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` (-x^2*cos*k*x/k);" "6#-%!G6#,$*,%\"xG\"\"#%$cosG\"\"\"%\"kGF+F(F+F,!\" \"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, \+ ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-1/Pi;" "6#,$*&\"\"\"F%%#PiG!\"\"F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(2*x*``(-cos*k*x/k),x = 0 .. Pi);" "6#-%$IntG6$*(\" \"#\"\"\"%\"xGF(-%!G6#,$**%$cosGF(%\"kGF(F)F(F0!\"\"F1F(/F);\"\"!%#PiG " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -Pi^2*cos*k*Pi/(k*Pi)+2/(k*Pi);" " 6#/%!G,&*,%#PiG\"\"#%$cosG\"\"\"%\"kGF*F'F**&F+F*F'F*!\"\"F-*&F(F**&F+ F*F'F*F-F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*cos*k*x,x = 0 .. Pi) ;" "6#-%$IntG6$**%\"xG\"\"\"%$cosGF(%\"kGF(F'F(/F';\"\"!%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int(x*co s*k*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$**%\"xG\"\"\"%$cosGF)%\"k GF)F(F)F(-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxG!\"\"F)F(" }{TEXT -1 8 ", w here " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "v = sin*k*x/k;" "6#/%\"vG**%$sinG\"\"\"%\"kGF'%\"xGF'F( !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "Applying the in tegration 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$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 17 " to the integral " }{XPPEDIT 18 0 "Int(x*cos*k*x,x = 0 .. Pi)" "6#-%$IntG6$**%\"xG\"\"\"%$cosGF(%\"kGF(F'F(/F';\"\"!%#PiG" }{TEXT -1 7 " with " }{XPPEDIT 18 0 "PIECEWISE([u = x, v = sin*k*x/k],[du/dx = \+ 1, dv/dx = cos*k*x])" "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%\"vG**%$sinG\" \"\"%\"kGF.F)F.F/!\"\"7$/*&%#duGF.%#dxGF0F./*&%#dvGF.F5F0*(%$cosGF.F/F .F)F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k] = -Pi^2*cos(k*Pi)/(k*P i)+2/(k*Pi);" "6#/&%\"bG6#%\"kG,&*(%#PiG\"\"#-%$cosG6#*&F'\"\"\"F*F0F0 *&F'F0F*F0!\"\"F2*&F+F0*&F'F0F*F0F2F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(x*sin*k*x/k);" "6#-%!G6#*,%\"xG\"\"\"%$sinGF(%\"kGF(F'F(F*!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-2/(k*Pi);" "6#,$*&\"\"#\"\"\"*&%\"kGF&%#PiGF&!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*k*x/k,x = 0 .. Pi);" "6#-%$Int G6$**%$sinG\"\"\"%\"kGF(%\"xGF(F)!\"\"/F*;\"\"!%#PiG" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=(-1)^(k+1)*``(Pi/k)+0+2/(Pi*k^3)" "6#/%!G,(*&),$\"\" \"!\"\",&%\"kGF)F)F)F)-F$6#*&%#PiGF)F,F*F)F)\"\"!F)*&\"\"#F)*&F0F)*$F, \"\"$F)F*F)" }{XPPEDIT 18 0 "``(cos*k*x);" "6#-%!G6#*(%$cosG\"\"\"%\"k GF(%\"xGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, `` ],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = (-1)^(k+1)*``(Pi/k)+2/(Pi*k^3);" "6#/%!G,&*&),$ \"\"\"!\"\",&%\"kGF)F)F)F)-F$6#*&%#PiGF)F,F*F)F)*&\"\"#F)*&F0F)*$F,\" \"$F)F*F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(cos*k*Pi-1);" "6#-%!G6#, &*(%$cosG\"\"\"%\"kGF)%#PiGF)F)F)!\"\"" }{TEXT -1 3 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (-1)^(k+1)*``(Pi/k)+2/(Pi*k^3);" "6#/%!G,&*&),$\"\"\"!\"\",&% \"kGF)F)F)F)-F$6#*&%#PiGF)F,F*F)F)*&\"\"#F)*&F0F)*$F,\"\"$F)F*F)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``((-1)^k-1);" "6#-%!G6#,&),$\"\"\"!\" \"%\"kGF)F)F*" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([-Pi/k , `if k is even`],[Pi/k-4/(Pi*k^3), `if k is odd`]);" "6#/%!G-%*PIECEW ISEG6$7$,$*&%#PiG\"\"\"%\"kG!\"\"F.%-if~k~is~evenG7$,&*&F+F,F-F.F,*&\" \"%F,*&F+F,*$F-\"\"$F,F.F.%,if~k~is~oddG" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8], [b[k], `|`, Pi-4/Pi, -Pi/ 2, Pi/3-4/27/Pi, -Pi/4, Pi/5-4/125/Pi, -Pi/6, Pi/7-4/343/Pi, -Pi/8]]); " "6#-%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"( \"\")7,&%\"bG6#F(F),&%#PiGF**&F-F*F7!\"\"F9,$*&F7F*F+F9F9,&*&F7F*F,F9F **(F-F*\"#FF9F7F9F9,$*&F7F*F-F9F9,&*&F7F*F.F9F**(F-F*\"$D\"F9F7F9F9,$* &F7F*F/F9F9,&*&F7F*F0F9F**(F-F*\"$V$F9F7F9F9,$*&F7F*F1F9F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "interface(showassumed=0): k := 'k': assume(k,integer ):\nb[k]=1/Pi*Int(x^2*sin(k*x),x=0..Pi);\n``=expand(value(rhs(%)));\nb b := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..8)],['b'[ k],`|`,seq(bb(k),k=1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#%#k|irG,$-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$sinG6#*&F'F0F.F0F0/F.;\"\"! %#PiG*$F8!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"#\"\"\"% #PiG!\"\"%#k|irG!\"$F***F'F(F)F*F+F,)F*F+F(F(*(F)F(F+F*F.F(F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\" #\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&%\"bG6#F(F),&*&F-F*%#PiG!\"\"F9F8F*, $*&F+F9F8F*F9,&*(F-F*\"#FF9F8F9F9*&F,F9F8F*F*,$*&F-F9F8F*F9,&*(F-F*\"$ D\"F9F8F9F9*&F.F9F8F*F*,$*&F/F9F8F*F9,&*(F-F*\"$V$F9F8F9F9*&F0F9F8F*F* ,$*&F1F9F8F*F9Q)pprint466\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " F(x) = Pi^2/6+``" "6#/-%\"FG6#%\"xG,&*&%#PiG\"\"#\"\"'!\"\"\"\"\"%!GF. " }{XPPEDIT 18 0 "Sum(``(2*(-1)^k/(k^2))*cos*k*x-``((k^2*Pi^2*(-1)^k-2 *(-1)^k+2)/(Pi*k^3))*sin*k*x,k = 1 .. infinity);" "6#-%$SumG6$,&**-%!G 6#*(\"\"#\"\"\"),$F-!\"\"%\"kGF-*$F1F,F0F-%$cosGF-F1F-%\"xGF-F-**-F)6# *&,(*(F1F,%#PiGF,),$F-F0F1F-F-*&F,F-),$F-F0F1F-F0F,F-F-*&F;F-*$F1\"\"$ F-F0F-%$sinGF-F1F-F4F-F0/F1;F-%)infinityG" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi^2/6-2*cos*x-(Pi-4/Pi)*sin*x+cos*2*x/2-Pi/2*sin*2*x-2/9*cos* 3*x-(Pi/3-4/27/Pi)*sin*3*x+` . . . `;" "6#/%!G,2*&%#PiG\"\"#\"\"'!\"\" \"\"\"*(F(F+%$cosGF+%\"xGF+F**(,&F'F+*&\"\"%F+F'F*F*F+%$sinGF+F.F+F*** F-F+F(F+F.F+F(F*F+*,F'F+F(F*F3F+F(F+F.F+F**,F(F+\"\"*F*F-F+\"\"$F+F.F+ F***,&*&F'F+F8F*F+*(F2F+\"#FF*F'F*F*F+F3F+F8F+F.F+F*%(~.~.~.~GF+" } {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 fi nite truncated Fourier series as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "FS := (x,n) -> Pi^2/ 6+sum(2*(-1)^k/(k^2)*cos(k*x)-\n (k^2*Pi^2*(-1)^k-2*(-1)^k+2)/(Pi*k^ 3)*sin(k*x),k=1..n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\" xG%\"nG6\"6$%)operatorG%&arrowGF),&*&\"\"'!\"\"%#PiG\"\"#\"\"\"-%$sumG 6$,&**F2F3)F0%\"kGF3F:!\"#-%$cosG6#*&F:F39$F3F3F3**,(*()F:F2F3)F1F2F3F 9F3F3*&F2F3F9F3F0F2F3F3F1F0F:!\"$-%$sinGF>F3F0/F:;F39%F3F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The trunc ated Fourier series involving terms as far as the terms in " } {XPPEDIT 18 0 "cos(4*x);" "6#-%$cosG6#*&\"\"%\"\"\"%\"xGF(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin(4*x);" "6#-%$sinG6#*&\"\"%\"\"\"%\"xGF( " }{TEXT -1 10 " are . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "FS(x,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*&\"\"'!\"\"%#PiG\"\"#\"\"\"*&F(F)-%$cosG6#%\"xGF)F&* (,&*$)F'F(F)F&\"\"%F)F)F'F&-%$sinGF-F)F&*&#F)F(F)-F,6#,$*&F(F)F.F)F)F) F)*&#F)F(F)*&F'F)-F5F9F)F)F&*&#F(\"\"*F)-F,6#,$*&\"\"$F)F.F)F)F)F&*&#F )\"#FF)*(,&*&FBF)F2F)F&F3F)F)F'F&-F5FDF)F)F&*&#F)\"\")F)-F,6#,$*&F3F)F .F)F)F)F)*&#F)F3F)*&F'F)-F5FSF)F)F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture compares the gra phs of some truncated 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 0 329 "f := x -> piecewise(x<0,0,x^2):\n'f(x)'=f(x);\nf_ := x -> f(x-2*Pi*floor((x+ Pi)/(2*Pi))):\nFS := (x,n) -> Pi^2/6+sum(2*(-1)^k/(k^2)*cos(k*x)-\n \+ (k^2*Pi^2*(-1)^k-2*(-1)^k+2)/(Pi*k^3)*sin(k*x),k=1..n):\nplot([f_(x),F S(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5)],x=-6.28..10,\n color=[black, red,blue,green,magenta,coral],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"!2F'F,7$*$)F'\"\"#\" \"\"%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 664 297 297 {PLOTDATA 2 "6*-%'CURVESG6%7^r7$$!3E++++++!G'!#<$\"3C,A$G==Y,\"!#A7$$!3/n;a#e&G \">'F*$\"3#*p`_)QfbW)!#?7$$!3(HL$3l6d-hF*$\"3=cl^Ko9iK!#>7$$!3y**\\iZn &Q,'F*$\"3GU[<*H(y`sF97$$!3emm;IB9DfF*$\"3=TF)QtZ>G\"!#=7$$!3i*\\7'\\C wqdF*$\"37(=s&\\=xDEFD7$$!3bL$e!pDQ;cF*$\"35***fc\"*eiW%FD7$$!3]mm^D* \\\"p_F*$\"3RjUEbxEG5F*7$$!31nm1iFQ%4&F*$\"3EapE\"\\^KT\"F*7$$!3vmmh)f :'>\\F*$\"31l.6]BKf=F*7$$!3e*\\()H2zcu%F*$\"3mJ&z,UDRO#F*7$$!3IL$etaU< d%F*$\"3\"oeW,KO!HHF*7$$!3\")**\\nA6[5WF*$\"31))4mi4-2NF*7$$!3An;*zp># \\UF*$\"3C#=f$*4;q8%F*7$$!3EM$eL$GC#3%F*$\"3b7/2tx9W[F*7$$!3V+]sofE:RF *$\"3L.sU$*>/2cF*7$$!3kL$ekDyDu$F*$\"3)4auh=%oakF*7$$!3%om\">W0*)pNF*$ \"3Opw&3Zo>O(F*7$$!3S++!RfBQ[$F*$\"3@=cx,gUOyF*7$$!3]L$3OkcxR$F*$\"3O' eeDd)pD$)F*7$$!3jmmJ$p*o6LF*$\"3Fql?$='yH))F*7$$!3=+]-VFiDKF*$\"3En&>P $))o[$*F*7$$!30@!)oL,\\.KF*$\"31UsihB_%[*F*7$$!3!>/^V_d8=$F*$\"3)G/FVf N8i*F*7$$!3M_Do>7HqJF*$\"3!>*R(**f4,p*F*7$$!3yiS,:\\AfJF*$\"3cr*==`G\" f(*F*7$$!3?tbM5'e\"[JF*$\"3$=)>')*Q#RG)*F*7$$!3k$3xcI#4PJF*$\"\"!Fat7$ $!3ODJ+(3FG4$F*F`t7$$!3km\"H$o=c[IF*F`t7$$!3m\\7)4VJ+'HF*F`t7$$!35LLj$ *4]rGF*F`t7$$!3')**\\#)yca:FF*F`t7$$!3imm,k.ffDF*F`t7$$!3`*****3(GX3AF *F`t7$$!3E++]tN(e&=F*F`t7$$!3x****4*3)4;:F*F`t7$$!3^m;>)R\\v?\"F*F`t7$ $!3\"GLLBh^lS)FDF`t7$$!3UILLphZ)H&FDF`t7$$!3#=+]#\\b/$o\"FDF`t7$$!3uqs ;H%>6H)F3F`t7$$\"3TcmmgJA<:FD$\"3v=LE>h'>I#F97$$\"33F$3d:CGF$FD$\"3U#z =azP62\"FD7$$\"3u(**\\2:D%G]FD$\"3#R^p'\\f]GDFD7$$\"3B++]hC<+nFD$\"3E( H%Q,6B*[%FD7$$\"3s-+Ds(>>P)FD$\"3C`!*HnS!*3qFD7$$\"33nmTl00'=\"F*$\"3r &>DP%fr19F*7$$\"3#p;H$36BY8F*$\"36xJ.(>QB\"=F*7$$\"3wm;C^;T1:F*$\"3'[k '*H1w#pAF*7$$\"3?+DT1!)=z;F*$\"3%oez3Os'>GF*7$$\"3mLLehV'>&=F*$\"3hkZd '*>xHMF*7$$\"3cLepO/VJ?F*$\"3uCU8>'4n7%F*7$$\"3YL$3=^'*3@#F*$\"35ci#fQ j!))[F*7$$\"3%ommxDArO#F*$\"3ZiV@$ynKg&F*7$$\"3B+]s.![L_#F*$\"3)R%H!z9 &GnjF*7$$\"3b$e9l/7xg#F*$\"3&zFPv6i,!oF*7$$\"3'o;/$*3w?p#F*$\"3Q;KgqOF ZsF*7$$\"3=]P4K,WwFF*$\"3+h252)>'3xF*7$$\"3]LL)[@*F* 7$$\"3S]i&H&\\pyIF*$\"3A*GeLhi$y%*F*7$$\"3^L3d3JFAJF*$\"3qwX]k$*e[(*F* 7$$\"3izWZZw;LJF*$\"30l2=2&Rn\")*F*7$$\"3HD\"yj=iS9$F*F`t7$$\"3'4x\"GD n&\\:$F*F`t7$$\"3i;a=k7&e;$F*F`t7$$\"3S3F*>MSw=$F*F`t7$$\"3=++!)>%H%4K F*F`t7$$\"3Q**\\d8.'*zLF*F`t7$$\"3c)**\\t?\"\\]NF*F`t7$$\"33+](Q-'[!)Q F*F`t7$$\"3!****\\]+*)oC%F*F`t7$$\"3Emm1!Q=hd%F*F`t7$$\"3Z****\\55kF\\ F*F`t7$$\"3MK$3-wshC&F*F`t7$$\"3E++!z47Wf&F*F`t7$$\"3tk;uv\"y?#fF*F`t7 $$\"3SK$ex!4L$4'F*F`t7$$\"35+]xROekiF*F`t7$$\"3i;Hij!*H[jF*$\"3P]8;y7! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "f := x -> piecewise(x<0,0,x ^2):\n'f(x)'=f(x);\nFourierSeries(f(x),x=-Pi..Pi,numterms=8,info=1):\n F := unapply(%,x):\n'F(x)'=F(x);\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2* Pi))):\nplot([f_(x),F(x)],x=-6..10,numpoints=80,\n color= [black,COLOR(RGB,.8,0,1)],linestyle=[3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"!2F'F,7$*$)F'\"\"#\" \"\"%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coeffic ient~-->G,$*&\"\"'!\"\"%#PiG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%!)!#>7$$!3YM()*=b)4)y&!#<$\"35 ?_a_*36X#!#=7$$!3!*3n$\\BCPg&F1$\"3g0W$p[tmh%F47$$!3ej]SKgP'R&F1$\"3g[ 6E)*oIkyF47$$!3y/CR\"R`w=&F1$\"3)>L3&y,>+7F17$$!3EG#eExA*z\\F1$\"3wLq) QB$\\)p\"F17$$!31>:s(*)Hty%F1$\"3(>'\\+1MePAF17$$!3)3CR686ze%F1$\"33s# \\>`aR(GF17$$!3Ca$ou&>n\"Q%F1$\"3AI+8>Iv:OF17$$!3sPIW&>%4wTF1$\"3/k\"f J&H$)RWF17$$!3Y75[)>KY'RF1$\"3o?e&y^)ov`F17$$!3Ma$ouF/:(QF1$\"3tF!p!*Q 0i\"eF17$$!3E'pbkNw$yPF1$\"3O9lizy1uiF17$$!35WNouj`tOF1$\"3RKx(3bn-\"o F17$$!3S#R6HR'poNF1$\"37=\\ol+XotF17$$!3/m75GfUjMF1$\"3'Q:+f&H/^zF17$$ !3AR6Hja:eLF1$\"3#>2Qwe*zb&)F17$$!3W(z.V'yqcKF1$\"3%R(fB7dcf\"*F17$$!3 mbkJl-EbJF1$\"3EnW^n]\"Ry*F17$$!3'*pbk1D]\\JF1$\"3c:FO%*y'*>)*F17$$!3% QouzuWP9$F1$\"3Y*y!yCq3c)*F17$$!3r(z.$*)p)z8$F1$F*F*7$$!3f6HjI#HA8$F1F ar7$$!3MR6H8Pr?JF1Far7$$!33n$\\f>)>4JF1Far7$$!3eAeEhr;'3$F1Far7$$!3_yA eEh8jIF1Far7$$!3&**=:s0uq,$F1Far7$$!3S,\"[y)>,rHF1Far7$$!3W'pbkZn9'GF1 Far7$$!3/\"Hj]'H#>v#F1Far7$$!3QE,\"[#eKmDF1Far7$$!3wf2')GAV]BF1Far7$$! 3oWNo9)H$f@F1Far7$$!3?[yAy,m\\>F1Far7$$!3S^@x,^+]F1$\"3R;B7,d*3h$F17$$\"3jY;`ib&=5#F1$\"3PC_u0oz D^#F1$\"3+gxc#G_F J'F17$$\"349\"Hj=mkg#F1$\"3GOqZ!)fm$z'F17$$\"3p^@xhKT+FF1$\"3#=\"\\N%y JAH(F17$$\"3F)*=:7s33GF1$\"3_?50\"z``)yF17$$\"3'[kJD;hd\"HF1$\"3@))y#p :j;])F17$$\"3Urm*F17$$\"3uID?Eg?@JF1$\"33$*H+eq#>u*F17$$\"3$R;`-;TS8$F1$\"3])*4;%*R@ A)*F17$$\"3u![ysse/9$F1$\"3'edRx,\"[i)*F17$$\"37(z.VHwo9$F1Far7$$\"3]8 \"H8'QH`JF1Far7$$\"3')HWNG9rfJF1Far7$$\"3!epbkp\"Q&=$F1Far7$$\"3thpbk> 06KF1Far7$$\"3i$\\f2]#RiKF1Far7$$\"3[D?'p.LPJ$F1Far7$$\"3V\"Hj]#>,7MF1 Far7$$\"3QdX;83H5NF1Far7$$\"3hvz.jHaGPF1Far7$$\"3Lk75)GLv\"RF1Far7$$\" 3oa$ou**Q?8%F1Far7$$\"3ZE,\"[mPvK%F1Far7$$\"3auz.$[#QTXF1Far7$$\"3+m$ \\fR%*es%F1Far7$$\"3Q3')3F>.O\\F1Far7$$\"34m753W,R^F1Far7$$\"3c?'pbGk= M&F1Far7$$\"3#)4n$\\\"z'Ra&F1Far7$$\"3GM()*=BE\"QdF1Far7$$\"3/!QIW^;![ fF1Far7$$\"3sG#eEMSx=\"F17$$\"3cz.VaRjmvF1 $\"3I\"*)H,VSsk\"F17$$\"39\\f213#\\w(F1$\"3m`'*o)3Sb>#F17$$\"3ES#R60`Q )zF1$\"3M>[Mv2F#*GF17$$\"3G7\"HjEy?=)F1$\"3WVwIqWz0OF17$$\"3%Q,\"[y1![ Q)F1$\"3(yV!*)ory;WF17$$\"3;=M()\\T'4f)F1$\"3C#G&y'=VeK&F17$$\"3aM10Cj r&o)F1$\"3_Lu-(Gb@x&F17$$\"39\\yA)\\o/y)F1$\"3P%>#HOLUOiF17$$\"3yP\\fZ 7Z\")))F1$\"3_(3g\"G)*3^nF17$$\"3UE?'p*RZ#)*)F1$\"3q-uyI%fhG(F17$$\"3E Va$o!zq#3*F1$\"3w()Q)fyEt$yF17$$\"37g)3n\"=%H=*F1$\"3B)RdX\"ye3%)F17$$ \"3#)ouzV4;!H*F1$\"3v?N@zC!>/*F17$$\"3]xg)32!Q(R*F1$\"3OHpHE!4#)p*F17$ $\"34mJD+r?4%*F1$\"3qf4x]A,s(*F17$$\"3)GD?'HT.@%*F1$\"3#zF#\\K_4Y)*F17 $$\"31)z.VkZpU*F1Far7$$\"3XTt)*e6'GV*F1Far7$$\"3&[)3ntYxQ%*F1Far7$$\"3 -IWN)=)oW%*F1Far7$$\"3=2')3ZAMo%*F1Far7$$\"3I%yAeI'*>\\*F1Far7$$\"3fQ6 HBWIR&*F1Far7$$\"3)G\\f2a7me*F1Far7$$\"3sx.VM+y&p*F1Far7$$\"3Nk75Gv%\\ !)*F1Far7$$\"#5F*Far-%'COLOURG6&%$RGBGF*F*F*-%*LINESTYLEG6#\"\"$-F$6%7 a]l7$F($!3'e,YTCkkO&F-7$$!3An$\\fF\\S*eF1$\"3o!=*=:p%)*o\"F47$F/$\"3!z c/K4TN/%F47$$!3HG#eE(\\+UdF1$\"3X$4[%z9([![F47$$!37AxT$R6fp&F1$\"3*ecU oSNWK&F47$$!33:s<9y\")\\cF1$\"3CHO!z#\\F)f&F47$F6$\"3Kj1#fcogm&F47$$!3 !yz.V=()=b&F1$\"3AQN`MhK%f&F47$$!3!e)3nL,0+bF1$\"3j#zFI&*4>^&F47$$!3!) 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Limit(f(x),x=Pi,left)=Pi^2" "6#/-% &LimitG6%-%\"fG6#%\"xG/F*%#PiG%%leftG*$F,\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "Limit(f(x),x=Pi,right)=0" "6#/-%&LimitG6%-%\"fG6#%\"xG /F*%#PiG%&rightG\"\"!" }{TEXT -1 34 ", the Fourier series converges to " }{XPPEDIT 18 0 "Pi^2/2" "6#*&%#PiG\"\"#F%!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "It follow s that " }{XPPEDIT 18 0 "Sum(2/(k^2),k = 1 .. infinity)=Pi^2/2-Pi^2/ 6" "6#/-%$SumG6$*&\"\"#\"\"\"*$%\"kGF(!\"\"/F+;F)%)infinityG,&*&%#PiGF (F(F,F)*&F2F(\"\"'F,F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Pi^2/3" "6#* &%#PiG\"\"#\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "We may conclude that:" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(k^2),k = 1 .. infinity)=1 +1/4+1/9+1/16+1/25+1/36+` . . . ` " "6#/-%$SumG6$*&\"\"\"F(*$%\"kG\"\" #!\"\"/F*;F(%)infinityG,0F(F(*&F(F(\"\"%F,F(*&F(F(\"\"*F,F(*&F(F(\"#;F ,F(*&F(F(\"#DF,F(*&F(F(\"#OF,F(%(~.~.~.~GF(" }{XPPEDIT 18 0 "`` = Pi^2 /6" "6#/%!G*&%#PiG\"\"#\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{TEXT 262 28 "____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Maple \"knows\" this result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Sum(1/(k^2),k=1..infinity);\n``=value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*$)%\"kG\"\"#F'!\" \"/F*;F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"'! \"\"%#PiG\"\"#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Theoretical considerations suggest that the Fourier se ries converges to " }{XPPEDIT 18 0 "Pi^2/2" "6#*&%#PiG\"\"#F%!\"\"" } {TEXT -1 6 " when " }{XPPEDIT 18 0 "x=Pi" "6#/%\"xG%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 60 "We can investigate the convergence \+ of the Fourier series at " }{XPPEDIT 18 0 "x=Pi" "6#/%\"xG%#PiG" } {TEXT -1 14 " numerically. " }}{PARA 0 "" 0 "" {TEXT -1 41 "A sum of t erms up as far as the terms in " }{XPPEDIT 18 0 "sin(100*x)" "6#-%$sin G6#*&\"$+\"\"\"\"%\"xGF(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(100* x)" "6#-%$cosG6#*&\"$+\"\"\"\"%\"xGF(" }{TEXT -1 56 " gives a value wh ich agrees with the numerical value of " }{XPPEDIT 18 0 "Pi^2/2" "6#*& %#PiG\"\"#F%!\"\"" }{TEXT -1 18 " to just 2 digits." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "f := x -> piecewise(x<0,0,x^2):\neval(FourierSeries(f(x),x=-Pi..Pi,numterms=100 ),x=Pi);\nevalf(%);\nPi^2/2;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"'!\"\"%#PiG\"\"#\"\"\"#\"]p,*p.4J\"=Q![$GV*))f)4Pa)\\$Q- (fQb<&GzH7J(y.LTp3&*e\"\"\\p+Si#3$**eFNx,O\"*37ty*>z()43s:JO$))*G[H:0> " 0 "" {MPLTEXT 1 0 83 "eval(FourierSeries(f(x),x=-P i..Pi,numterms=4000),x=Pi):\nevalf(%);\nPi^2/2;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+kAIM\\!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"%#PiGF%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+-A![$\\!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 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 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "f(x) = PIECEWISE([0, -Pi <= x and x < 0],[sin*x, 0 <= x and x < Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"!31,$%#PiG!\"\"F '2F'F,7$*&%$sinG\"\"\"F'F631F,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 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 0 195 "f := x - > piecewise(x<0,0,sin(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..5*Pi,color=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"!2F'F,7$-%$sinGF&% *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 625 171 171 {PLOTDATA 2 "6'-%'CURVESG6# 7iw7$$!3)****>YH&=$G'!#<$\"3)fq*)fiefD\"!#D7$$!37$$!3Ex7\"4=\\L;'F*$\"3'QBiU5&\\&>\"!#=7$$!3N;p0C6V.hF*$\"3MG!\\ jAxyy\"F97$$!3VbD?nI^VgF*$\"3_,43zC%QP#F97$$!3_%>[.,&f$)fF*$\"3!p'3$[x (G^HF97$$!3hLQ\\`pnBfF*$\"3xRd%\\gS\"=NF97$$!3qs%Rm*)eP'eF*$\"3#e,'=yk OsSF97$$!3y6^yR3%Q!eF*$\"3KuurAi(>h%F97$$!33rt+y?d*p&F*$\"3-@*[7%oU5bF 97$$!3SI'HiJ.`f&F*$\"3;BKj'\\A!\\jF97$$!3q*)=XaX.\"\\&F*$\"3`F()HbUl=r F97$$!3+\\Tn#zlnQ&F*$\"3tIcOyA'4\"yF97$$!3M6d#\\m/&p_F*$\"3%3m$zJp$y[) F97$$!3ots'**F97$$!3:--vlOmSZF*$\"3+ac8xH+'***F9 7$$!3;O,WgWk\"o%F*$\"3/BlsWUF&***F97$$!310+#)\\ggjXF*$\"3Eg>i@A_*))*F9 7$$!3'R()*>RwcXWF*$\"3G*GfEuQhk*F97$$!3zX3aW-4GVF*$\"3moBE4dhq#*F97$$! 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Pi);" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$%#PiG!\"\"F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sin*x,x = 0 .. Pi);" "6#-%$IntG6$*&%$sinG\"\"\"%\"x GF(/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#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 36 "The coeffic ients of the cosine terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 59 "The Fourier coefficients of the cos ine terms are given by: " }}{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 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*x*cos*k*x,x = 0 .. Pi);" "6#-%$IntG6$*,%$sinG\"\"\"%\"xGF(%$cos GF(%\"kGF(F)F(/F);\"\"!%#PiG" }{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 "Int(sin((k+1)*x)-sin( (k-1)*x),x=0..Pi)" "6#-%$IntG6$,&-%$sinG6#*&,&%\"kG\"\"\"F-F-F-%\"xGF- F--F(6#*&,&F,F-F-!\"\"F-F.F-F3/F.;\"\"!%#PiG" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-cos((k+1)*x)/(k+1)+cos((k-1)*x)/ (k-1));" "6#-%!G6#,&*&-%$cosG6#*&,&%\"kG\"\"\"F.F.F.%\"xGF.F.,&F-F.F.F .!\"\"F1*&-F)6#*&,&F-F.F.F1F.F/F.F.,&F-F.F.F1F1F." }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG 6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "k< >1" "6#0%\"kG\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/ %!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(-cos((k+1)*Pi)/ (k+1)+cos((k-1)*Pi)/(k-1)-(-1/(k+1)+1/(k-1)))" "6#-%!G6#,(*&-%$cosG6#* &,&%\"kG\"\"\"F.F.F.%#PiGF.F.,&F-F.F.F.!\"\"F1*&-F)6#*&,&F-F.F.F1F.F/F .F.,&F-F.F.F1F1F.,&*&F.F.,&F-F.F.F.F1F1*&F.F.,&F-F.F.F1F1F.F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF &!\"\"" }{XPPEDIT 18 0 "``(-(-1)^(k+1)/(k+1)+(-1)^(k-1)/(k-1)-2/(k^2-1 ));" "6#-%!G6#,(*&),$\"\"\"!\"\",&%\"kGF*F*F*F*,&F-F*F*F*F+F+*&),$F*F+ ,&F-F*F*F+F*,&F-F*F*F+F+F**&\"\"#F*,&*$F-F5F*F*F+F+F+" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = ``((-1)^(k+1)-1)/(Pi*(k^2-1));" "6#/%!G*&-F$6#,&) ,$\"\"\"!\"\",&%\"kGF+F+F+F+F+F,F+*&%#PiGF+,&*$F.\"\"#F+F+F,F+F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Note: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[1] = 1/Pi;" "6#/&%\"aG6#\"\"\"*&F 'F'%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*x,x = -Pi \+ .. Pi);" "6#-%$IntG6$*(-%\"fG6#%\"xG\"\"\"%$cosGF+F*F+/F*;,$%#PiG!\"\" F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*x*cos*x,x = 0 .. Pi);" "6#-%$IntG6$**%$sinG\"\"\"%\"xGF( %$cosGF(F)F(/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*2*x/2,x = 0 .. Pi);" "6#-%$IntG 6$**%$sinG\"\"\"\"\"#F(%\"xGF(F)!\"\"/F*;\"\"!%#PiG" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G* &\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-cos*2*x/4)" " 6#-%!G6#,$**%$cosG\"\"\"\"\"#F)%\"xGF)\"\"%!\"\"F-" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG 6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{XPPEDIT 18 0 "``= 0" "6#/%!G\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "It follows that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[k] = PIECEWISE([-2/(Pi*(k^2-1)), `if k is even`],[0, \+ `if k is odd`]);" "6#/&%\"aG6#%\"kG-%*PIECEWISEG6$7$,$*&\"\"#\"\"\"*&% #PiGF/,&*$F'F.F/F/!\"\"F/F4F4%-if~k~is~evenG7$\"\"!%,if~k~is~oddG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k,`|`, 1, 2, 3, 4, 5, 6, 7, 8 \+ ],[a[k],`|`, 0,-2/(3*Pi), 0, -2/(15*Pi), 0, -2/(35*Pi), 0, -2/(63*Pi) ]])" "6#-%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\" \"(\"\")7,&%\"aG6#F(F)\"\"!,$*&F+F**&F,F*%#PiGF*!\"\"F;F6,$*&F+F**&\"# :F*F:F*F;F;F6,$*&F+F**&\"#NF*F:F*F;F;F6,$*&F+F**&\"#jF*F:F*F;F;" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "interface(showassumed=0): k := 'k': assume(k,in teger):\na[k]=1/Pi*Int(sin(x)*cos(k*x),x=0..Pi);\n``=value(rhs(%));\na a := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..8)],['a'[ k],`|`,0,seq(aa(k),k=2..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6#%#k|irG,$-%$IntG6$*&-%$sinG6#%\"xG\"\"\"-%$cosG6#*&F'F1F0F1F1/F0 ;\"\"!%#PiG*$F9!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(%#PiG! \"\",&\"\"\"F*)F(%#k|irGF*F*,&F*F(*$)F,\"\"#F*F*F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\" \"&\"\"'\"\"(\"\")7,&%\"aG6#F(F)\"\"!,$*(F+F*F,!\"\"%#PiGF9F9F6,$*(F+F *\"#:F9F:F9F9F6,$*(F+F*\"#NF9F:F9F9F6,$*(F+F*\"#jF9F:F9F9Q)pprint506\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "The coeffi cients of the sine terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 56 "The Fourier coefficients of the sine terms are given by:" }}{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 "Int(sin*x*si n*k*x,x = 0 .. Pi);" "6#-%$IntG6$*,%$sinG\"\"\"%\"xGF(F'F(%\"kGF(F)F(/ F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos((k-1)*x)-cos((k+1)*x),x=0..P i)" "6#-%$IntG6$,&-%$cosG6#*&,&%\"kG\"\"\"F-!\"\"F-%\"xGF-F--F(6#*&,&F ,F-F-F-F-F/F-F./F/;\"\"!%#PiG" }{TEXT -1 10 ", where " }{XPPEDIT 18 0 "k<>1" "6#0%\"kG\"\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#P iGF&!\"\"" }{XPPEDIT 18 0 "``(sin((k-1)*x)/(k-1)-sin((k+1)*x)/(k+1)); " "6#-%!G6#,&*&-%$sinG6#*&,&%\"kG\"\"\"F.!\"\"F.%\"xGF.F.,&F-F.F.F/F/F .*&-F)6#*&,&F-F.F.F.F.F0F.F.,&F-F.F.F.F/F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG %!G7$F(F(7$\"\"!F(" }{TEXT -1 10 " = 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b[1]=1/Pi " "6#/&%\"bG6#\"\"\"*&F'F'%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Int(sin^2*x,x = 0 .. Pi);" "6#-%$IntG6$*&%$sinG\"\"#%\"xG\"\"\"/F);\" \"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(``(1/2-cos*2*x/2),x = 0 .. Pi);" "6#-%$IntG6$-%! G6#,&*&\"\"\"F+\"\"#!\"\"F+**%$cosGF+F,F+%\"xGF+F,F-F-/F0;\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{XPPEDIT 18 0 "``(x/2-sin*2*x/4 );" "6#-%!G6#,&*&%\"xG\"\"\"\"\"#!\"\"F)**%$sinGF)F*F)F(F)\"\"%F+F+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " } {XPPEDIT 18 0 "``(1/Pi)*``(Pi/2) = 1/2;" "6#/*&-%!G6#*&\"\"\"F)%#PiG! \"\"F)-F&6#*&F*F)\"\"#F+F)*&F)F)F/F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "interface (showassumed=0): k := 'k': assume(k,integer):\nb[k]=1/Pi*Int(sin(x)*si n(k*x),x=0..Pi);\n``=value(rhs(%));\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$-%$IntG6$*&-%$sinG6#%\"xG\"\"\"-F.6#* &F'F1F0F1F1/F0;\"\"!%#PiG*$F8!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%!G\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "F(x) = 1/Pi+sin*x/2+``" "6#/-%\"FG6#%\"xG,(*&\"\"\"F*%# PiG!\"\"F**(%$sinGF*F'F*\"\"#F,F*%!GF*" }{XPPEDIT 18 0 "Sum(-``(2/(Pi* (4*k^2-1)))*cos*2*k*x,k = 1 .. infinity)" "6#-%$SumG6$,$*,-%!G6#*&\"\" #\"\"\"*&%#PiGF-,&*&\"\"%F-*$%\"kGF,F-F-F-!\"\"F-F5F-%$cosGF-F,F-F4F-% \"xGF-F5/F4;F-%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/Pi+sin*x/ 2-2/(3*Pi)*cos*2*x-2/(15*Pi)*cos*4*x-2/(35*Pi)*cos*6*x-2/(63*Pi)*cos*8 *x-` . . . `" "6#/%!G,0*&\"\"\"F'%#PiG!\"\"F'*(%$sinGF'%\"xGF'\"\"#F)F '*,F-F'*&\"\"$F'F(F'F)%$cosGF'F-F'F,F'F)*,F-F'*&\"#:F'F(F'F)F1F'\"\"%F 'F,F'F)*,F-F'*&\"#NF'F(F'F)F1F'\"\"'F'F,F'F)*,F-F'*&\"#jF'F(F'F)F1F'\" \")F'F,F'F)%(~.~.~.~GF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The Fourier series converges to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " for all values \+ of " }{TEXT 268 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture compares the g raphs of some truncated 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 0 284 "f := x -> piecewise(x<0,0,sin(x)):\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi ))):\nFS := (x,n) -> 1/Pi+1/2*sin(x)+sum(-2/(Pi*(4*k^2-1))*cos(2*k*x), k=1..n);\nplot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5)],x=-2*Pi ..3*Pi,\n color=[black,red,blue,green,magenta,coral],linestyle=[3,1$ 5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)ope ratorG%&arrowGF),(*&\"\"\"F/%#PiG!\"\"F/*&#F/\"\"#F/-%$sinG6#9$F/F/-%$ sumG6$,$**F4F/F0F1,&*&\"\"%F/)%\"kGF4F/F/F/F1F1-%$cosG6#,$*(F4F/FBF/F8 F/F/F/F1/FB;F/9%F/F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 629 153 153 {PLOTDATA 2 "6*-%'CURVESG6%7ct7$$!3)****>YH&=$G'!#<$\"3)fq*)fiefD\"!#D 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inestyle=[3,1],thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 624 153 153 {PLOTDATA 2 "6&-%'CURVESG6&7`u7$$!3)****>YH&=$G'!#<$\"3)fq*)fi efD\"!#D7$$!3@jRnao&!#>7$$!3aDzs9%o\"zhF*$\"3WL3$>D %HQ5!#=7$$!3!z)=yu*fr7'F*$\"3St/&>eIRb\"F97$$!3C]e$[`^^2'F*$\"3?Wy$pmk `1#F97$$!37!*p\"Q+ZY)fF*$\"3]?,F$zM7%HF97$$!3'38)zsC9%*eF*$\"3oR7[*\\H Iz$F97$$!3+7S4i1O#z&F*$\"3.BHA&RLNr%F97$$!3;$*)*Q^)y0p&F*$\"3qr5b2%\\_ e&F97$$!3ATaxkB7)e&F*$\"3:g449yL/kF97$$!3G*)4;yem&[&F*$\"3ev*y>bci:(F9 7$$!3G%zU$\\jp$Q&F*$\"3*RD+XQ\"4IyF97$$!3I*fC0#os\"G&F*$\"3\"G@*\\K6eA %)F97$$!3qq=I9!)=(=&F*$\"3ihq!=8YQ*))F97$$!3+V\"z!3#\\E4&F*$\"3-F:b.3o &G*F97$$!3Yvo8](fZ*\\F*$\"30%>O[ivRg*F97$$!3+2Y>#Hqo*[F*$\"33%yrHH:.$) *F97$$!3U,lZ$e^i%[F*$\"3:WqCGx`5**F97$$!3s'ReZ(Gj&z%F*$\"39H&zLFs`'**F 97$$!3.#HSg;9]u%F*$\"3%*o%F*$\"3KmxV.G!Ro)F97$ $!3yQ(HsQ6@5%F*$\"3'>-<=i*)[>)F97$$!3w#R)ziIKX]oF97$$!3e'Hfy_4Hz$F*$\"3+#>%fHAMigF97$$!3'o a84\\g&*o$F*$\"3^&*HUBo`4_F97$$!3EdaqVX'**e$F*$\"35;rS9$*)\\L%F97$$!3k 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\\&F*$!3)pEzGOX->$Fhv7$$\"3QA`\\R8wsbF*$\"35,tltzIikFc\\l7$$\"3]0;?=!z `l&F*$\"3Y[!eF5v'Fhv7$$\"3h))y!pp '*zt&F*$\"3S5`f+QjqnFhv7$$\"3GH5EObIzdF*$\"3r+f1&)f*\\J'Fhv7$$\"3$3<9c P91#eF*$\"3%fK')3GyQv%Fhv7$$\"3)=O'od#)Q,fF*$!3?QByBc6j5Fhv7$$\"3%Hbe( R@;#)fF*$!3ZErj0od.$)Fhv7$$\"3!*[Yz!3\\D-'F*$!3Ot>9i2z56F-7$$\"3)RuI=- OH1'F*$!3/<&H;$f?X7F-7$$\"31Ro'G'HK.hF*$!3S]k-XQ&*f6F-7$$\"3.NH!R!*4P9 'F*$!3YE2n![x`$yFhv7$$\"33Q()psEFGiF*$\"3R'oV#R%Gh@\"F-7$$\"3CSX\\Ta$G J'F*$\"3Z8LzhF=V^F-7$$\"3QU.H5#)R(R'F*$\"3\\>e#ytRD5\"F87$$\"3WXh3z4'> ['F*$\"3%o(=PrU]^=F87$$\"3'z(*ff(\\SjlF*$\"3SLI+_C\"om#F87$$\"3g4Q$G(* [[k'F*$\"3BHt[ts^4NF87$$\"3BTwqpHHEnF*$\"3m@vlF**)RK%F87$$\"3wt9empt2o F*$\"3'30lMLY]2&F87$$\"3cx?g;_-\"*oF*$\"39rIfZKamdF87$$\"3E#oAmY8V(pF* $\"3s?*>dvK3R'F87$$\"3%pGVmr,w0(F*$\"3u/dX>(3$opF87$$\"3v!*Qmm**)39(F* $\"38H4Oc'Qj^(F87$$\"3m%*H#za)[BsF*$\"32KpqP=yF*$\"3s!f4^:zs&**F87$$\"3%)e )fcxr='yF*$\"3gOg%)e6Qh**F87$$\"3%z**)*>[m`!zF*$\"3KA]j9-)ew)F*$\"3]F\")3Ue)f:'F87$$\"3_4=0%Rz_%))F*$\"3eH#y*4)R 'RbF87$$\"35R&f$o&yY#*)F*$\"3!3#yS\"))3t&[F87$$\"3oosmUx2/!*F*$\"3F:oM lr=0TF87$$\"3E)*\\(p\"pZ$3*F*$\"3[e?:_!z&*H$F87$$\"3c[P@t@!)o\"*F*$\"3 BLO(z$3Q=CF87$$\"3!*)\\_%Hu7a#*F*$\"3!f[e!p_,)e\"F87$$\"3?\\7p&o_%R$*F *$\"3(f34E%zxC%*F*$\"3`!HA^Xxn`$F--%*THICKNESSG6# \"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fchn-%'COLOURG6&%$RGBG$\"*++++\"!\")$\" \"!F]inF\\in-%%VIEWG6$;$!+3`=$G'!\"*$\"+izxC%*Fdin%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Define a fun ction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " by: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([cos *x, -Pi <= x and x < 0],[1-2*x/Pi, 0 <= x and x < Pi]);" "6#/-%\"fG6#% \"xG-%*PIECEWISEG6$7$*&%$cosG\"\"\"F'F.31,$%#PiG!\"\"F'2F'\"\"!7$,&F.F .*(\"\"#F.F'F.F2F3F331F5F'2F'F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " } {XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "In order \+ to draw the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 ", we start by defining " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "-Pi<=x" "6#1,$%#PiG !\"\"%\"xG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 202 "f := x - > piecewise(x<0,cos(x),1-2*x/Pi):\n'f(x)'=f(x);\nf_ := x -> f(x-2*Pi*f loor((x+Pi)/(2*Pi))):\n'f_(x)'='f(x-2*Pi*floor((x+Pi)/(2*Pi)))';\nplot (f_(x),x=-2*Pi..5*Pi,color=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$-%$cosGF&2F' \"\"!7$,&\"\"\"F2*(\"\"#F2%#PiG!\"\"F'F2F6%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-% &floorG6#,$*(F.!\"\",&F'F,F/F,F,F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 665 129 129 {PLOTDATA 2 "6'-%'CURVESG6#7]t7$$!3)****>YH&=$G' !#<$\"25>V+#********F*7$$!3VbD?nI^VgF*$\"3_U*H98*>u%)!#=7$$!3y6^yR3%Q! eF*$\"3y,n\"GM)R[pF27$$!3SI'HiJ.`f&F*$\"3\"R!o0Ns!3i&F27$$!3+\\Tn#zlnQ &F*$\"3;2pHFh@$H%F27$$!3otsQ,#y5?+GF27$$!3C*R!o\"G@x\"\\ F*$\"3Nie5Hg=28F27$$!3;O,WgWk\"o%F*$!3?9_')GpDd>!#>7$$!3'R()*>RwcXWF*$ !3u+*y[TP')p\"F27$$!3h<=))\\Gh5UF*$!3oTA!)fgS%>$F27$$!3QgPcg!ec(RF*$!3 G*eDZqu,p%F27$$!3%oQfX*[#yv$F*$!3[EPG5P%p2'F27$$!3K8]bG<**RNF*$!3#f'=% er7PY(F27$$!3V$)oa-\"QWJ$F*$!3k[;I'pJ'**))F27$$!3a`(QlZ%)))3$F*$!3W7([ fX7h)**F27$$!35\\L'Qf;c&GF*$!3a)RZ5pnQf*F27$$!35Xz=6([Bi#F*$!3kwr7;g%> o)F27$$!3;x1d@*G)*Q#F*$!3Z04SL]&[I(F27$$!3y3M&>84t:#F*$!3lrVF*$!38j\"4;HcRS$F27$$!3')p '[gNY&)z\"F*$!3gqNDuA'yD#F27$$!3aB/3k(e*y;F*$!3ur&*e)f;&z5F27$$!3-4>)z %fit:F*$!3+%[**4T<'HG!#?7$$!3]%R$)=8$Ho9F*$\"3&=-?j!zBB5F27$$!3'*z[y:. 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Pi);" "6#-%$In tG6$-%\"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&!\"\"" }{XPPEDIT 18 0 "``(Int(cos*x,x = -Pi .. 0)+Int(1- 2*x/Pi,x = 0 .. Pi));" "6#-%!G6#,&-%$IntG6$*&%$cosG\"\"\"%\"xGF,/F-;,$ %#PiG!\"\"\"\"!F,-F(6$,&F,F,*(\"\"#F,F-F,F1F2F2/F-;F3F1F," }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0;" "6# /%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "The coefficients of the cosine terms" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 59 "The Fourier coeff icients of the cosine terms are given by: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = 1/Pi;" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#P iG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*k*x,x = -Pi .. P i);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF+%\"kGF+F*F+/F*;,$%#PiG! \"\"F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " } {XPPEDIT 18 0 "1/Pi" "6#*&\"\"\"F$%#PiG!\"\"" }{XPPEDIT 18 0 "``(Int(c os*x*cos*k*x,x = -Pi .. 0)+Int((1-2*x/Pi)*cos*k*x,x = 0 .. Pi));" "6#- %!G6#,&-%$IntG6$*,%$cosG\"\"\"%\"xGF,F+F,%\"kGF,F-F,/F-;,$%#PiG!\"\"\" \"!F,-F(6$**,&F,F,*(\"\"#F,F-F,F2F3F3F,F+F,F.F,F-F,/F-;F4F2F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "k<>1" "6#0 %\"kG\"\"\"" }{TEXT -1 6 ", then" }}{PARA 256 "" 0 "" {TEXT -1 4 " \+ " }{XPPEDIT 18 0 "Int(cos*x*cos*k*x,x = -Pi .. 0) = 1/2;" "6#/-%$IntG6 $*,%$cosG\"\"\"%\"xGF)F(F)%\"kGF)F*F)/F*;,$%#PiG!\"\"\"\"!*&F)F)\"\"#F 0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos((k+1)*x)+cos((k-1)*x),x = - Pi .. 0);" "6#-%$IntG6$,&-%$cosG6#*&,&%\"kG\"\"\"F-F-F-%\"xGF-F--F(6#* &,&F,F-F-!\"\"F-F.F-F-/F.;,$%#PiGF3\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\" " }{XPPEDIT 18 0 "``(sin((k+1)*x)/(k+1)+sin((k-1)*x)/(k-1));" "6#-%!G6 #,&*&-%$sinG6#*&,&%\"kG\"\"\"F.F.F.%\"xGF.F.,&F-F.F.F.!\"\"F.*&-F)6#*& ,&F-F.F.F1F.F/F.F.,&F-F.F.F1F1F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIEC EWISE([0, ``],[``, ``],[-Pi, ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F( 7$,$%#PiG!\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= 0" "6#/%!G\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "while," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos^2*x,x = -Pi .. 0) = Int(``(1/2+cos*2*x/2),x = -Pi .. 0);" "6# /-%$IntG6$*&%$cosG\"\"#%\"xG\"\"\"/F*;,$%#PiG!\"\"\"\"!-F%6$-%!G6#,&*& F+F+F)F0F+**F(F+F)F+F*F+F)F0F+/F*;,$F/F0F1" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " } {XPPEDIT 18 0 "``(x/2+sin*2*x/4);" "6#-%!G6#,&*&%\"xG\"\"\"\"\"#!\"\"F )**%$sinGF)F*F)F(F)\"\"%F+F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWIS E([0, ``],[``, ``],[-Pi, ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$ %#PiG!\"\"F(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Pi/2;" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "No w " }{XPPEDIT 18 0 "Int((1-2*x/Pi)*cos*k*x,x) = Int(u*``(dv/dx),x);" " 6#/-%$IntG6$**,&\"\"\"F)*(\"\"#F)%\"xGF)%#PiG!\"\"F.F)%$cosGF)%\"kGF)F ,F)F,-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxGF.F)F," }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = 1-2*x/Pi;" "6#/%\"uG,&\"\"\"F&*(\"\"#F&%\"xGF&%#Pi G!\"\"F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = sin*k*x/k;" "6#/%\"v G**%$sinG\"\"\"%\"kGF'%\"xGF'F(!\"\"" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the integration by parts formula: " } {XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$In tG6$*&%\"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 "Int((1-2*x/Pi)*cos*k*x,x = \+ 0 .. Pi)" "6#-%$IntG6$**,&\"\"\"F(*(\"\"#F(%\"xGF(%#PiG!\"\"F-F(%$cosG F(%\"kGF(F+F(/F+;\"\"!F," }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECE WISE([u = 1-2*x/Pi, v = sin*k*x/k],[du/dx = -2/Pi, dv/dx = cos*k*x]); " "6#-%*PIECEWISEG6$7$/%\"uG,&\"\"\"F**(\"\"#F*%\"xGF*%#PiG!\"\"F//%\" vG**%$sinGF*%\"kGF*F-F*F4F/7$/*&%#duGF*%#dxGF/,$*&F,F*F.F/F//*&%#dvGF* F9F/*(%$cosGF*F4F*F-F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1-2*x/Pi) *``(sin*k*x/k);" "6#/%!G*&,&\"\"\"F'*(\"\"#F'%\"xGF'%#PiG!\"\"F,F'-F$6 #**%$sinGF'%\"kGF'F*F'F1F,F'" }{XPPEDIT 18 0 "PIECEWISE([Pi , ``],[0 , ``])" "6#-%*PIECEWISEG6$7$%#PiG%!G7$\"\"!F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-Int(``(-2/Pi)*``(sin*k*x/k),x = 0 .. Pi);" "6#,$-%$Int G6$*&-%!G6#,$*&\"\"#\"\"\"%#PiG!\"\"F0F.-F)6#**%$sinGF.%\"kGF.%\"xGF.F 5F0F./F6;\"\"!F/F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0+``(2/Pi)*``(-c os*k*x/(k^2));" "6#/%!G,&\"\"!\"\"\"*&-F$6#*&\"\"#F'%#PiG!\"\"F'-F$6#, $**%$cosGF'%\"kGF'%\"xGF'*$F4F,F.F.F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G 7$F(F(7$\"\"!F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -``(2/(k^2*Pi))* (cos*k*Pi-1);" "6#/%!G,$*&-F$6#*&\"\"#\"\"\"*&%\"kGF*%#PiGF+!\"\"F+,&* (%$cosGF+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 "`` = 2*(1-( -1)^k)/(k^2*Pi);" "6#/%!G*(\"\"#\"\"\",&F'F'),$F'!\"\"%\"kGF+F'*&F,F&% #PiGF'F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = PIECEWISE([2* (1-(-1)^k)/(k^2*Pi^2), `if`*k <> 1],[1/2+4/(Pi^2), `if`*k = 1]);" "6#/ &%\"aG6#%\"kG-%*PIECEWISEG6$7$*(\"\"#\"\"\",&F.F.),$F.!\"\"F'F2F.*&F'F -%#PiGF-F20*&%#ifGF.F'F.F.7$,&*&F.F.F-F2F.*&\"\"%F.*$F4F-F2F./*&F7F.F' F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8], [a[k], `|`, 1/2+4/(Pi^2), 0, 4/9/(Pi^2), 0, 4/25/(Pi^2), 0, 4/ 49/(Pi^2), 0]]);" "6#-%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"% \"\"&\"\"'\"\"(\"\")7,&%\"aG6#F(F),&*&F*F*F+!\"\"F**&F-F**$%#PiGF+F8F* \"\"!*(F-F*\"\"*F8*$F;F+F8F<*(F-F*\"#DF8*$F;F+F8F<*(F-F*\"#\\F8*$F;F+F 8F<" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 268 "f := x -> piecewise(x<0,cos(x),1-2*x/Pi): \ninterface(showassumed=0): k := 'k': assume(k,integer):\na[k]=1/Pi*In t('f(x)'*cos(k*x),x=-Pi..Pi);\n``=value(rhs(%));\naa := unapply(rhs(%) ,k):\nk := 'k':\nmatrix([[k,`|`,seq(k,k=1..8)],['a'[k],`|`,1/2+4/(Pi^2 ),seq(aa(k),k=2..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%# k|irG,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#*&F'F1F0F1F1/F0;,$%#PiG! \"\"F9*$F9F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\"%#P iG!\"#,&F(!\"\")F,%#k|irGF(F(F.F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\") 7,&%\"aG6#F(F),&#F*F+F**&F-F*%#PiG!\"#F*\"\"!,$*(F-F*\"\"*!\"\"F9F:F*F ;,$*(F-F*\"#DF?F9F:F*F;,$*(F-F*\"#\\F?F9F:F*F;Q)pprint536\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "The coefficients of the sine \+ terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 57 "The Fourier coefficients of the sine terms are given by : " }}{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! \"\"" }{XPPEDIT 18 0 "``(Int(cos*x*sin*k*x,x = -Pi .. 0)+Int((1-2*x/Pi )*sin*k*x,x = 0 .. Pi));" "6#-%!G6#,&-%$IntG6$*,%$cosG\"\"\"%\"xGF,%$s inGF,%\"kGF,F-F,/F-;,$%#PiG!\"\"\"\"!F,-F(6$**,&F,F,*(\"\"#F,F-F,F3F4F 4F,F.F,F/F,F-F,/F-;F5F3F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "k<>1" "6#0%\"kG\"\"\"" }{TEXT -1 6 ", then" }} {PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "Int(cos*x*sin*k*x, x = -Pi .. 0) = 1/2;" "6#/-%$IntG6$*,%$cosG\"\"\"%\"xGF)%$sinGF)%\"kGF )F*F)/F*;,$%#PiG!\"\"\"\"!*&F)F)\"\"#F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin((k+1)*x)+sin((k-1)*x),x = -Pi .. 0)" "6#-%$IntG6$,&-%$sinG6 #*&,&%\"kG\"\"\"F-F-F-%\"xGF-F--F(6#*&,&F,F-F-!\"\"F-F.F-F-/F.;,$%#PiG F3\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(-cos( (k+1)*x)/(k+1)-cos((k-1)*x)/(k-1));" "6#-%!G6#,&*&-%$cosG6#*&,&%\"kG\" \"\"F.F.F.%\"xGF.F.,&F-F.F.F.!\"\"F1*&-F)6#*&,&F-F.F.F1F.F/F.F.,&F-F.F .F1F1F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([0, ``],[``, ``],[- Pi, ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$%#PiG!\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" } {XPPEDIT 18 0 "``(-1/(k+1)-1/(k-1)+cos((k+1)*Pi)/(k+1)+cos((k-1)*Pi)/( k-1));" "6#-%!G6#,**&\"\"\"F(,&%\"kGF(F(F(!\"\"F+*&F(F(,&F*F(F(F+F+F+* &-%$cosG6#*&,&F*F(F(F(F(%#PiGF(F(,&F*F(F(F(F+F(*&-F06#*&,&F*F(F(F+F(F4 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 "`` = (-k)/(k^2-1)+(-1 )^(k+1)*``(k/(k^2-1));" "6#/%!G,&*&,$%\"kG!\"\"\"\"\",&*$F(\"\"#F*F*F) F)F**&),$F*F),&F(F*F*F*F*-F$6#*&F(F*,&*$F(F-F*F*F)F)F*F*" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -k*(1+(-1 )^k)/(k^2-1);" "6#/%!G,$*(%\"kG\"\"\",&F(F(),$F(!\"\"F'F(F(,&*$F'\"\"# F(F(F,F,F," }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "while," }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*x*sin*x,x = - Pi .. 0) = Int(sin*2*x/2,x = -Pi .. 0);" "6#/-%$IntG6$**%$cosG\"\"\"% \"xGF)%$sinGF)F*F)/F*;,$%#PiG!\"\"\"\"!-F%6$**F+F)\"\"#F)F*F)F5F0/F*;, $F/F0F1" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = -cos*2*x/4;" "6#/%!G,$**%$cosG\"\"\"\"\"#F(%\"xGF( \"\"%!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([0, ``],[``, ` `],[-Pi, ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$%#PiG!\"\"F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 " ``= 0" "6#/%!G\"\"!" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " } {XPPEDIT 18 0 "Int((1-2*x/Pi)*sin*k*x,x) = Int(u*``(dv/dx),x);" "6#/-% $IntG6$**,&\"\"\"F)*(\"\"#F)%\"xGF)%#PiG!\"\"F.F)%$sinGF)%\"kGF)F,F)F, -F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxGF.F)F," }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "u = 1-2*x/Pi;" "6#/%\"uG,&\"\"\"F&*(\"\"#F&%\"xGF&%#PiG !\"\"F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*k*x/k;" "6#/%\"v G,$**%$cosG\"\"\"%\"kGF(%\"xGF(F)!\"\"F+" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the integration 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 "Int((1-2*x/Pi)*sin*k*x,x = \+ 0 .. Pi)" "6#-%$IntG6$**,&\"\"\"F(*(\"\"#F(%\"xGF(%#PiG!\"\"F-F(%$sinG F(%\"kGF(F+F(/F+;\"\"!F," }{TEXT -1 11 " ... " }{XPPEDIT 18 0 "P IECEWISE([u = 1-2*x/Pi, v = -cos*k*x/k],[du/dx = -2/Pi, dv/dx = sin*k* x]);" "6#-%*PIECEWISEG6$7$/%\"uG,&\"\"\"F**(\"\"#F*%\"xGF*%#PiG!\"\"F/ /%\"vG,$**%$cosGF*%\"kGF*F-F*F5F/F/7$/*&%#duGF*%#dxGF/,$*&F,F*F.F/F//* &%#dvGF*F:F/*(%$sinGF*F5F*F-F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1-2*x/Pi)*``(-cos*k*x/k);" "6#/%!G*&,&\"\"\"F'*(\"\"#F'%\"xGF'%#PiG! \"\"F,F'-F$6#,$**%$cosGF'%\"kGF'F*F'F2F,F,F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG 6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-Int(``(- 2/Pi)*``(-cos*k*x/k),x = 0 .. Pi);" "6#,$-%$IntG6$*&-%!G6#,$*&\"\"#\" \"\"%#PiG!\"\"F0F.-F)6#,$**%$cosGF.%\"kGF.%\"xGF.F6F0F0F./F7;\"\"!F/F0 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (cos*k*Pi+1)/k+``(2/(k*Pi))*``(sin *k*x/k);" "6#/%!G,&*&,&*(%$cosG\"\"\"%\"kGF*%#PiGF*F*F*F*F*F+!\"\"F**& -F$6#*&\"\"#F**&F+F*F,F*F-F*-F$6#**%$sinGF*F+F*%\"xGF*F+F-F*F*" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ((-1)^k+1)/k;" "6#/%!G *&,&),$\"\"\"!\"\"%\"kGF)F)F)F)F+F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k] = PIECEWISE([``(-k/(Pi*(k^2 -1))+1/(Pi*k))*(1+(-1)^k), `if`*k <> 1],[0, `if`*k = 1]);" "6#/&%\"bG6 #%\"kG-%*PIECEWISEG6$7$*&-%!G6#,&*&F'\"\"\"*&%#PiGF2,&*$F'\"\"#F2F2!\" \"F2F8F8*&F2F2*&F4F2F'F2F8F2F2,&F2F2),$F2F8F'F2F20*&%#ifGF2F'F2F27$\" \"!/*&F@F2F'F2F2" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 6 " = " }{XPPEDIT 18 0 "PIECEWISE([-(1+(-1)^k)/(Pi*k*(k ^2-1)), `if`*k <> 1],[0, `if`*k = 1]);" "6#-%*PIECEWISEG6$7$,$*&,&\"\" \"F*),$F*!\"\"%\"kGF*F**(%#PiGF*F.F*,&*$F.\"\"#F*F*F-F*F-F-0*&%#ifGF*F .F*F*7$\"\"!/*&F6F*F.F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8], [b[k], `|`, 0, -1/(3*Pi), 0, -1/(30*Pi) , 0, -1/(105*Pi), 0, -1/(252*Pi)]]);" "6#-%'matrixG6#7$7,%\"kG%\"|grG \"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&%\"bG6#F(F)\"\"!,$*&F*F**& F,F*%#PiGF*!\"\"F;F6,$*&F*F**&\"#IF*F:F*F;F;F6,$*&F*F**&\"$0\"F*F:F*F; F;F6,$*&F*F**&\"$_#F*F:F*F;F;" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "f := x -> p iecewise(x<0,cos(x),1-2*x/Pi):\ninterface(showassumed=0): k := 'k': as sume(k,integer):\nb[k]=1/Pi*Int('f(x)'*sin(k*x),x=-Pi..Pi);\n``=value( rhs(%));\nbb := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1 ..8)],['b'[k],`|`,0,seq(bb(k),k=2..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG 6#*&F'F1F0F1F1/F0;,$%#PiG!\"\"F9*$F9F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**%#PiG!\"\",&)F(%#k|irG\"\"\"F,F,F,F+F(,&*$)F+\"\"#F,F,F, F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,%\"kG%\"|grG \"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&%\"bG6#F(F)\"\"!,$*&F*F**& F,F*%#PiGF*!\"\"F;F6,$*&F*F**&\"#IF*F:F*F;F;F6,$*&F*F**&\"$0\"F*F:F*F; F;F6,$*&F*F**&\"$_#F*F:F*F;F;Q)pprint556\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series o f " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = (1/2+4/(Pi^2)) *cos*x+``" "6#/-%\"FG6#%\"xG,&*(,&*&\"\"\"F,\"\"#!\"\"F,*&\"\"%F,*$%#P iGF-F.F,F,%$cosGF,F'F,F,%!GF," }{XPPEDIT 18 0 "Sum(``(2*(1-(-1)^k)/(k^ 2*Pi^2))*cos*k*x-``((1+(-1)^k)/(Pi*k*(k^2-1)))*sin*k*x,k = 2 .. infini ty)" "6#-%$SumG6$,&**-%!G6#*(\"\"#\"\"\",&F-F-),$F-!\"\"%\"kGF1F-*&F2F ,%#PiGF,F1F-%$cosGF-F2F-%\"xGF-F-**-F)6#*&,&F-F-),$F-F1F2F-F-*(F4F-F2F -,&*$F2F,F-F-F1F-F1F-%$sinGF-F2F-F6F-F1/F2;F,%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "T he following picture compares the graphs of some truncated Fourier ser ies 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 0 350 "f := x -> piecewise(x<0,cos(x),1-2*x/Pi) :\n'f(x)'=f(x);\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nFS := (x, n) -> (1/2+4/(Pi^2))*cos(x)+sum(2*(1-(-1)^k)/(k^2*Pi^2)*cos(k*x)-\n(1+ (-1)^k)/(Pi*k*(k^2-1))*sin(k*x),k=2..n);\nplot([f_(x),FS(x,1),FS(x,2), FS(x,3),FS(x,4),FS(x,5)],x=-2*Pi..3*Pi,\n color=[black,red,blue,gree n,magenta,coral],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$-%$cosGF&2F'\"\"!7$,&\"\"\"F2*(\"\"#F 2F'F2%#PiG!\"\"F6%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FS Gf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&*&,&#\"\"\"\"\"#F1*&\"\"%F1 %#PiG!\"#F1F1-%$cosG6#9$F1F1-%$sumG6$,&*,F2F1,&F1F1)!\"\"%\"kGFBF1FCF6 F5F6-F86#*&FCF1F:F1F1F1*,,&F1F1FAF1F1F5FBFCFB,&*$)FCF2F1F1F1FBFB-%$sin GFEF1FB/FC;F29%F1F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 646 200 200 {PLOTDATA 2 "6*-%'CURVESG6%7]r7$$!3)****>YH&=$G'!#<$\"25>V+#********F* 7$$!3!QHkk%3*>6'F*$\"3y'R))Q@U,\"*)!#=7$$!3j(e3$)R'zSfF*$\"3i.Ot2XG?yF 27$$!3+[+MC5%=z&F*$\"3S1e/X^+soF27$$!3Z2:P]c)Gk&F*$\"3t/!eByDP#fF27$$! 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kGcWF27$Fejl$\"3Q.ytH\\hsMF27$Fjjl$\"3U$[MpVpM\\#F27$F_[m$\"3=Npn\\![V a\"F27$Fd[m$\"3%*)4nT*)RyP&Fen7$Fi[m$!3XbDWmL;WkFen7$F^\\m$!3-hI-$G5vy \"F27$Fc\\m$!3ML<7<&pYu#F27$Fh\\m$!3Cc'=Oyc9m$F27$F]]m$!3s[(p*G5LfYF27 $Fb]m$!3'>stmi$3TdF27$Fg]m$!3Mw=0S2ecoF27$F\\^m$!3_C-TsMupzF27$Fa^m$!3 O')=4*)QX)**)F27$Ff^m$!3%\\)3HjvFl'*F2-F[_m6&F]_mF]jn$\")AR!)\\F_jnF`j nFajn-%+AXESLABELSG6$Q\"x6\"Q!F`ht-%%VIEWG6$;$!+3`=$G'!\"*$\"+izxC%*Fh ht%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "A truncated Fo urier series with a fairly large number of terms appears to give a goo d match with the graph of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 323 "f := x -> piecewise(x<0,cos(x),1-2 *x/Pi):\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nFS := (x,n) -> (1 /2+4/(Pi^2))*cos(x)+sum(2*(1-(-1)^k)/(k^2*Pi^2)*cos(k*x)-\n(1+(-1)^k)/ (Pi*k*(k^2-1))*sin(k*x),k=2..n): \nplot([f_(x),FS(x,30)],x=-2*Pi..3*Pi ,numpoints=80,\n color=[black,COLOR(RGB,.8,0,1)],linestyle=[3,1],thi ckness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 623 238 238 {PLOTDATA 2 " 6&-%'CURVESG6&7fq7$$!3)****>YH&=$G'!#<$\"25>V+#********F*7$$!3C]e$[`^^ 2'F*$\"3)fhd?(ehv')!#=7$$!3'38)zsC9%*eF*$\"3g#R[LRwK_(F27$$!3;$*)*Q^)y 0p&F*$\"3)\\1;,8]tA'F27$$!3G*)4;yem&[&F*$\"3ytXREO$G#\\F27$$!3I*fC0#os \"G&F*$\"3s'>,>E#Hq o*[F*$\"3gB]I2XWu6F27$$!3U'=AtX&R%p%F*$!3MbThbK]X6!#>7$$!3!z18='*pD\\% F*$!3[58lN8T*R\"F27$$!3WM^&F27$$!3'oa84\\g&*o$F*$!3w g>Q,%G:^'F27$$!3knt\\'fo.\\$F*$!399*)='QB'zxF27$$!3!p&HWtK[4LF*$!3MnU1 =^?$F*$!3fL#)*GJGeh*F27$$!3CS#G(GJR%4$F*$!3%)**p0# 4j)))**F27$$!3g;(HstS)[IF*$!3/%Q)zQi,d**F27$$!3_#>JdM)G.IF*$!3?h(Gk%>^ /**F27$$!3))oEBaftdHF*$!3SfmeV\"f9$)*F27$$!3DXTtiN=7HF*$!3d_f6!R4!Q(*F 27$$!3IWN&=/2i!GF*$!3(o`,F*$!31:w\"4rN>H$F27$$!3WD,?+1M7= 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45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The err or in approximating " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 59 " by the previous truncated Fourier series is plotted below." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "f := x -> piecewise(x < 0,cos(x),1-2*x/Pi):\nf_ := x -> f(x-2*Pi* floor((x+Pi)/(2*Pi))):\nFS := (x,n) -> \n(1/2+4/(Pi^2))*cos(x)+sum(2*( 1-(-1)^k)/(k^2*Pi^2)*cos(k*x)-\n (1+(-1)^k)/(Pi*k*(k^2-1))*sin(k *x),k=2..n): \nplot(f_(x)-FS(x,35),x=-3.5..3.5,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 395 221 221 {PLOTDATA 2 "6&-%'CURVESG6#7\\^m7$$!3 ++++++++N!#<$\"31r%3v))zo.\"!#@7$$!3[mT5!\\F4[$F*$!3#)=*>\"[C)=2#F-7$$ !3QL$3-)\\&=Y$F*$!3NHoj[4+0XF-7$$!3+vVt_o3dMF*$!3yh0+sY6Z[F-7$$!31Y\\n+W/0&F-7$$!3oeky(f]vW$F*$!30Q#[hn%z/^F-7$$!3H+DJqCyUMF *$!3hp)Gly0T+&F-7$$!3_$ek`@YKV$F*$!358.v=qsNVF-7$$!3vmmTg*4PU$F*$!3-$* 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" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "1/2+4*1/(Pi^2)+4/ Pi^2*Sum(1/(2*k+1)^2,k=1..100);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\"\"\"\"#F%*&F%F%*$)%#PiGF&F%!\"\"\"\"%*&-%$SumG6$ *&F%F%*$),&%\"kGF&F%F%F&F%F+/F5;F%\"$+\"F%*$)F*F&F%F+F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+z#o**)**!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 12 " and taking " }{XPPEDIT 18 0 "n = 100 00" "6#/%\"nG\"&++\"" }{TEXT -1 13 " gives . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "1/2+4*1/(Pi^ 2)+4/Pi^2*Sum(1/(2*k+1)^2,k=1..10000);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\"\"\"\"#F%*&F%F%*$)%#PiGF&F%!\"\"\"\"%*&-%$SumG6$ *&F%F%*$),&%\"kGF&F%F%F&F%F+/F5;F%\"&++\"F%*$)F*F&F%F+F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+(o)*)****!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "In fact Maple sums this series analy tically to 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "1/2+4*1/(Pi^2)+4/Pi^2*Sum(1/(2*k+1)^2,k=1..infin ity);\n``=value(%);\n``=normal(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\"\"\"\"#F%*&\"\"%F%%#PiG!\"#F%*(F(F%F)F*-%$SumG6$*&F%F%*$) ,&*&F&F%%\"kGF%F%F%F%F&F%!\"\"/F4;F%%)infinityGF%F%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,(#\"\"\"\"\"#F'*&\"\"%F'%#PiG!\"#F'*(F*F'F+F,,&F '!\"\"*&\"\")F/F+F(F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\" \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Th is follows from the fact that " }{XPPEDIT 18 0 "Sum(1/(2*k+1)^2,k=0..i nfinity)=Pi^2/8" "6#/-%$SumG6$*&\"\"\"F(*$,&*&\"\"#F(%\"kGF(F(F(F(F,! \"\"/F-;\"\"!%)infinityG*&%#PiGF,\"\")F." }{TEXT -1 100 ", or converse ly, the convergence of the Fourier series can be used to deduce this s ummation result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Sum(1/(2*k+1)^2,k=0..infinity);\n``=value(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*$),&*&\"\"#F'% \"kGF'F'F'F'F,F'!\"\"/F-;\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\")!\"\"%#PiG\"\"#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Truncated Fourier series \+ can be obtained using the procedure " }{TEXT 0 13 "FourierSeries" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "f := x -> piecewise(x<0,cos(x),1-2*x/Pi):\n'f(x )'=f(x);\nFourierSeries(f(x),x=-Pi..Pi,numterms=8,info=1):\nF := unapp ly(%,x):\n'F(x)'=F(x);\nplot(F(x),x=-2*Pi..3*Pi,color=red,thickness=2) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$-% $cosGF&2F'\"\"!7$,&\"\"\"F2*(\"\"#F2F'F2%#PiG!\"\"F6%*otherwiseG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%YH&=$G'!#<$\"3)*)))[gu))zu*!#=7$$!3!QHkk%3*>6 'F*$\"3g+'G!QGV()*)F-7$$!3j(e3$)R'zSfF*$\"3=1BW$[M)pyF-7$$!3+[+MC5%=z& F*$\"3*\\Vke#H=VoF-7$$!3Z2:P]c)Gk&F*$\"3Sd5qq%R2)eF-7$$!3WWwe`(p`Z&F*$ \"3s6-xW8Fg[F-7$$!3I#y.o&Q&yI&F*$\"3\\Qt>$*f!G#QF-7$$!3#p\\7$$!3.`Y+c:&\\j%F*$!3O$GGuMbhx%Fen7$$!3wY(HuVc$zWF*$!3_19]- *)yd9F-7$$!3]S[&)=8wBVF*$!3aQy()Gz*RZ#F-7$$!31C:*fe^E;%F*$!3B:kQx6%)HN F-7$$!3l2#GJ&=a,SF*$!3%fi%[&*RkUXF-7$$!3$=6Jp$>#\\$QF*$!3')47R6AAabF-7 $$!3c:St??IoOF*$!3*yb!)eL)f2mF-7$$!3A.8sck@-NF*$!3#>Wm/tnIu(F-7$$!3J\" f3F*38OLF*$!3IzZXV:$)[))F-7$$!3m\\'fT=62D$F*$!3?X9vmM+@$*F-7$$!3+32hv9 HlJF*$!3!3YDacX]n*F-7$$!3SPiL@;eAJF*$!3#ps>0R:zz*F-7$$!3!owhqwr)zIF*$! 3C:G!39E8))*F-7$$!3v&H(y7>;PIF*$!3(3#)yrk.U#**F-7$$!39DG^e?X%*HF*$!3#= n,@\\)oE**F-7$$!3AwJ%pgwR%GF*$!3mR$*op(4>k*F-7$$!3uFNPb6]$p#F*$!3]SrqE Q)o,*F-7$$!3)[()>pV,T_#F*$!3w)R]F9%*=5)F-7$$!3+AiY=&Q6^9?F*$!3)>1Kg>?JJ%F-7 $$!3!\\\")R(3Gf]=F*$!3*yt59ZVgt#F-7$$!3YM)f*yWn'o\"F*$!3d4e89h'>@\"F*$\"3B7GS*f,n]$F-7$$!3;XJ#\\&['\\.\"F*$\"3U:_EI&4L9&F- 7$$!3?V]Sic@]))F-$\"33OF-$\"3c?<2&Qhb))*F-7$$!3EAM9cU_Y:F-$\"3Er#40zHI#**F-7$$!3Q*Fen$\"3/&**)*>xnGS*F-7$$\"3)Q)p\"4:eKh#F-$\"3mXL!\\jOGU)F- 7$$\"3sHp&)=YEEUF-$\"3\")fcPc6D7tF-7$$\"3dvoz'3r#ReF-$\"3_(>sS#))*fB'F -7$$\"3'Q)>g?TFAvF-$\"3q2&F-7$$\"3;#42W:x_?*F-$\"3M=\\*HB9\"oTF -7$$\"3?5FhJ*z]2\"F*$\"3uDaLX*H)yJF-7$$\"3>6ZyZ@jH7F*$\"3)Q&f-n+Mh@F-7 $$\"3y$Q0AfPjR\"F*$\"3;@/([BRV3\"F-7$$\"3QcgiOI/j:F*$\"3k$y16-[Ax%!#?7 $$\"3xAvmTJ?OF*$!3]]*[K\"RFW@F-7$$ \"3/^H=F'*4g?F*$!3*e.$*Q`so8$F-7$$\"3!H\"pl2g$3@#F*$!3YyJP)F-7$$\"3'[TUQ >\\t.$F*$!39a?+k,tW$*F-7$$\"3;Kv_w!))=?$F*$!3]%Hh>X$>z)*F-7$$\"36C\\)Q .)oTKF*$!3J:0#Q<0<#**F-7$$\"3];BC\"*z[\"G$F*$!3gM6hu&p!H**F-7$$\"3))3( *f[zG@LF*$!3*z#Hy>UA-**F-7$$\"3#35df!z3hLF*$!3X!Q3LQ:H%)*F-7$$\"3;&)=n ?yoSMF*$!3[@+fQb8P'*F-7$$\"3]pmQNxG?NF*$!3p@%=[)G>O$*F-7$$\"3)3LSJ+_qp $F*$!3%=O&*QVFRX)F-7$$\"3G#*R*3F;Q(QF*$!3e`MLQhj/uF-7$$\"3))p!zt#okKSF *$!3\\lcHcB34jF-7$$\"3YZT'QQx9>%F*$!3[f/]J&GG,&F-7$$\"3aUah/H1hVF*$!3V 6MMF-7$$\"3iPnOD%[1`%F*$!37W`(3A^Dx\"F-7$$\"3!)Q<`7\"=Vo%F*$!3$eb 9*>QeEFFen7$$\"3(*Rnp*z()z$[F*$\"3dI:[NPhA7F-7$$\"3nY9Dd$*)f+&F*$\"3kc [x$4$RrGF-7$$\"3O`h![\"4*R<&F*$\"3=oI+xjgyWF-7$$\"3v'f(z(Gn?L&F*$\"3kO )y-!*4O%eF-7$$\"39S!*ygO9!\\&F*$\"30ag$\\E\"41qF-7$$\"3]0;?=!z`l&F*$\" 3?[-))[drT!)F-7$$\"3$3<9cP91#eF*$\"3S*3)=\"=j^%*)F-7$$\"3%Hbe(R@;#)fF* $\"3_CzZ[I'3j*F-7$$\"3.NH!R!*4P9'F*$\"3,]L**F-7$$\"33Q()psEFGiF*$\"35u%\\s?K2()*F-7$$\"3;Rm4dSbqiF *$\"3)RWO4lE?y*F-7$$\"3CSX\\Ta$GJ'F*$\"37-!fb:1\\l*F-7$$\"3QU.H5#)R(R' F*$\"3q`ZGZaz&H*F-7$$\"3WXh3z4'>['F*$\"3*\\RW`I-)ew)F*$!3![$HW:@_ldF-7$$\"35R &f$o&yY#*)F*$!3_,)4\\K^Wy'F-7$$\"3E)*\\(p\"pZ$3*F*$!3#*\\vJg]QxyF-7$$ \"3!*)\\_%Hu7a#*F*$!3G(R#4>ae!**)F-7$$\"3^***H>%zxC%*F*$!3i22GG())zu*F --%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fb_m-%'COLOURG6&%$RGBG$ \"*++++\"!\")$\"\"!F\\`mF[`m-%%VIEWG6$;$!+3`=$G'!\"*$\"+izxC%*Fc`m%(DE FAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 44.000000 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 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T asks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 63 "(a) Find form ulas for the Fourier coefficients of the function " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " defined by: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([0, -Pi <= x and x < 0],[Pi-x, 0 <= x and x < Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\" \"!31,$%#PiG!\"\"F'2F'F,7$,&F0\"\"\"F'F131F,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 w ith period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 " (b) Compare the graphs of some truncated Fourier series with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "(c) To wh at value does the Fourier series convergence when " }{XPPEDIT 18 0 "x= 0" "6#/%\"xG\"\"!" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 100 " \+ Perform some numerical calculations to investigate the convergence of the Fourier series when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 17 " (a) \+ " }{XPPEDIT 18 0 "c = Pi/4;" "6#/%\"cG*&%#PiG\"\"\"\"\"%!\"\"" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "a[k] = (1-(-1)^k)/(k^2*Pi);" "6#/&% \"aG6#%\"kG*&,&\"\"\"F*),$F*!\"\"F'F-F**&F'\"\"#%#PiGF*F-" }{XPPEDIT 18 0 "``=PIECEWISE([0, `k even`],[2/(k^2*Pi), `k odd`])" "6#/%!G-%*PIE CEWISEG6$7$\"\"!%'k~evenG7$*&\"\"#\"\"\"*&%\"kGF-%#PiGF.!\"\"%&k~oddG " }{TEXT -1 5 ", " }{XPPEDIT 18 0 "b[k] = 1/k;" "6#/&%\"bG6#%\"kG*& \"\"\"F)F'!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ Fourier series: " }{XPPEDIT 18 0 "Pi/4+2/Pi*cos*x+sin*x+sin*2* x/2+2/(9*Pi)*cos*3*x+sin*3*x/3+sin*4*x/4+2/(25*Pi)*cos*5*x+sin*5*x/5+s in*6*x/6+2/(49*Pi)*cos*7*x+sin*7*x/7+sin*8*x/8+` . . . `" "6#,>*&%#PiG \"\"\"\"\"%!\"\"F&**\"\"#F&F%F(%$cosGF&%\"xGF&F&*&%$sinGF&F,F&F&**F.F& F*F&F,F&F*F(F&*,F*F&*&\"\"*F&F%F&F(F+F&\"\"$F&F,F&F&**F.F&F3F&F,F&F3F( F&**F.F&F'F&F,F&F'F(F&*,F*F&*&\"#DF&F%F&F(F+F&\"\"&F&F,F&F&**F.F&F9F&F ,F&F9F(F&**F.F&\"\"'F&F,F&F " 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" }}{PARA 0 "" 0 " " {TEXT -1 63 "(a) Find formulas for the Fourier coefficients of the f unction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " defin ed by: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = PI ECEWISE([Pi^2, -Pi <= x and x < 0],[Pi^2-x^2, 0 <= x and x < Pi]);" "6 #/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*$%#PiG\"\"#31,$F-!\"\"F'2F'\"\"!7$,&* $F-F.\"\"\"*$F'F.F231F4F'2F'F-" }{TEXT -1 2 ", " }}{PARA 256 "" 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 75 "(b) Compa re the graphs of some truncated Fourier series with the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "(c) To what value does the Fourier series convergence when " }{XPPEDIT 18 0 "x = Pi;" "6#/% \"xG%#PiG" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 100 " Perf orm some numerical calculations to investigate the convergence of the \+ Fourier series when " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 2 ". " }}{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 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 16 "Code for picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 521 "p1 := plot([[[0,1],[0,-1]], [t,1/(t^2+1),t=0..1],[-t,-1/(t^2+1),t=0..1]],\n style=[point, line,line],symbol=circle,symbolsize=13,color=red):\np2 := plot([[[0,0] ]$4],color=red,style=point,symbol=[circle$2,diamond,cross],\n symbo lsize=[13,10$3]):\np3 := plot([[0,-1],[0,1]],color=black,linestyle=2): \nt1 := plots[textplot]([[0,-1.15,`x = d`],[.3,.15,`f(d-) + f(d+)`],\n [.3,.07,`_________`],[.3,-.15,`2`]],color=black):\nt2 := plots[tex tplot]([.7,.9,`y = f(x)`],color=red):\nplots[display]([p1,p2,p3,t1,t2] ,axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 \+ 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }