{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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 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 "Purple \+ Emphasis" -1 266 "Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Grey Emphasis" -1 267 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple 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 }{PSTYLE "Normal" -1 257 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 "Normal " -1 258 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" -1 259 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 62 "Fourier series of some functions \+ with infinite discontinuities" }}{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 267 9 "fourier.m" }{TEXT -1 37 " contains the code fo r the procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 25 " used in t his worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/procdrs/fourier.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x) = 1/2" "6#/-%\"fG6#%\"xG*&\" \"\"F)\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1/(2-2*cos*x));" "6#-%#lnG6#*&\"\"\"F',&\"\"#F'*(F)F'%$cosGF'%\"xGF'!\"\"F-" }{XPPEDIT 18 0 "``=-ln(2-2*cos*x)/2" "6#/%!G,$*&-%#lnG6#,&\"\"#\"\"\"*(F+F,%$cos GF,%\"xGF,!\"\"F,F+F0F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "f := x -> -ln(2-2*cos (x))/2:\n'f(x)'=f(x);\nplot(f(x),x=-Pi..6*Pi,y=-1..5,numpoints=50);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*&#\"\"\"\"\"#F+-%#ln G6#,&F,F+*&F,F+-%$cosGF&F+!\"\"F+F4" }}{PARA 13 "" 1 "" {GLPLOT2D 615 273 273 {PLOTDATA 2 "6%-%'CURVESG6$7fq7$$!3*****4tk#fTJ!#<$!3'GX*f0=ZJ p!#=7$$!35!ynBtI?n#F*$!3)*Gn\"pT\"H`mF-7$$!3Qq/\"4UnME#F*$!3jjx`x;$\\$ fF-7$$!3n'*Q(oC&*R!=F*$!3SI5Nw)R_]%F-7$$!3(*HN,$Rx9M\"F*$!3)ee]R4@j<#F -7$$!3w'pix0y:\"))F-$\"3'pI4r>@3f\"F-7$$!3g%Que9-xn'F-$\"3g#)o=ucgCUF- 7$$!3Wsg)RBEQa%F-$\"3u!=%f::LuzF-7$$!3S#=k+yt!RMF-$\"3ydM!fy:B2\"F*7$$ !3P#HUhK@VL#F-$\"3yY6jo`8d9F*7$$!3OZ8=*4X>y\"F-$\"3zFWqwH?Ex'!#>$\"3y1)4(ocd#p#F*7$$!37B^) H=k\"[7Fbo$\"3%z7I\"fG]$Q%F*7$$\"3'pco!e'=XY%Fbo$\"3-8'*)=$=44JF*7$$\" 3qDA\"*\\,s<5F-$\"3Ab81;3+*F-$\"3MQ'3sqkDR\"F-7$$\"3=HB]w+no8F*$!3?63yi7JXBF-7$$\"3kWbmVFS \"y\"F*$!320\"3LwA\\T%F-7$$\"3)*z\\Ur>/YAF*$!3Q\"f'4\\!\\N*eF-7$$\"3V8 _\\_!*e7FF*$!39v]Ai0j*p'F-7$$\"3ij#yTh$>iJF*$!3lPnu'HT4$pF-7$$\"3iZK2H wZqNF*$!3#G'>re(f(*p'F-7$$\"3Ox(eG\"o'f0%F*$!3k*)[A*Q'zZeF-7$$\"3o'>YK $)RsY%F*$!3g%\\))p9s@b%F-7$$\"3gm!3v7[c%\\F*$!3w^\"z$z8W^@F-7$$\"3!pl& 4J07p`F*$\"3C.'R!ff7\\7F-7$$\"3\"\\:YGqG9g&F*$\"3oLo&RvW_-%F-7$$\"3\"H l'fuotLeF*$\"3#\\$3$['>l\"3)F-7$$\"33]%)RKIMWfF*$\"3ki8%=04q3\"F*7$$\" 39[-?!>\\\\0'F*$\"3O'>Nm,Y&z9F*7$$\"3gZ65pAD5hF*$\"3;]wr4s4c'F*$\"3IoO9zF_3CF*7$$\"3xW H!pUe3A'F*$\"3S$oojMCbx#F*7$$\"31%R`j'*4&[iF*$\"33'Q0LewT$Q')H,0jF*$\"3pw>PyifCQF*7$$\"3W!)H' pYkQL'F*$\"3sC]`?vM#)HF*7$$\"3)*[DaZfrijF*$\"3+-#y)yk(=`#F*7$$\"3_<@7G uc\"R'F*$\"3\"QGFEj\"eAAF*7$$\"3ia7G*Qq#\\kF*$\"3')[5S(o/kz\"F*7$$\"3r \"RS/Ntp]'F*$\"3)GT7T\\V\"*\\\"F*7$$\"3*emeFFzBi'F*$\"3)e,(4E;)f3\"F*7 $$\"32Sp2&>&yPnF*$\"3A8gc)pL'pzF-7$$\"3e7K6?Du\\pF*$\"3xx;i*p2@C%F-7$$ \"35&[\\^%)*phrF*$\"3rex1P/!*=;F-7$$\"3aLF8k!\\*=wF*$!3&*\\YKF0@S@F-7$ $\"3YwW5&\\-R4)F*$!3V'puc05<`%F-7$$\"3$fmwH&3N2&)F*$!3i$3)4=CFSeF-7$$ \"3KBlE1k)Q&*)F*$!3Ga&48*))p^mF-7$$\"3us-NE8?:%*F*$!3:xGDlrNJpF-7$$\"3 9,<+wz]m)*F*$!3#4REo%eb&o'F-7$$\"3y-&Hl.<..\"!#;$!3MRe=,UIMfF-7$$\"3Q& fq*\\4!)y5Fi]l$!3V\")Q$)*)Hg-WF-7$$\"3S\"e1E!fOA6Fi]l$!3iVc=&\\$4%=#F- 7$$\"3')\\%)H)y!))o6Fi]l$\"32!o15.N!H;F-7$$\"3QXDD)[b**=\"Fi]l$\"3RCV \\KZPQUF-7$$\"33Tm?)=I5@\"Fi]l$\"3gFAhfV$z$zF-7$$\"3G8/,&H]DA\"Fi]l$\" 3/Eq]Op5\"3\"F*7$$\"3m&=9=SqSB\"Fi]l$\"3)[#)G.S>3\\\"F*7$$\"3&=2;_XI)R 7Fi]l$\"3Sa\"3?c]8D\"Fi]l$\"3ea'3D:[\"F-7$$\"3u(Q7#[15!R\"Fi]l$!3Sw)zC4dI8#F-7$$\"3#3u%4n$ *[O9Fi]l$!3w\")3=*fbN[%F-7$$\"3[n%3o@n6[\"Fi]l$!3I(3/)R+x\"*eF-7$$\"3d nb5rn&o_\"Fi]l$!3%R&p@Z%p\")o'F-7$$\"3'*H\\#G*z;s:Fi]l$!3JyMH?mBJpF-7$ $\"3t!eV(*H/Qh\"Fi]l$!3wFa;boX)p'F-7$$\"3ozr71U_h;Fi]l$!3W'ocS:r^'eF-7 $$\"3t/)p#fa?/XF-7$$\"3%[oUsd7(\\.m;cG6F*7$$\"3Ei\\.yQHj=Fi]l$\"3TB()\\*pz:`\"F*7$$\"3m@Pi0$4(o=Fi]l$ \"3NT<:CeS==F*7$$\"33\"[7KtCT(=Fi]l$\"3Wf:)ekfKA#F*7$$\"3igo+ZC$o(=Fi] l$\"3F!)=3Gxs5DF*7$$\"3]S7!3;S&z=Fi]l$\"30'=a#[)Rg\"HF*7$$\"3WI%)p " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Note that the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " has vertical asy mptotes " }{XPPEDIT 18 0 "x=n*Pi" "6#/%\"xG*&%\"nG\"\"\"%#PiGF'" } {TEXT -1 8 ", where " }{TEXT 263 1 "n" }{TEXT -1 17 " is any integer. \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "For example, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x=0)=infinity" "6#/-%&LimitG 6$-%\"fG6#%\"xG/F*\"\"!%)infinityG" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f := x -> -ln(2-2*cos(x))/2:\nLimit(f(x),x= 0);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&# \"\"\"\"\"#F)-%#lnG6#,&F*F)*&F*F)-%$cosG6#%\"xGF)!\"\"F)F4/F3\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G%)infinityG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 22 "The cosine series of " }{XPPEDIT 18 0 "f(x) = 1/2 " "6#/-%\"fG6#%\"xG*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1/(2-2*cos*x));" "6#-%#lnG6#*&\"\"\"F',&\"\"#F'*(F)F'%$cosGF'%\" xGF'!\"\"F-" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, Pi]; " "6#7$\"\"!%#PiG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c+``" "6#,&%\"cG\"\"\"%!GF%" }{XPPEDIT 18 0 "Sum(a [k]*cos*x,k = 1 .. infinity);" "6#-%$SumG6$*(&%\"aG6#%\"kG\"\"\"%$cosG F+%\"xGF+/F*;F+%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1 /Pi*Int(f(x),x = 0 .. Pi);" "6#/%\"cG*(\"\"\"F&%#PiG!\"\"-%$IntG6$-%\" fG6#%\"xG/F/;\"\"!F'F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = 2/Pi; " "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*k*x,x = 0 .. Pi)" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\" %$cosGF+%\"kGF+F*F+/F*;\"\"!%#PiG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "k =1,2,3,` . . . `" "6&/%\"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 3 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The \+ constant coefficient " }{TEXT 262 1 "c" }{TEXT -1 7 " is 0. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "f := x -> -ln(2-2*cos(x))/2:\n'f(x)'=f(x);\nc=Int('f(x)',x=0..Pi);\n``= value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*&# \"\"\"\"\"#F+-%#lnG6#,&F,F+*&F,F+-%$cosGF&F+!\"\"F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG-%$IntG6$-%\"fG6#%\"xG/F+;\"\"!%#PiG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The cosine coefficients can be \+ calculated as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "f := x -> -ln(2-2*cos(x))/2:\n'f(x )'=f(x);\nfor k to 5 do\n a[k]=2/Pi*Int(f(x)*cos(k*x),x=0..Pi);\n \+ ``=value(rhs(%));\nend do;\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*&#\"\"\"\"\"#F+-%#lnG6#,&F,F+*&F,F+-%$cosGF&F+!\" \"F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\",$,$-%$IntG6$ ,$*&#F'\"\"#F'*&-%#lnG6#,&F0F'*&F0F'-%$cosG6#%\"xGF'!\"\"F'F7F'F'F;/F: ;\"\"!%#PiG*&F0F'F?F;F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"#,$,$-%$IntG6$,$*&#\" \"\"F'F0*&-%#lnG6#,&F'F0*&F'F0-%$cosG6#%\"xGF0!\"\"F0-F86#,$*&F'F0F:F0 F0F0F0F;/F:;\"\"!%#PiG*&F'F0FCF;F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%!G#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"$,$,$ -%$IntG6$,$*&#\"\"\"\"\"#F0*&-%#lnG6#,&F1F0*&F1F0-%$cosG6#%\"xGF0!\"\" F0-F96#,$*&F'F0F;F0F0F0F0F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of \+ " }{XPPEDIT 18 0 "f(x) = 1/2" "6#/-%\"fG6#%\"xG*&\"\"\"F)\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1/(2-2*cos*x));" "6#-%#lnG6#*&\"\"\" F',&\"\"#F'*(F)F'%$cosGF'%\"xGF'!\"\"F-" }{TEXT -1 16 " appears to be: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x)=Sum(cos*k*x /k,k=1..infinity)" "6#/-%\"FG6#%\"xG-%$SumG6$**%$cosG\"\"\"%\"kGF-F'F- F.!\"\"/F.;F-%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 19 " may als o be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 90 "f := x -> -ln(2-2*cos(x))/2:\n'f(x)'=f(x);\nFourier Series(f(x),x=0..Pi,type=cos,numterms=8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*&#\"\"\"\"\"#F+-%#lnG6#,&F,F+*&F,F+-%$ cosGF&F+!\"\"F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2-%$cosG6#%\"xG \"\"\"*&#F(\"\"#F(-F%6#,$*&F+F(F'F(F(F(F(*&#F(\"\"$F(-F%6#,$*&F2F(F'F( F(F(F(*&#F(\"\"%F(-F%6#,$*&F9F(F'F(F(F(F(*&#F(\"\"&F(-F%6#,$*&F@F(F'F( F(F(F(*&#F(\"\"'F(-F%6#,$*&FGF(F'F(F(F(F(*&#F(\"\"(F(-F%6#,$*&FNF(F'F( F(F(F(*&#F(\"\")F(-F%6#,$*&FUF(F'F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "We can set up a procedure to gr aph truncated Fourier series of the form " }{XPPEDIT 18 0 "F[n](x) = S um(cos*k*x/k,k = 1 .. n);" "6#/-&%\"FG6#%\"nG6#%\"xG-%$SumG6$**%$cosG \"\"\"%\"kGF0F*F0F1!\"\"/F1;F0F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "f := x -> - ln(2-2*cos(2*Pi*x))/2:\nFS := (x,n) -> Sum(cos(2*k*Pi*x)/k,k=1..n);\np lot([f(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5),FS(x,6)],\n x=-1/2. .2,y=-1..4,color=[black,red,blue,green,magenta,coral,\n COLOR(RGB,.4 ,0,.9)],linestyle=[2,1$6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf *6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF)-%$SumG6$*&-%$cosG6#,$*(%\"kG\" \"\"%#PiGF79$F7\"\"#F7F6!\"\"/F6;F79%F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 523 334 334 {PLOTDATA 2 "6+-%'CURVESG6%7ir7$$!1+++++++]!#;$! 1`%*f0=ZJpF*7$$!1+++tb`FZF*$!1!\\!*zb#z%*oF*7$$!1+++Y62bWF*$!16b())o7U y'F*7$$!1+++08+=UF*$!1JVSgChEmF*7$$!1+++k9$4)RF*$!1-&>(p2**4kF*7$$!1++ +-=rZMF*$!1[%[zdG?p&F*7$$!1+++:w&4\"HF*$!1A)z()*z\"Gg%F*7$$!1+++EXvwBF *$!1,/Q\"*[:jIF*7$$!1+++!))z9)=F*$!1m+;q*oU3\"F*7$$!1+++wd1D;F*$\"1YZ_ voN)H#!#<7$$!1+++s;lo8F*$\"1oc4z7()==F*7$$!1+++RyY.6F*$\"1-IaSZgjQF*7$ $!1+++]+%GQ)FZ$\"1Y4bzZ#p_'F*7$$!1+++DJq#p0\"FZ$\"1)>c>A6@r# 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" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Sum(cos(k*Pi)/k,k=1..inf inity);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Su mG6$*&-%$cosG6#*&%\"kG\"\"\"%#PiGF,F,F+!\"\"/F+;F,%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+1=ZJp!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 22 ", the serie s becomes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+1/2+ 1/3+1/4+1/5+` . . . `;" "6#,.\"\"\"F$*&F$F$\"\"#!\"\"F$*&F$F$\"\"$F'F$ *&F$F$\"\"%F'F$*&F$F$\"\"&F'F$%(~.~.~.~GF$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = infinity;" "6#/%!G%)in finityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = 1/2;" "6#/- %\"fG6#%\"xG*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cot( x/2) = sin*x/(2-2*cos*x);" "6#/-%$cotG6#*&%\"xG\"\"\"\"\"#!\"\"*(%$sin GF)F(F),&F*F)*(F*F)%$cosGF)F(F)F+F+" }{TEXT -1 2 ". 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$\"3p#3\")f)4J;;F-7$$\"3=H&f$>a;Z>Fjx$\"3i?wmbSGb:F-7$$\"3'eCs.4]&\\>F jx$\"3iR:kGE\"R\\\"F-7$$\"3G4'o))o8?&>Fjx$\"3T'p\\m8R\\V\"F-7$$\"3wpYO RQUa>Fjx$\"3+G67f`9\"Q\"F-7$$\"3oPs*Qybn&>Fjx$\"3K.p\"*e**RK8F-7$$\"3a &*\\r5]Mf>Fjx$\"3;TqQXmq\"G\"F-7$$\"3?m7Fjx$\"34_HFlQ%*Q7F-7$$\" 3A&))f$>c:k>Fjx$\"3=iS*fMDf>\"F-7$$\"3#Rku)elSm>Fjx$\"3%Q%3?UT4f6F-7$$ \"3#GJ:.Vn)o>Fjx$\"3sOu#f\"e'47\"F-7$$\"3'oyg&=H=r>Fjx$\"3MA@#)eD'p3\" F-7$$\"3?=\\&4F.O(>Fjx$\"3u$\\\\'[??`5F-7$$\"3G@.#ygpf(>Fjx$\"3.x!fAm= =-\"F-7$$\"3_6x@VpWy>Fjx$\"3*ek`I)3\\0**F^dl7$$\"3y%pV;\"H$3)>Fjx$\"3e &\\a]b-&='*F^dl7$$\"31=B(*3HF$)>Fjx$\"34nE.8kKQ$*F^dl7$$\"37K6*=q#p&)> Fjx$\"3)eK;3>-G2*F^dl7$$\"3q^NFSi\"z)>Fjx$\"30V]5_2*)Q))F^dl7$$\"3;#)f =mYY!*>Fjx$\"3[>I0Mw&=e)F^dl7$$\"3il?q3Su#*>Fjx$\"3%)*)*>b,48O)F^dl7$$ \"3!HpJKEu^*>Fjx$\"3cV.#*3FBN\")F^dl7$$\"3gO+\"pK+v*>Fjx$\"3y;$=tGnq#z F^dl7$$\"#?F*$\"3-*f%yE_v6xF^dl-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fj gr-F$6)F&F^^lF]dmFginFa_pF[eq-Fdgr6&FfgrFjgrFggrFjgr-F$6%7$7$$!3C'ezrI &=$G'F-$FigrF*7$Fchr$\"\")F*-Fdgr6&FfgrF*F*F*-%*LINESTYLEG6#\"\"$-F$6% 7$7$$\"3C'ezrI&=$G'F-Fehr7$FcirFghrFihrF[ir-F$6%7$7$$\"3E " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Note that the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " has vertical asymptotes " } {XPPEDIT 18 0 "x = 2*n*Pi;" "6#/%\"xG*(\"\"#\"\"\"%\"nGF'%#PiGF'" } {TEXT -1 8 ", where " }{TEXT 265 1 "n" }{TEXT -1 17 " is any integer. \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "For example, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = 0,right) = infinity;" "6 #/-%&LimitG6%-%\"fG6#%\"xG/F*\"\"!%&rightG%)infinityG" }{TEXT -1 1 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = 0,left) = infinity" "6#/-%&LimitG6% -%\"fG6#%\"xG/F*\"\"!%%leftG%)infinityG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "f := x -> cot(x/2)/2:\nLimit(f(x),x =0,right);\n``=value(%);\nLimit(f(x),x=0,left);\n``=value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6%,$*&#\"\"\"\"\"#F)-%$cotG6# ,$*&F*!\"\"%\"xGF)F)F)F)/F1\"\"!%&rightG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6%,$ *&#\"\"\"\"\"#F)-%$cotG6#,$*&F*!\"\"%\"xGF)F)F)F)/F1\"\"!%%leftG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$%)infinityG!\"\"" }}}{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 20 "The sin e series of " }{XPPEDIT 18 0 "f(x) = 1/2;" "6#/-%\"fG6#%\"xG*&\"\"\"F )\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cot(x/2)" "6#-%$cotG6#*&% \"xG\"\"\"\"\"#!\"\"" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, Pi];" "6#7$\"\"!%#PiG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k]*sin*x,k = 1 .. infinity);" "6# -%$SumG6$*(&%\"aG6#%\"kG\"\"\"%$sinGF+%\"xGF+/F*;F+%)infinityG" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k] = 2/Pi" "6#/&%\"bG6#%\"kG*&\" \"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin*k*x, x = 0 .. Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$sinGF+%\"kGF+F*F+/F* ;\"\"!%#PiG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "k=1,2,3,` . . . `" "6&/ %\"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "f := x -> cot(x/2)/2:\n'f(x)'=f(x);\nfor k to 5 d o\n a[k]=2/Pi*Int(f(x)*sin(k*x),x=0..Pi);\n ``=map(combine@simplif y,rhs(%));\n ``=value(rhs(%));\nend do;\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*&#\"\"\"\"\"#F+-%$cotG6#,$*&F,!\"\" F'F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\",$,$-%$In tG6$,$*&#F'\"\"#F'*&-%$cotG6#,$*&F0!\"\"%\"xGF'F'F'-%$sinG6#F8F'F'F'/F 8;\"\"!%#PiG*&F0F'F?F7F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$,$-% $IntG6$,&#\"\"\"\"\"#F,*&F+F,-%$cosG6#%\"xGF,F,/F2;\"\"!%#PiG*&F-F,F6! \"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"#,$,$-%$IntG6$,$*&#\"\"\"F'F0*&-%$cotG6 #,$*&F'!\"\"%\"xGF0F0F0-%$sinG6#,$*&F'F0F8F0F0F0F0F0/F8;\"\"!%#PiG*&F' F0FAF7F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$,$-%$IntG6$,(-%$cosG 6#%\"xG\"\"\"*&#F/\"\"#F/-F,6#,$*&F2F/F.F/F/F/F/F1F//F.;\"\"!%#PiG*&F2 F/F:!\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"$,$,$-%$IntG6$,$*&#\"\"\"\"\"#F0*& -%$cotG6#,$*&F1!\"\"%\"xGF0F0F0-%$sinG6#,$*&F'F0F9F0F0F0F0F0/F9;\"\"!% #PiG*&F1F0FBF8F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$,$-%$IntG6$, *#\"\"\"\"\"#F,*&F+F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,-F06#F5F,-F06#,$ *&F-F,F5F,F,F,/F5;\"\"!%#PiG*&F-F,F?!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6# \"\"%,$,$-%$IntG6$,$*&#\"\"\"\"\"#F0*&-%$cotG6#,$*&F1!\"\"%\"xGF0F0F0- %$sinG6#,$*&F'F0F9F0F0F0F0F0/F9;\"\"!%#PiG*&F1F0FBF8F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$,$-%$IntG6$,,-%$cosG6#,$*&\"\"#\"\"\"%\"xGF1 F1F1*&#F1F0F1-F,6#,$*&\"\"%F1F2F1F1F1F1F4F1-F,6#F2F1-F,6#,$*&\"\"$F1F2 F1F1F1/F2;\"\"!%#PiG*&F0F1FD!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"&,$,$-%$In tG6$,$*&#\"\"\"\"\"#F0*&-%$cotG6#,$*&F1!\"\"%\"xGF0F0F0-%$sinG6#,$*&F' F0F9F0F0F0F0F0/F9;\"\"!%#PiG*&F1F0FBF8F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$,$-%$IntG6$,.#\"\"\"\"\"#F,*&F+F,-%$cosG6#,$*&\"\"&F,%\"xG F,F,F,F,-F06#,$*&\"\"$F,F5F,F,F,-F06#F5F,-F06#,$*&\"\"%F,F5F,F,F,-F06# ,$*&F-F,F5F,F,F,/F5;\"\"!%#PiG*&F-F,FI!\"\"F," }}{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 "Th e Fourier series of " }{XPPEDIT 18 0 "f(x) = 1/2;" "6#/-%\"fG6#%\"xG*& \"\"\"F)\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cot(x/2)" "6#-%$co tG6#*&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 16 " appears to be: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = Sum(sin*k*x,k = 1 .. infinity);" "6#/-%\"FG6#%\"xG-%$SumG6$*(%$sinG\"\"\"%\"kGF-F'F-/F.;F- %)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 19 " may also be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "f := x -> cot(x/2)/2:\n'f(x)'=f(x);\nFourierSeries(f(x),x=0..Pi, type=sine,numterms=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#% \"xG,$*&#\"\"\"\"\"#F+-%$cotG6#,$*&F,!\"\"F'F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6-%$sinG6#%\"xG\"\"\"-F%6#,$*&\"\"#F(F'F(F(F(-F%6#, $*&\"\"$F(F'F(F(F(-F%6#,$*&\"\"%F(F'F(F(F(-F%6#,$*&\"\"&F(F'F(F(F(-F%6 #,$*&\"\"'F(F'F(F(F(-F%6#,$*&\"\"(F(F'F(F(F(-F%6#,$*&\"\")F(F'F(F(F(-F %6#,$*&\"\"*F(F'F(F(F(-F%6#,$*&\"#5F(F'F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "We can set up a procedure to graph truncated Fourier series of the form " }{XPPEDIT 18 0 "F[n]( x) = Sum(sin*k*x,k = 1 .. n);" "6#/-&%\"FG6#%\"nG6#%\"xG-%$SumG6$*(%$s inG\"\"\"%\"kGF0F*F0/F1;F0F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "FS := (x,n) -> sum(sin(k*x),k=1..n);\nplot([cot(x/2)/2,FS(x,1),FS(x,2),FS(x,3),FS (x,4),FS(x,5),FS(x,6)],\n x=-7..14,y=-5..5,color=[black,red,blue,gree n,magenta,coral,\n COLOR(RGB,.4,0,.9)],linestyle=[2,1$6],discont=tru e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)oper atorG%&arrowGF)-%$sumG6$-%$sinG6#*&%\"kG\"\"\"9$F5/F4;F59%F)F)F)" }} {PARA 13 "" 1 "" {GLPLOT2D 728 422 422 {PLOTDATA 2 "6+-%'CURVESG6)7gn7 $$!\"(\"\"!$!3>LW[U#3[L\"!#<7$$!3U/Z&*zaP%)pF-$!3iU)**[hHsO\"F-7$$!3'y 92'o1yqpF-$!3#e*yRgue'R\"F-7$$!3s'))49)=\\bpF-$!3x*y;RYk4V\"F-7$$!36on ?]<5SpF-$!3Wxu2gF7n9F-7$$!3Q%y5cw%yCpF-$!38Tr;Fxw/:F-7$$!3[+)z,'Re5pF- $!3N=,u`MDT:F-7$$!3yp()p[)ze*oF-$!3ePTIcWv!e\"F-7$$!39Q1dNGn!)oF-$!3D1 k#oo&eB;F-7$$!3eqk_\"f9b'oF-$!3EX@xNvVo;F-7$$!3=7?MuC#*\\oF-$!3`SQ$4\\ .qr\"F-7$$!3du*Q!H*)=OoF-$!3kb^1zh)>w\"F-7$$!358\"4)4#G2#oF-$!3V.l!H;/ 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" }}{PARA 0 "" 0 "" {TEXT -1 66 "However if the partial sums a re averaged, we do have convergence. " }}{PARA 0 "" 0 "" {TEXT -1 22 " This is the so-called " }{TEXT 266 11 "Cesaro sum " }{TEXT -1 15 "of t he series. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 75 "xx := Pi/3;\nseq(FS(xx,n),n=1..20);\nadd(FS(xx,n),n =1..198)/198;\ncot(xx/2)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG, $*&\"\"$!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "66,$*&\"\" #!\"\"\"\"$#\"\"\"F%F)*$F'F(F*F#\"\"!F+F#F*F*F#F+F+F#F*F*F#F+F+F#F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"\"\"$#\"\"\"F%F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"\"\"$#\"\"\"F%F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "xx := 1.;\nadd(FS(xx,n),n=1..1000)/1000;\ncot(xx/2)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6Ye^\"*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5 'QC:*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 90 ": This example can be related to the previous example via differen tiation or integration. " }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sin*x/(2-2*cos*x),x) = ln(2-2*cos*x)/2+c;" "6#/-%$I ntG6$*(%$sinG\"\"\"%\"xGF),&\"\"#F)*(F,F)%$cosGF)F*F)!\"\"F/F*,&*&-%#l nG6#,&F,F)*(F,F)F.F)F*F)F/F)F,F/F)%\"cGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 259 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sin*k*x,x)=-cos*k*x/k+c" "6#/-%$IntG6$*(%$sinG\"\" \"%\"kGF)%\"xGF)F+,&**%$cosGF)F*F)F+F)F*!\"\"F/%\"cGF)" }{TEXT -1 2 ". " }}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 }