{VERSION 6 0 "IBM INTEL LINUX" "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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "# FOURIER SINE SERIE S OF A SQUARE WAVE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# f(x) = 1 on (0,Pi); f(x) = -1 on (-Pi,0); f has period 2*Pi." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# First we need to calculate the Fourier coefficients." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "bn := (2/Pi)*Int(sin(n*x), x=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bnG,$,$-%$IntG6$-%$sinG6#*&%\"nG\"\"\"%\" xGF//F0;\"\"!%#PiG*&\"\"#F/F4!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"# \"\"\"%#PiG!\"\",&F&F(-%$cosG6#*&F'F&%\"nGF&F&F&F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# This is 0 unless n is odd." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 94 "# Note that, unlike the coefficients of the triangl e wave, these decrease like 1/n, not 1/n^2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "bodd := k -> (4/Pi)*(1/(2*k+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%bod dGj+6#%\"kG6\"6$%)operatorG%&arrowGF(,$*(\"\"%\"\"\"%#PiG!\"\",&*&\"\" #F/9$F/F/F/F/F1F/F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "partialsum := K -> sum(bodd(k)*sin((2*k+1)*x), k=0..K);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%+partialsumGj+6#%\"KG6\"6$%)operator G%&arrowGF(-%$sumG6$*&-%%boddG6#%\"kG\"\"\"-%$sinG6#*&,&*&\"\"#F4F3F4F 4F4F4F4%\"xGF4F4/F3;\"\"!9$F(F(F(6#F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(partialsum(0), x=-8..8);" }}{PARA 13 "" 1 "" {GLPLOT2D 387 387 387 {PLOTDATA 2 "6%-%'CURVESG6$7]s7$$!\")\"\"!!3WL2 ^V+pf7!#<7$$!3Anmm\"p0k&zF-$!3M_'>@!ocm7F-7$$!3cLLLQ6G\"zF-!3O]2mxo. r7F-7$$!3*)*****\\2<#pyF-$!3+fHEw<4t7F-7$$!37nmmmFiDyF-!3#G[%GfFFF\" F-7$$!3YLLLe%G?y(F-$!3\"\\DK2-X*p7F-7$$!3z******\\TVQxF-!305q%fM\\ZE \"F-7$$!3+nmmT)R[p(F-$!3gxkbT/:d7F-7$$!3NLLLLbC^wF-!3a\"y:nuirC\"F-7 !3A+++N3_*\$$F-!3gQ#>p/(3%>\"F-7!3?mmmOhzZtF-!3#oI3ppvN6\"F-7!3 K*****\\%e;xrF-!3gnHqNg!e#**!#=7!3LLLLb1qF-!3p[5'R(ovU)F_o7!3 oKLL8UxMoF-!3_st\"G:*HsmF_o7!3\"HLLL(G,jmF-!3@H[YYrm?ZF_o7!3++++ &)y1#\\'F-!3^m!H)GtFSEF_o7!30nmm'*G7@jF-!3DEE\"zI0#H[!#>7!3o+++5 ]jihF-\"3&*f9AY\"z6\"F_o7!3XLLLBr9/gF-\"36NG4w%)*o]F_o7!3=mmm' 4U+%eF-\"3C*zF)f:TfaF_o7!3!)******pqfn&F-\"3rif7P)H_E(F_o7!39nm m;%>i]&F-\"35&R[v&o#p#*)F_o7!3fLLLj<]OF-\"30Uwyb3@L5F-7!3)pmmmR Gt;&F-\"3;0VftV6F-7!3S+++I]:)*\\F-\"3ozV>wXg@7F-7!3dnm;Hs96\\F -\"3k]\\tmE<[7F-7!3(QLLG%RT#[F-\"3Xsqn.wHl7F-7!3-nm\"z_N1y%F- \"363Gt!ev-F\"F-7!3<++]F;8PZF-\"3/%>*o#)\\GF\"F-7!3KLL3Fxip%F- \"3y*pN4Z:IF\"F-7!3YmmmEQ7]YF-\"3(\\h8%*>s2F\"F-7!37F-7!3HLLLxdVVF-\"3\">:YYaAw=\"F-7!3zmmm;'G5<%F- \"39?G\"=3x64\"F-7!3K+++![z%)*RF-\"3%*ROThICB'*F_o7!3@+++S=ADQF- \"3&GIny*)=>/)F_o7!36++++U'>lF-\"3MTBz*Rz>iF_o7!33+++g)*\$F- \"3%)oHEJXpG%F_o7!3/+++?D.=LF-\"3o4!>&Hk[B#F_o7!3rmmmT:TmJF- \"3es=\"pc9(fJFip7!3SLLLj0z9IF-!3zZho>8<5;F_o7!39+++b&)\\MGF-!3z @!*es()[QF_o7!3*ommma1Ul#F-!3Ytt_%4G'fF_o7!3/nmmmbZ,DF-!3W,IhG U#\\g(F_o7!3>nmm'eW([BF-!3[(y7nR&)*p!*F_o7!3zLLL[@3r@F-!3H.YP:Wk ]5F-7!3S+++5(>M*>F-!3Ic\\=[\\@h6F-7!3\"RLL[eh=F-!3Rv]VJJnG7F- 7!3Unmm')p*)y;F-!3Ob5awB\"F-7!3x******>i^p6F-!3\"3#=\"Qh&4s6F-7 !3*)******Hn@05F-!3=!o01o\\2\"F-7!3\"=LLL[uyL)F_o!3!QD1EH6\"G% *F_o7!3qkmmm;eBmF_o!3MbxA\"4W,yF_o7!33mmm;dK\$F_o!3+XL5yiFfhF_ o7!3Xnmmm(p]ZF_o!3N2V1G#zgL%F_o7!38,++]?-x'3lubo]AF_o7 !3?\"[LLLLu*y!#?!3Qpa!)*)>_05Fip7\"3Pfmm;+#[o\"F_o\"3PlC^b\\/N@F_ o7\"3bmmmVh[MF_o\"3*4lF_o\"37W;LcE6DxF_o7\"3*\\KLL)>lx\")F_o\"3^A CpAC!)*G*F_o7\"3AemmmK\"f)*F_o\"3r[jF@B'*f5F-7\"3jKLLVf!\\:\"F- \"3x@-eSZqk6F-7\"3W******f0AE8F-\"33%)R<6[B\"F-7\"3;*****\\_>+T \"F-\"3wloqiXs7F-7\"3h)****\\X;F-\"3J.kg+(GkTh;F-\"3#Hf/!*>:!o7F -7\"3a)****\\.wN#=F-\"3UyI8DsxK7F-7\"3v)*****\\ct&)>F-\"3Ge3-MM> l6F-7\"3m)****\\D'yl@F-\"3%[aQagQW0\"F-7\"3e)*****foeM#F-\"3'4n 07+If4*F_o7\"3RlmmO.i2DF-\"3e'H80WK?a(F_o7\"3?KLL8QSpEF-\"3kfp%= sg6z&F_o7\"3&fmmm?U@%GF-\"3/())=_&\\*fvF_o7\"3p*******f!)[,F- \"3-VTeMT.4;F_o7\"3\"GLLesSrJF-!3U&Gp5+^bzFip7\"3%fmmm\"RzKF- !31'o#=!)R')eBF_o7\"3%GLL)*)e!*\\F-!3z>iGs(3bWF_o7\"3t******z Q=qOF-!3Tlhj0y;@kF_o7\"3plmm^\"*>JQF-!3S_8zzy25)F_o7\"3mJLLBW@# *RF-!3F?fN&>A3d*F_o7\"3Mlmmw<_gTF-!3m=2[2MA%3\"F-7\"3.******H\"H )GVF-!3:w&ylQs!=\"F-7\"3VKL3t/6T%F-!3G/4:(z)e@\"F-7\"3%emm;L! 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