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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 Output" -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 "Norma l" -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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 54 "Further symmetry properties relat ed to Fourier series " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, \+ Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2 007" }}{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 Fou rier transform procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m- file " }{TEXT 369 9 "fourier.m" }{TEXT -1 37 " contains the code for t he procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 1 " " }{TEXT -1 24 "used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can b e read into a Maple session by a command similar to the one that follo ws, 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 60 "Functions w hich are even or odd with respect to an interval " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " is defined on the interva l " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 25 " satisfies the condition " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(a+b-x)=f(x)" "6#/-%\"fG6#,(%\"aG\"\"\"%\"bGF)% \"xG!\"\"-F%6#F+" }{TEXT -1 10 ", for all " }{TEXT 421 1 "x" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "a<=x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b" "6#1%!G%\"bG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "then " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " co uld be called " }{TEXT 262 33 "even with respect to the interval" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 59 "The graph of such a function is symme trical about the line " }{XPPEDIT 18 0 "x=(a+b)/2" "6#/%\"xG*&,&%\"aG \"\"\"%\"bGF(F(\"\"#!\"\"" }{TEXT -1 36 ", which is the vertical line \+ in the " }{TEXT 420 3 "x-y" }{TEXT -1 39 " plane which bisects the int erval from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 8 " on the " }{TEXT 422 1 "x" }{TEXT -1 7 " axis. 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" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following formulas holds: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Int(f(x),x = a .. m) = Int(f(x),x = m . . b), ``],[Int(f(x),x = a .. b) = 2*Int(f(x),x = a .. m), ``]);" "6#-% *PIECEWISEG6$7$/-%$IntG6$-%\"fG6#%\"xG/F.;%\"aG%\"mG-F)6$-F,6#F./F.;F2 %\"bG%!G7$/-F)6$-F,6#F./F.;F1F9*&\"\"#\"\"\"-F)6$-F,6#F./F.;F1F2FEF:" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 418 16 "___ _____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "m=(a+b)/2" "6#/%\"mG*&,&%\"aG\"\"\"%\"bGF(F(\"\"#!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Exp lanation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x) ,x = m .. b);" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"mG%\"bG" }{TEXT -1 10 " ... " }{XPPEDIT 18 0 "PIECEWISE([x = a+b-t, t = a+b-x],[dx = -dt , `x =`*m*` implies t` = m],[``, `x =`*b*` implies t` = a]);" "6#-%* PIECEWISEG6%7$/%\"xG,(%\"aG\"\"\"%\"bGF+%\"tG!\"\"/F-,(F*F+F,F+F(F.7$/ %#dxG,$%#dtGF./*(%$x~=GF+%\"mGF+%,~implies~~tGF+F97$%!G/*(F8F+F,F+F:F+ F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = -Int(f(a+b-t),t = m .. a);" "6 #/%!G,$-%$IntG6$-%\"fG6#,(%\"aG\"\"\"%\"bGF.%\"tG!\"\"/F0;%\"mGF-F1" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f(a+b-t),t = a .. m);" "6#/%!G-%$IntG6$-%\"fG6#,(%\"aG\"\"\"%\"bG F-%\"tG!\"\"/F/;F,%\"mG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "( since switching the limits changes the sign of the integral )" } }{PARA 256 "" 0 "" {TEXT -1 3 "= " }{XPPEDIT 18 0 "Int(f(t),t = a .. \+ m);" "6#-%$IntG6$-%\"fG6#%\"tG/F);%\"aG%\"mG" }{TEXT -1 2 " " }} {PARA 257 "" 0 "" {TEXT -1 26 "( because, by hypothesis, " }{XPPEDIT 18 0 "f(a+b-t) = f(t);" "6#/-%\"fG6#,(%\"aG\"\"\"%\"bGF)%\"tG!\"\"-F%6 #F+" }{TEXT -1 9 " for all " }{TEXT 419 1 "t" }{TEXT -1 3 " ) " }} {PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "Int(f(x),x = a .. m) ;" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"mG" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 75 "(since it does not matter what variable we us e in the definite integral ). " }}{PARA 0 "" 0 "" {TEXT -1 31 "This pr oves the first formula. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = a .. b) = Int( f(x),x = a .. m)+Int(f(x),x = m .. b);" "6#/-%$IntG6$-%\"fG6#%\"xG/F*; %\"aG%\"bG,&-F%6$-F(6#F*/F*;F-%\"mG\"\"\"-F%6$-F(6#F*/F*;F6F.F7" } {XPPEDIT 18 0 "``=2*Int(f(x),x=a..m)" "6#/%!G*&\"\"#\"\"\"-%$IntG6$-% \"fG6#%\"xG/F.;%\"aG%\"mGF'" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Example " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x) = sin*x;" "6#/-%\"fG6#%\"xG*&%$sinG\"\"\"F'F*" }{TEXT -1 4 " is " }{TEXT 262 4 "even" }{TEXT -1 30 " with respect to the interval " }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 7 " since " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(Pi-x) = sin(Pi-x); " "6#/-%\"fG6#,&%#PiG\"\"\"%\"xG!\"\"-%$sinG6#,&F(F)F*F+" }{XPPEDIT 18 0 "`` = sin*x;" "6#/%!G*&%$sinG\"\"\"%\"xGF'" }{XPPEDIT 18 0 "`` = \+ f(x);" "6#/%!G-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "It is clear from the following \+ picture that " }{XPPEDIT 18 0 "Int(sin*x,x = Pi/2 .. Pi) = Int(sin*x,x = 0 .. Pi/2);" "6#/-%$IntG6$*&%$sinG\"\"\"%\"xGF)/F*;*&%#PiGF)\"\"#! \"\"F.-F%6$*&F(F)F*F)/F*;\"\"!*&F.F)F/F0" }{TEXT -1 9 " so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*x,x = 0 .. Pi ) = Int(sin*x,x = 0 .. Pi/2)+Int(sin*x,x = Pi/2 .. Pi);" "6#/-%$IntG6$ *&%$sinG\"\"\"%\"xGF)/F*;\"\"!%#PiG,&-F%6$*&F(F)F*F)/F*;F-*&F.F)\"\"#! \"\"F)-F%6$*&F(F)F*F)/F*;*&F.F)F6F7F.F)" }{XPPEDIT 18 0 "`` = 2*Int(si n*x,x = 0 .. Pi/2);" "6#/%!G*&\"\"#\"\"\"-%$IntG6$*&%$sinGF'%\"xGF'/F- ;\"\"!*&%#PiGF'F&!\"\"F'" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 579 218 218 {PLOTDATA 2 "6--%'CURVESG6%7S7$$\"\"!F )F(7$$\"3%)eD2LzxZo!#>$\"3v9-&QTFC%oF-7$$\"3)\\$px*G*f!G\"!#=$\"3q>Km$ *>5x7F37$$\"3+5@exGm]>F3$\"3/`P()fcJQ>F37$$\"3[99!=3o^i#F3$\"3/)Q8J`>^ f#F37$$\"35!\\D0[nkH$F3$\"3'yFM)o\")3PKF37$$\"37\"=Za&z%)=RF3$\"3b2`lK +J>QF37$$\"3edXa()oGjXF3$\"33g-l[Vb1WF37$$\"3W%3**Hbm(H_F3$\"3OWs$HJ6Y *\\F37$$\"37PRr4)3T*eF3$\"3g4tHMSrebF37$$\"3y\"[)yykYxlF3$\"32tCIi;N8h F37$$\"3[s'ocGo$zrF3$\"3iG>vb;KylF37$$\"3UQ0;gr'p&yF3$\"3S'e5DeyJ2(F37 $$\"3vFMt?$[t`)F3$\"3Cf#G%e3SPvF37$$\"3u\"p30k@I>*F3$\"3C$=wEj'y^zF37$ $\"3#*R7\\HeV)y*F3$\"3=)oD&\\m_)H)F37$$\"3#G[))*)3W'\\5!#<$\"3Y4SV'zgC n)F37$$\"30'[@XS@'46Fdp$\"3oX3_YFIb*)F37$$\"3G9w$3G*Qz6Fdp$\"3=c,/>?tV #*F37$$\"3>%3*3tc9T7Fdp$\"3#*zOG!*[bh%*F37$$\"3Ey0DBA!*38Fdp$\"3'*))o< xI,f'*F37$$\"3qjE.#[AMP\"Fdp$\"3W%*yQT$\\e!)*F37$$\"3*R:_MgU2W\"Fdp$\" 3!f%R/z#\\b\"**F37$$\"3\"Qu#)**[jD]\"Fdp$\"3!>/b+VIn(**F37$$\"3=hw\"ym X#p:Fdp$\"274'Gxz)*****Fdp7$$\"3mwM!*4(4&Q;Fdp$\"39vB0ZK3x**F37$$\"3CD L:iU!))p\"Fdp$\"3[oG=d;==**F37$$\"3o(R1/.CRw\"Fdp$\"3Vz\\$>Q(39)*F37$$ \"32Id)H7*>J=Fdp$\"3?VTu'[jGm*F37$$\"3Gfb7wY,(*=Fdp$\"3Sm0JN*4EZ*F37$$ \"3cM5'zg%pg>Fdp$\"3ADf%RFx%\\#*F37$$\"3**oJ8:.SJ?Fdp$\"3?af@Y>%y&*)F3 7$$\"3)**p!zPD$\\4#Fdp$\"3?%>'G5ccd')F37$$\"3k-G %)H)F37$$\"3wak3@YBCAFdp$\"3cJr!4)G)*RzF37$$\"39/bF37$$\"3H.6)e58)4IFdp$\"3M.OA\"o%)RJ\"F37$$\"3Kt2RXCLtIFd p$\"3ls_wI6s?oF-7$$\"3!)***\\/l#fTJFdp$\"3pawpOMzRJ!#E-%'COLOURG6&%$RG BG$\"#5!\"\"F(F(-%*THICKNESSG6#\"\"#-%)POLYGONSG677&F'7$$\"+\\Z#)\\')! #6$\"+GD/R')Fj[l7$Fh[lF(F'7&Fg[l7$$\"+,\"*f<;!#5$\"+.Rb5;Fb\\l7$F`\\lF (F]\\l7&F_\\l7$$\"+6_*RY#Fb\\l$\"+X\"Q\"RCFb\\l7$Fh\\lF(Fe\\l7&Fg\\l7$ $\"+%y,gJ$Fb\\l$\"+%\\kbD$Fb\\l7$F_]lF(F\\]l7&F^]l7$$\"+>*eR;%Fb\\l$\" +1%pY/%Fb\\l7$Ff]lF(Fc]l7&Fe]l7$$\"+GP7]\\Fb\\l$\"+*QD/v%Fb\\l7$F]^lF( Fj]l7&F\\^l7$$\"+_=:kdFb\\l$\"+#e=-X&Fb\\l7$Fd^lF(Fa^l7&Fc^l7$$\"+-4-1 mFb\\l$\"+uQ#f8'Fb\\l7$F[_lF(Fh^l7&Fj^l7$$\"+o+>XuFb\\l$\"+)f\"=wnFb\\ l7$Fb_lF(F__l7&Fa_l7$$\"+-(y$3$)Fb\\l$\"+.d'\\Q(Fb\\l7$Fi_lF(Ff_l7&Fh_ l7$$\"+?dno!*Fb\\l$\"+^NxvyFb\\l7$F``lF(F]`l7&F_`l7$$\"+(3!fC**Fb\\l$ \"+Gost$)Fb\\l7$Fg`lF(Fd`l7&Ff`l7$$\"+#*=Sy5!\"*$\"+uW.7))Fb\\l7$F^alF (F[al7&F]al7$$\"+&yB7;\"F`al$\"+z7\"H<*Fb\\l7$FfalF(Fcal7&Feal7$$\"+)* RVO7F`al$\"+Vw>Y%*Fb\\l7$F]blF(Fjal7&F\\bl7$$\"+;i'eK\"F`al$\"+[Fa,(*F b\\l7$FdblF(Fabl7&Fcbl7$$\"+Oqi,9F`al$\"+\"o\\s&)*Fb\\l7$F[clF(Fhbl7&F jbl7$$\"+!)[v*[\"F`al$\"+)Qzr'**Fb\\l7$FbclF(F_cl7&Facl7$$\"+Cjzq:F`al $\"\"\"F)7$FiclF(Ffcl-%&COLORG6&F[[lF[dl$\"#&)!\"#Fadl-%&STYLEG6#%,PAT CHNOGRIDG-Fd[l677&7$$\"+Ijzq:F`alF[dl7$$\"+xXHd;F`al$\"+#e8E'**Fb\\l7$ F_elF(7$F\\elF(7&F^el7$$\"+SibK<=F`al$\"+<\"pzp*Fb\\l7$F^flF(F[fl7&F]fl7$$\"+3lR->F`al$\"+^g Ab%*Fb\\l7$FeflF(Fbfl7&Fdfl7$$\"+@A>()>F`al$\"+'=Eb9*Fb\\l7$F\\glF(Fif l7&F[gl7$$\"+-(3e1#F`al$\"+8$G'*z)Fb\\l7$FcglF(F`gl7&Fbgl7$$\"+9:@Z@F` al$\"+V$=UQ)Fb\\l7$FjglF(Fggl7&Figl7$$\"+>%)RJAF`al$\"+b$Hi*yFb\\l7$Fa hlF(F^hl7&F`hl7$$\"+N`J:BF`al$\"+g\"RTN(Fb\\l7$FhhlF(Fehl7&Fghl7$$\"+) >M;S#F`al$\"+:BdUnFb\\l7$F_ilF(F\\il7&F^il7$$\"++RmxCF`al$\"+pz:ihFb\\ l7$FfilF(Fcil7&Feil7$$\"+P`DjDF`al$\"+wSKmaFb\\l7$F]jlF(Fjil7&F\\jl7$$ \"+?#)>\\EF`al$\"+YAPFZFb\\l7$FdjlF(Fajl7&Fcjl7$$\"+8,-KFF`al$\"++E<#) RFb\\l7$F[[mF(Fhjl7&Fjjl7$$\"+E.B2GF`al$\"+*yo;G$Fb\\l7$Fb[mF(F_[m7&Fa [m7$$\"+WDm'*GF`al$\"+8_)[U#Fb\\l7$Fi[mF(Ff[m7&Fh[m7$$\"+jLUsHF`al$\"+ ibj$o\"Fb\\l7$F`\\mF(F]\\m7&F_\\m7$$\"+27bgIF`al$\"+lmF&4)Fj[l7$Fg\\mF (Fd\\m7&Ff\\m7$$\"+]EfTJF`al$\"+PMzRJF37$F^]mF(F[]m-F_dl6&F[[lFadlF[dl FadlFddl-F$6$7$7$$\"3c'*[zEjzq:FdpF(7$Fi]mF[dl-Fiz6&F[[lF)F)F)-%%TEXTG 6&7$$\"#NF^[l$!\"&FcdlQ\"x6\"F\\^m-%%FONTG6$%*HELVETICAG\"\"*-F_^m6&7$ $!\"'Fcdl$\"$<\"FcdlQ\"yFg^mF\\^mFh^m-F_^m6&7$$F\\_mF^[l$\"#(*FcdlQ*y~ =~sin~xFg^m-F_dl6&F[[lFh_mF(F(Fh^m-%+AXESLABELSG6%Q!Fg^mFa`mFh^m-Fi^m6 $%'SYMBOLGF\\_m-%*AXESTICKSG6$7%/F)%\"0G/$\"+Fjzq:F`al%$p/2G/$\"+aEfTJ F`al%\"pG\"\"$-%%VIEWG6$;$F^[lF^[lFb^m;Fd^mFb_m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curv e 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose t hat " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " is define d on the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 25 " satisfies the condition " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "f(a+b-x) = -f(x);" "6#/-%\"fG6#,(%\"a G\"\"\"%\"bGF)%\"xG!\"\",$-F%6#F+F," }{TEXT -1 10 ", for all " }{TEXT 424 1 "x" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "a<=x" "6#1%\"aG% \"xG" }{XPPEDIT 18 0 "``<=b" "6#1%!G%\"bG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "then " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 17 " could be called " }{TEXT 262 32 "odd with respect to the interval" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 "The graph of such a fun ction is symmetrical about the point" }{XPPEDIT 18 0 "``((a+b)/2,0);" "6#-%!G6$*&,&%\"aG\"\"\"%\"bGF)F)\"\"#!\"\"\"\"!" }{TEXT -1 46 ", whic h is the mid-point of the interval from " }{XPPEDIT 18 0 "x=a" "6#/%\" xG%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" } {TEXT -1 8 " on the " }{TEXT 425 1 "x" }{TEXT -1 7 " axis. 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" }{XPPEDIT 18 0 "PIECEWISE([x = a+b- t, t = a+b-x],[dx = -dt, `x =`*m*` implies t` = m],[``, `x =`*b*` imp lies t` = a]);" "6#-%*PIECEWISEG6%7$/%\"xG,(%\"aG\"\"\"%\"bGF+%\"tG! \"\"/F-,(F*F+F,F+F(F.7$/%#dxG,$%#dtGF./*(%$x~=GF+%\"mGF+%,~implies~~tG F+F97$%!G/*(F8F+F,F+F:F+F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = -Int( f(a+b-t),t = m .. a);" "6#/%!G,$-%$IntG6$-%\"fG6#,(%\"aG\"\"\"%\"bGF.% \"tG!\"\"/F0;%\"mGF-F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f(a+b-t),t = a .. m);" "6#/%!G-%$IntG6$ -%\"fG6#,(%\"aG\"\"\"%\"bGF-%\"tG!\"\"/F/;F,%\"mG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 63 "( since switching the limits changes the \+ sign of the integral )" }}{PARA 256 "" 0 "" {TEXT -1 3 "= " } {XPPEDIT 18 0 "-Int(f(t),t = a .. m);" "6#,$-%$IntG6$-%\"fG6#%\"tG/F*; %\"aG%\"mG!\"\"" }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 26 "( b ecause, by hypothesis, " }{XPPEDIT 18 0 "f(a+b-t) = -f(t);" "6#/-%\"fG 6#,(%\"aG\"\"\"%\"bGF)%\"tG!\"\",$-F%6#F+F," }{TEXT -1 9 " for all " } {TEXT 426 1 "t" }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "-Int(f(x),x = a .. m);" "6#,$-%$IntG6$-%\"fG6#%\"xG/F* ;%\"aG%\"mG!\"\"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 75 "(s ince it does not matter what variable we use in the definite integral \+ ). " }}{PARA 0 "" 0 "" {TEXT -1 31 "This proves the first formula. " } }{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = a .. b) = Int(f(x),x = a .. m)-Int(f(x),x = m .. b);" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;%\"aG%\"bG,&-F%6$-F(6#F*/F* ;F-%\"mG\"\"\"-F%6$-F(6#F*/F*;F6F.!\"\"" }{XPPEDIT 18 0 "`` = 0;" "6#/ %!G\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Example " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x) = cos*x;" " 6#/-%\"fG6#%\"xG*&%$cosG\"\"\"F'F*" }{TEXT -1 4 " is " }{TEXT 262 3 "o dd" }{TEXT -1 30 " with respect to the interval " }{XPPEDIT 18 0 "[0,P i]" "6#7$\"\"!%#PiG" }{TEXT -1 7 " since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(Pi-x)=cos(Pi-x)" "6#/-%\"fG6#,&%#PiG\"\"\"% \"xG!\"\"-%$cosG6#,&F(F)F*F+" }{XPPEDIT 18 0 "``=-cos*x" "6#/%!G,$*&%$ cosG\"\"\"%\"xGF(!\"\"" }{XPPEDIT 18 0 "``=-f(x)" "6#/%!G,$-%\"fG6#%\" xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 44 "It is clear from the following picture that " } {XPPEDIT 18 0 "Int(cos*x,x = Pi/2 .. Pi) = -Int(cos*x,x = 0 .. Pi/2); " "6#/-%$IntG6$*&%$cosG\"\"\"%\"xGF)/F*;*&%#PiGF)\"\"#!\"\"F.,$-F%6$*& F(F)F*F)/F*;\"\"!*&F.F)F/F0F0" }{TEXT -1 9 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*x,x=0..Pi)=Int(cos*x,x = 0 .. Pi/2)+Int(cos*x,x =Pi/2 .. Pi)" "6#/-%$IntG6$*&%$cosG\"\"\"%\"xGF) /F*;\"\"!%#PiG,&-F%6$*&F(F)F*F)/F*;F-*&F.F)\"\"#!\"\"F)-F%6$*&F(F)F*F) /F*;*&F.F)F6F7F.F)" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 2 ". 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Then, bec ause of the periodicity, if " }{TEXT 412 1 "n" }{TEXT -1 8 " is any " }{TEXT 262 11 "odd integer" }{TEXT -1 9 " we have " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "f(n*L-x) = -f(x);" "6#/-%\"fG6#,&*&% \"nG\"\"\"%\"LGF*F*%\"xG!\"\",$-F%6#F,F-" }{TEXT -1 9 " for all " } {TEXT 413 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "The periodic function " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 36 " with this property could be call ed " }{TEXT 262 10 "pseudo-odd" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 33 " is symmetrical about the points " }{XPPEDIT 18 0 "` . . \+ . `,``(-3*L/2,0),``(-L/2,0),``(L/2,0),``(3*L/2,0),``(5*L/2,0),` . . . \+ `;" "6)%(~.~.~.~G-%!G6$,$*(\"\"$\"\"\"%\"LGF*\"\"#!\"\"F-\"\"!-F%6$,$* &F+F*F,F-F-F.-F%6$*&F+F*F,F-F.-F%6$*(F)F*F+F*F,F-F.-F%6$*(\"\"&F*F+F*F ,F-F.F#" }{TEXT -1 2 ". 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L) = 0, Int(f(x),x = -L .. 0) = 0],[Int(f(x),x = -L .. L) = 0,``])" "6#-%*PIEC EWISEG6$7$/-%$IntG6$-%\"fG6#%\"xG/F.;\"\"!%\"LGF1/-F)6$-F,6#F./F.;,$F2 !\"\"F1F17$/-F)6$-F,6#F./F.;,$F2F;F2F1%!G" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 410 21 "_____________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The first two results follow from the fact that " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 46 " is odd with resp ect to each of the intervals " }{XPPEDIT 18 0 "[0,L]" "6#7$\"\"!%\"LG " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "[-L,0]" "6#7$,$%\"LG!\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "The formula " } {XPPEDIT 18 0 "Int(f(x),x = -L .. L) = 0" "6#/-%$IntG6$-%\"fG6#%\"xG/F *;,$%\"LG!\"\"F.\"\"!" }{TEXT -1 49 " follows immediately from the fir st two results. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examp le " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x ) = exp(sin*x)*cos*5*x;" "6#/-%\"fG6#%\"xG**-%$expG6#*&%$sinG\"\"\"F'F .F.%$cosGF.\"\"&F.F'F." }{TEXT -1 25 " is periodic with period " } {XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 13 " and is al so " }{TEXT 262 10 "pseudo-odd" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 366 "f := x -> e xp(sin(x))*cos(5*x):\n'f(x)'=f(x);\np1 := plot(f(x),x=-Pi-.4..3*Pi+.4) :\np2 := plot([[[-Pi,-3.3],[-Pi,3.3]],[[Pi,-3.3],[Pi,3.3]],\n [[2*P i,-3.3],[2*Pi,3.3]],[[3*Pi,-3.3],[3*Pi,3.3]]],color=navy,linestyle=2): \np3 := plot([[[-Pi/2,0],[Pi/2,0],[3*Pi/2,0],[5*Pi/2,0]]$3],color=blac k,\n style=point,symbol=[circle,diamond,cross]):\nplots[display ](p1,p2,p3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&-%$exp G6#-%$sinGF&\"\"\"-%$cosG6#,$*&\"\"&F.F'F.F.F." }}{PARA 13 "" 1 "" {GLPLOT2D 706 216 216 {PLOTDATA 2 "6,-%'CURVESG6$7c\\l7$$!3')***4pk#fT N!#<$\"3V')fi(HMG9'!#=7$$!3wrV'=/u^]$F*$\"3YDezVEm!\\$F-7$$!3mV(=oVb(o MF*$\"3%Q=h!R5gf*)!#>7$$!3b:JxJoLKMF*$!3![*z#))Qfeb\"F-7$$!3)p[FnA=fR$ F*$!3Y57nI_Q!z$F-7$$!3NIij;53BLF*$!3S8XAormvtF-7$$!39u\\a1QC]KF*$!3k;N aQ(e2a*F-7$$!3H&[%y*)**RMKF*$!37M/*oh))3\")*F-7$$!3*f*R-thb=KF*$!34!=: ?sU4+\"F*7$$!3q1NEcBr-KF*$!3^be$*[Wt85F*7$$!3%y,.&R&oo=$F*$!3k2#f$)Q/' 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$$\"3mHIgPqWl$*F*$!31$f$4v2u95F*7$$\"3[k5:9yL#Q*F*$!3&pS!Hkz$*>5F*7$$ \"3K*4*p!fG#*R*F*$!3*3kgPJ;v,\"F*7$$\"3oQ7*op\"zm%*F*$!3IO=tmE*yP*F-7$ $\"3-yL3.[NM&*F*$!3_d?!e4R9^F-7$$\"3 /)o1BPm&z'*F*$!3Ee4_$\\`MF#F-7$$\"3#[M=p:s@v*F*$\"3M'p1nm%>%z%F87$$\"3 %)***H:%zxC)*F*$\"38Y.-\\&*=>GF--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F`cp F_cp-F$6%7$7$$!37$z*e`EfTJF*$!3#)*************H$F*7$Fecp$\"3#)******** *****H$F*-Fibp6&F[cp$\")!\\DP\"!\")F^dp$\")viobF`dp-%*LINESTYLEG6#\"\" #-F$6%7$7$$\"37$z*e`EfTJF*Fgcp7$F[epFjcpF\\dpFcdp-F$6%7$7$$\"3C'ezrI&= $G'F*Fgcp7$FbepFjcpF\\dpFcdp-F$6%7$7$$\"3Nz$p2'zxC%*F*Fgcp7$FiepFjcpF \\dpFcdp-F$6&7&7$$!3c'*[zEjzq:F*F_cp7$$\"3c'*[zEjzq:F*F_cp7$$\"3n*o%Q! )*)Q7ZF*F_cp7$$\"3y#[uRj\")R&yF*F_cp-%'SYMBOLG6#%'CIRCLEG-Fibp6&F[cpF` cpF`cpF`cp-%&STYLEG6#%&POINTG-F$6&F^fp-F\\gp6#%(DIAMONDGF_gpFagp-F$6&F ^fp-F\\gp6#%&CROSSGF_gpFagp-%+AXESLABELSG6%Q\"x6\"Q!Fchp-%%FONTG6#%(DE FAULTG-%%VIEWG6$;$!+aEfTN!\"*$\"+izxC)*F_ipFhhp" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 46.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curv e 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(Pi-x) = exp(sin(Pi-x))*cos(5*(Pi-x));" "6#/-%\"fG6#,& %#PiG\"\"\"%\"xG!\"\"*&-%$expG6#-%$sinG6#,&F(F)F*F+F)-%$cosG6#*&\"\"&F ),&F(F)F*F+F)F)" }{XPPEDIT 18 0 "`` = exp(sin*x)*cos(5*Pi-5*x);" "6#/% !G*&-%$expG6#*&%$sinG\"\"\"%\"xGF+F+-%$cosG6#,&*&\"\"&F+%#PiGF+F+*&F2F +F,F+!\"\"F+" }{XPPEDIT 18 0 "`` = exp(sin*x)*cos(Pi-5*x);" "6#/%!G*&- %$expG6#*&%$sinG\"\"\"%\"xGF+F+-%$cosG6#,&%#PiGF+*&\"\"&F+F,F+!\"\"F+ " }{XPPEDIT 18 0 "`` = -exp(sin*x)*cos*5*x;" "6#/%!G,$**-%$expG6#*&%$s inG\"\"\"%\"xGF,F,%$cosGF,\"\"&F,F-F,!\"\"" }{XPPEDIT 18 0 "``=-f(x)" "6#/%!G,$-%\"fG6#%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(sin*x)*cos*5*x,x = -Pi .. Pi) = 0;" "6#/-%$IntG6$**-%$expG6#*&%$sinG\"\"\"%\"xGF-F-%$cosGF-\"\"&F-F.F -/F.;,$%#PiG!\"\"F4\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "More generally, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(sin*x)*cos*k*x,x = -Pi .. Pi) = 0" "6#/-%$IntG6$**-%$expG6# *&%$sinG\"\"\"%\"xGF-F-%$cosGF-%\"kGF-F.F-/F.;,$%#PiG!\"\"F4\"\"!" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "for any odd integer " } {TEXT 409 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Int(exp(sin(x))*cos(5*x),x=- Pi..Pi);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$e xpG6#-%$sinG6#%\"xG\"\"\"-%$cosG6#,$*&\"\"&F.F-F.F.F./F-;,$%#PiG!\"\"F 8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "Maple probably does not obtain this result from symmetry considerations. The indefinite integral is \+ as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "Int(exp(sin(x))*cos(5*x),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"\"\"-%$c osG6#,$*&\"\"&F.F-F.F.F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(\"\" #\"\"\"-%$expG6#-%$sinG6#%\"xGF&-%$cosG6#,$*&\"\"%F&F-F&F&F&F&*(\"#)*F &F'F&-F/6#,$*&F%F&F-F&F&F&!\"\"*(\"$3%F&F'F&F*F&F:*(\"#;F&F'F&-F+6#,$* &\"\"$F&F-F&F&F&F&*&\"$d%F&F'F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 68 "The following definite integral takes a n oticable time to evaluate. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "st := time():\nInt(exp(sin(x))*cos( 47*x),x=-Pi..Pi);\nvalue(%);\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"\"\"-%$cosG6#,$*&\"#ZF.F-F.F .F./F-;,$%#PiG!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%&)R!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "f(x)" "6 #-%\"fG6#%\"xG" }{TEXT -1 36 " is a periodic function with period " } {XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 40 " which sati sfies the symmetry relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(L-x)=f(x)" "6#/-%\"fG6#,&%\"LG\"\"\"%\"xG!\"\"-F%6#F* " }{TEXT -1 22 " for all real numbers " }{TEXT 414 1 "x" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 37 "Then, because of the periodicity, i f " }{TEXT 415 1 "n" }{TEXT -1 8 " is any " }{TEXT 262 11 "odd integer " }{TEXT -1 9 " we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(n*L-x) = f(x);" "6#/-%\"fG6#,&*&%\"nG\"\"\"%\"LGF*F*% \"xG!\"\"-F%6#F," }{TEXT -1 9 " for all " }{TEXT 416 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " I shall say that the periodic function " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 23 " with this property is " }{TEXT 262 11 "pseud o-even" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 32 " is symmetrica l about the lines " }{XPPEDIT 18 0 "x = ` . . . `,-3*L/2,-L/2,L/2,3*L/ 2,5*L/2,` . . . `;" "6)/%\"xG%(~.~.~.~G,$*(\"\"$\"\"\"%\"LGF)\"\"#!\" \"F,,$*&F*F)F+F,F,*&F*F)F+F,*(F(F)F*F)F+F,*(\"\"&F)F*F)F+F,F%" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " matrix([[f(x), g(x), f(x)*`.`*g(x)], [`pseudo-even`, `pseudo-even`, `p seudo-even`], [`pseudo-even`, `pseudo-odd`, `pseudo-odd`], [`pseudo-od d`, `pseudo-even`, `pseudo-odd`], [`pseudo-odd`, `pseudo-odd`, `pseudo -even`]])" "6#-%'matrixG6#7'7%-%\"fG6#%\"xG-%\"gG6#F+*(-F)6#F+\"\"\"% \".GF2-F-6#F+F27%%,pseudo-evenGF7F77%F7%+pseudo-oddGF97%F9F7F97%F9F9F7 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 " " {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 17 " is pseudo-even, " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#% \"xG" }{TEXT -1 19 " is pseudo-odd and " }{XPPEDIT 18 0 "h(x) = f(x)*` .`*g(x)" "6#/-%\"hG6#%\"xG*(-%\"fG6#F'\"\"\"%\".GF,-%\"gG6#F'F," } {TEXT -1 7 ", then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(L-x) = f(L-x)*`.`*g(L-x);" "6#/-% \"hG6#,&%\"LG\"\"\"%\"xG!\"\"*(-%\"fG6#,&F(F)F*F+F)%\".GF)-%\"gG6#,&F( F)F*F+F)" }{XPPEDIT 18 0 "`` = f(x)*`.`*(-g(x))" "6#/%!G*(-%\"fG6#%\"x G\"\"\"%\".GF*,$-%\"gG6#F)!\"\"F*" }{XPPEDIT 18 0 "`` = - f(x)*`.`*g(x )" "6#/%!G,$*(-%\"fG6#%\"xG\"\"\"%\".GF+-%\"gG6#F*F+!\"\"" }{XPPEDIT 18 0 "`` = -h(x)" "6#/%!G,$-%\"hG6#%\"xG!\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "so " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"x G" }{TEXT -1 8 " is odd." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "The Fourie r series of a pseudo-even function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 36 " is a periodic fu nction with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" } {TEXT -1 40 " which satisfies the symmetry relation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(L-x) = -f(x);" "6#/-%\"fG6#,&%\" LG\"\"\"%\"xG!\"\",$-F%6#F*F+" }{TEXT -1 22 " for all real numbers " } {TEXT 365 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Then, \+ because of the periodicity, if " }{TEXT 366 1 "n" }{TEXT -1 8 " is any " }{TEXT 262 11 "odd integer" }{TEXT -1 9 " we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(n*L-x) = -f(x);" "6#/-%\"fG6#,&* &%\"nG\"\"\"%\"LGF*F*%\"xG!\"\",$-F%6#F,F-" }{TEXT -1 9 " for all " } {TEXT 367 1 "x" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 47 "Thus , in the terminology of the first section, " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 6 " is a " }{TEXT 262 11 "pseudo-even" } {TEXT -1 11 " function. 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3#=`#\\mvgw#*F37$$\"3=`:Y(fnk2\"F_\\m$!3Cyh8q?hM(*F37$$\"3Ayb-,vk\"4\" F_\\m$!3OI'y\"HGto**F37$$\"3%4[f;(p*f4\"F_\\m$!3/mk>UAm$***F37$$\"3$QQ $HUkM+6F_\\m$!3YwSVI()o****F37$$\"3t'GFH\"fp/6F_\\m$!3#4jT[*3!o)**F37$ $\"3X*=hNQX!46F_\\m$!3%>^ZE6B]&**F37$$\"31&**G[KWx6\"F_\\m$!3ze&>Xvt]$ )*F37$$\"3o+o4mKWE6F_\\m$!3ZhDbfxuS'*F37$$\"3![JQ'RT+U6F_\\m$!30$=opD$ e7\"*F37$$\"3uG)zJ,lv:\"F_\\m$!3q(3P0$3@k$)F37$$\"3K-2%*\\h:u6F_\\m$!3 Nl6'[U#=WtF37$$\"3sv:q'GZ2>\"F_\\m$!3d1Wd'3dC7'F37$$\"39\"3j:7Fm?\"F_ \\m$!3U_NEow7&z%F37$$\"3c'eCk&p]A7F_\\m$!3)>#=21Q8ZLF37$$\"3G$Hsw+s&R7 F_\\m$!3aYV2KZB)p\"F37$$\"3+++#*eqjc7F_\\m$!36t.J_s\"f^#!#D-%&COLORG6& %$RGBG$\"\"&!\"\"$\"\"!Fjbm$\"\"\"Fjbm-%*THICKNESSG6#\"\"#-%+AXESLABEL SG6$Q\"x6\"Q!Fecm-%%VIEWG6$;$!+aEfTJ!\"*$\"+iqjc7!\")%(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 "" }}{PARA 0 "" 0 "" {TEXT 291 39 "Calculat ion of the constant coefficient" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "f := x -> p iecewise(x<0,sin(x),x " 0 "" {MPLTEXT 1 0 165 "f := x -> piecewise(x<0,sin(x),x " 0 "" {MPLTEXT 1 0 67 "a[1]=1/Pi*Int('f(x )'*cos(x),x=-Pi..Pi);\nvalue(%);\naa(1) := rhs(%):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\",$-%$IntG6$*&-%\"fG6#%\"xGF'-%$cosGF/ F'/F0;,$%#PiG!\"\"F6*$F6F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 #\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "matrix([[k,`|`,seq(k,k=1..12)],['a'[k],`|`,seq(a a(k),k=1..12)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$70% \"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#77 0&%\"aG6#F(F)\"\"!,$*&F*F**&F,F*%#PiGF*!\"\"F?F:,$*(F+F*\"#:F?F>F?F*F: ,$*(\"#F?F?F:,$*(F+F*\"#jF?F>F?F*F:,$*(\"#\\F*\"%vCF?F>F?F ?F:,$*(F+F*\"$V\"F?F>F?F*Q)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "a[k]=0" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 297 1 "k" }{TEXT -1 4 " is " }{TEXT 262 3 "odd" }{TEXT -1 15 ", as expected. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 49 "Calculation of the coefficients of the sine terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f := x -> piecewise(x<0,sin(x),x " 0 "" {MPLTEXT 1 0 77 "b[1]=1/Pi*In t('f(x)'*sin(x),x=-Pi..Pi);\nsimplify(value(%));\nbb(1) := rhs(%):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$-%$IntG6$*&-%\"fG6# %\"xGF'-%$sinGF/F'/F0;,$%#PiG!\"\"F6*$F6F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*(\"\"#!\"\",&%#PiGF'\"\"%F'F'F-F+F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "matrix([[k,`|`,seq(k,k=1..12)],['b'[k],`|`,seq(bb(k),k=1..12)] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$70%\"kG%\"|grG\" \"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#770&%\"bG6#F(F) ,$*(F+!\"\",&%#PiGF*F-F*F*F>FFFFFF " 0 "" {MPLTEXT 1 0 193 "FS := (x,n) ->(Pi^2-8)/(8*Pi)+(Pi+4)/(2*Pi)*sin(x)+\n sum(((1+(-1)^k)/(Pi*k^2*(k ^2-1))+2*cos(Pi*k/2)/(Pi*k^2))*cos(k*x)+\n \+ 2*sin(Pi*k/2)/(Pi*k^2)*sin(k*x),k=2..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),(*(\"\") !\"\",&*$)%#PiG\"\"#\"\"\"F6F/F0F6F4F0F6*&#F6F5F6*(,&F4F6\"\"%F6F6F4F0 -%$sinG6#9$F6F6F6-%$sumG6$,&*&,&**,&F6F6)F0%\"kGF6F6F4F0FI!\"#,&*$)FIF 5F6F6F6F0F0F6**F5F6-%$cosG6#,$*(F5F0F4F6FIF6F6F6F4F0FIFJF6F6-FP6#*&FIF 6F?F6F6F6*,F5F6-F=FQF6F4F0FIFJ-F=FUF6F6/FI;F59%F6F)F)F)" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 44 "The first few ter ms of the Fourier series of" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "FS(x,11);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#,:*(\"\")!\"\",&*$)%#PiG\"\"#\"\"\"F,F%F&F,F*F&F,*&#F ,F+F,*(,&F*F,\"\"%F,F,F*F&-%$sinG6#%\"xGF,F,F,*&#F,\"\"$F,*&F*F&-%$cos G6#,$*&F+F,F5F,F,F,F,F&*&#F+\"\"*F,*&F*F&-F36#,$*&F8F,F5F,F,F,F,F&*&#F +\"#:F,*&F*F&-F;6#,$*&F1F,F5F,F,F,F,F,*&#F+\"#DF,*&F*F&-F36#,$*&\"\"&F ,F5F,F,F,F,F,*&#\"#<\"$:$F,*&F*F&-F;6#,$*&\"\"'F,F5F,F,F,F,F&*&#F+\"# \\F,*&F*F&-F36#,$*&\"\"(F,F5F,F,F,F,F&*&#F+\"#jF,*&F*F&-F;6#,$*&F%F,F5 F,F,F,F,F,*&#F+\"#\")F,*&F*F&-F36#,$*&FAF,F5F,F,F,F,F,*&#F^o\"%vCF,*&F *F&-F;6#,$*&\"#5F,F5F,F,F,F,F&*&#F+\"$@\"F,*&F*F&-F36#,$*&\"#6F,F5F,F, F,F,F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 39 "Graphs of some truncated Fourier series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 354 "F S := (x,n)->(Pi^2-8)/(8*Pi)+(Pi+4)/(2*Pi)*sin(x)+\n sum(((1+(-1)^k)/ (Pi*k^2*(k^2-1))+2*cos(Pi*k/2)/(Pi*k^2))*cos(k*x)+\n \+ 2*sin(Pi*k/2)/(Pi*k^2)*sin(k*x),k=2..n):\nf_ :=x-> f(x-2 *Pi*floor((x+Pi)/(2*Pi))):\nplot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,7 )],x=-Pi..2*Pi,\n color=[black,red,blue,brown,magenta],linestyle=[3 ,1$4]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 623 321 321 {PLOTDATA 2 "6)-%' CURVESG6%7_o7$$!3*****4tk#fTJ!#<$!3=5KT_Kzzi!#E7$$!3S_J_4$fh$HF*$!3,$z j/69*R?!#=7$$!3I/4wgGTdFF*$!33xH&yV))zu$F37$$!37p-i%y$RcDF*$!3%*\\5t3< lBbF37$$!3mEnbBA/aBF*$!3if!pfCqi3(F37$$!3u\"zSTS_E:#F*$!3OK'*o[.Wa$)F3 7$$!3//4&=EQf'>F*$!32'GVHP>%H#*F37$$!3W9H8ACFp=F*$!3Cw:!HCdyb*F37$$!3k C\\T#e1Ex\"F*$!3gy'R\"*H`qz*F37$$!3%fzbvg?Es\"F*$!3/YZic\"o\\))*F37$$! 3+nmpKYjs;F*$!3G(Q*z(f*=[**F37$$!34Qv$yl[Ei\"F*$!3wd`g!pfl)**F37$$!3Q4 %yHoiEd\"F*$!2w.yNe#)*****F*7$$!3o&fuP,PG_\"F*$!3o4UlD<]))**F37$$!3?#y qXM6IZ\"F*$!37uO%*p&GA&**F37$$!3\\opOvc=B9F*$!3+emX\\JD\"*)*F37$$!3yaJ ;1+Ot8F*$!3H*H$H5os0)*F37$$!3![U/fNc3F\"F*$!3!=u\">sAa`&*F37$$!3/&pXcq _$o6F*$!3R9g&4am5?*F37$$!3sk6iP;#y()*F3$!37Ua@#>q![$)F37$$!3$QLxh,D]%y F3$!3WO3H\"pJZ1(F37$$!3#Rx)\\O:)Q!eF3$!3#oO2`r&[$[&F37$$!3%f+R\"z:'o$Q F3$!3a9;qh9TVPF37$$!3++n(R,>10#F3$!3%fs!eZwFO?F37$$\"3IXD63lhRt!#?F]s7 $$\"3U;GFHcrs=F3Fas7$$\"3=)okm\">vlRF3Fds7$$\"3pIqN#oV%=eF3Fgs7$$\"3w' =q^=S6&yF3Fjs7$$\"3')G\")pYzu'y*F3F]t7$$\"3)eUE(e^j!=\"F*F`t7$$\"3sEG8 =y4m8F*Fct7$$\"3=I_y%3@hY\"F*Fft7$$\"3kLwV^V9m:F*Fit7$$\"3Q8DY9//q;F*$ \"3uzs7RAbr9F*7$$\"39$R([xk$Rx\"F*$\"3)**R-hF* $\"3EYO`>Dx'=\"F*7$$\"3'Q5?'Q%z,:#F*$\"3g#*op\\@89**F37$$\"3=yj:;Z+_BF *$\"3K\\TLu$ze*yF37$$\"3I>&y`P^%\\DF*$\"3DQF6#y79#fF37$$\"31-Spq6\\SFF *$\"3i5z&*G[,6SF37$$\"3O&R)*>H3E&HF*$\"3axR\"fhV)*)=F37$$\"3J)[!yf\\?V JF*$!3!yMb@\\0Bh\"F_s7$$\"35V?o%e2nM$F*$!3d`Q3JozO?F37$$\"3jV+G476JNF* $!3?fyu)yIuz$F37$$\"3m\"pX$yIrKPF*$!3gx@6+o\"Hd&F37$$\"3WCa$fs/C#RF*$! 3(ew%o/]cQqF37$$\"3T-0$[:(o?TF*$!316!>'ok#**H)F37$$\"3AgP!=ZWXJ%F*$!3$ **Q$e:C)*=#*F37$$\"3;B(eVz>gT%F*$!3ObBh'oJSc*F37$$\"35'o8p6&\\(zWf&zG'[F*$!33\"\\stZwp))*F37$$\"3OM.Y4D&G\"\\F *$!3ck6XNQu*z*F37$$\"3_e7(p!3(>,&F*$!3UH>'G%)*fa&*F37$$\"3o#=#[/\"*36^ F*$!3'R\\;B6kc@*F37$$\"3#)[4DZzC$H&F*$!3](H$))pg\"*f$)F37$$\"3m8)yw`A? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "f := x -> piecewise(x<0,sin(x),xG,$*(\"\")! \"\",&*$)%#PiG\"\"#\"\"\"F-F&F'F-F+F'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 55 " be the \+ \"pseudo even\" function defined on the interval " }{XPPEDIT 18 0 "[-P i,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([2*sin*x, -Pi <= x and x < 0],[sin*x+1, 0 <= x and x < Pi])" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6 $7$*(\"\"#\"\"\"%$sinGF.F'F.31,$%#PiG!\"\"F'2F'\"\"!7$,&*&F/F.F'F.F.F. F.31F6F'2F'F3" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 53 "and the n extended to a periodic function with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 280 "f := x -> piecewise( x<0,2*sin(x),sin(x)+1):\n'f(x)'=f(x);\npi := evalf(Pi): pi2 := pi/2:\n p1 := plot(f(x),x=-Pi..Pi,thickness=2):\np2 := plot([[[-pi2,-2.2],[-pi 2,2.2]],[[pi2,-2.2],[pi2,2.2]]],color=blue,linestyle=3):\nplots[displa y]([p1,p2],view=[-3.2..3.2,-2.3..2.3],labels=[`x`,``]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,$*&\"\"#\"\"\"-% $sinGF&F/F/2F'\"\"!7$,&F0F/F/F/%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 440 282 282 {PLOTDATA 2 "6'-%'CURVESG6%7\\o7$$!3*****4tk#fTJ !#<$!3/UE[]'efD\"!#D7$$!3w\"*pr)3PY+$F*$!3%)))G6&[c0t#!#=7$$!3?3E[*ysa )GF*$!3=1HhK!yl1&F37$$!3gX&))>2g9v#F*$!3Lj-6a2A1wF37$$!3)4577.flh#F*$! 31kxF(>%[-5F*7$$!3u\"QM::*H#[#F*$!3^4MBUr6D7F*7$$!3y\\AhcI#yN#F*$!3V8X EZ&3>T\"F*7$$!3&e=d-FN*GAF*$!3![tmd\"QE#e\"F*7$$!3u%*GBP$Rc4#F*$!3?\\) QBB03t\"F*7$$!3mCj&f)3xi>F*$!3#y5Ub58$[=F*7$$!3(or4AN*4E=F*$!3&>@&)*GO F*7$$!3ixF*7$$!3QQQ*3**=dq\"F*$!3C-'3QZB=) >F*7$$!3f9&yM5fzj\"F*$!3%p#>\"o&3\\&*>F*7$$!3e!>jg@*>q:F*$!3y%yeMk**** *>F*7$$!3b.*R+5h@]\"F*$!3+Xlla5H&*>F*7$$!3`;m,%)H7M9F*$!3=g4dc%\\8)>F* 7$$!39d=2_cbo8F*$!3_+!*=HzBf>F*7$$!3v(4F,K))HI\"F*$!3YC()43lqG>F*7$$!3 S*4!R#[0R=\"F*$!3)RLL[It@&=F*7$$!3pY@QqWIU5F*$!3z^It6q8F?'3Wb59F*7$$!3'=[BP-8If' F3$!3a*GILZH^A\"F*7$$!3CB3<@?)yB&F3$!3#)o;Ty!G.+\"F*7$$!3_*)R$!33e)*\\Rt`%R\"F37$$!38,z7VKJ,J!#? $!3RP;IVlh-iFit7$$\"3ve@2&Q*oF7Fit$\"3Esm2*oF7+\"F*7$$\"3k=AF8?pcbFit$ \"389yTjmb05F*7$$\"3_yAZTYp&))*Fit$\"3g!yDO`&))45F*7$$\"3%QBnpsp9U\"Fc t$\"30Hld=U@95F*7$$\"3#eC2EDqsG#Fct$\"3*)y!H#32(G-\"F*7$$\"3!yDZ#y22`J Fct$\"3d?*zM[D:.\"F*7$$\"3u\"GF&H=n%)[Fct$\"3s)\\&)eHF)[5F*7$$\"3p0t!3 )GF;mFct$\"3KWW2oW6m5F*7$$\"3ONnL)\\Zz+\"F3$\"3-(oHW;C15\"F*7$$\"39SFf 3xEa8F3$\"3I#R]l=8]8\"F*7$$\"312**yIK@d>F3$\"3Y`w\"o8uW>\"F*7$$\"3'R2( )Hve,c#F3$\"3k-fa.$GKD\"F*7$$\"3IcwRF*7$$\"3 \\-)REfwoI\"F*$\"3aJTMp[Pl>F*7$$\"3cM2vQyFT9F*$\"3%fD`\"*>C;*>F*7$$\"3 !y32s$*Qxc\"F*$\"3%\\4)=E`****>F*7$$\"35+K?Bs#**p\"F*$\"3w#)er%=u;*>F* 7$$\"3<#H$eMa;H=F*$\"3y$>7M'z!o'>F*7$$\"3%>>CZ'eYk>F*$\"3rf-+?x]B>F*7$ $\"3kYuyfix%4#F*$\"3#zU$=iZ$e'=F*7$$\"3T4q))fu.GAF*$\"3WEZpI2o\"z\"F*7 $$\"3y?w'**=&>gBF*$\"3dXi'z?sUq\"F*7$$\"3!*>3a=Wj\"[#F*$\"3,]=XdQ38;F* 7$$\"3A?c*)yu\"3i#F*$\"3/\")p&**p_v\\\"F*7$$\"3-i-HnWIXFF*$\"39:#G2'o* fQ\"F*7$$\"3u=RWhN.yGF*$\"3)3EwtQ=0E\"F*7$$\"3ss(*RSA20IF*$\"3eLwJMn4O 6F*7$$\"3!)***\\/l#fTJF*$\"3a$zRJ++++\"F*-%'COLOURG6&%$RGBG$\"#5!\"\"$ \"\"!F^`lF]`l-%*THICKNESSG6#\"\"#-F$6%7$7$$!3/+++Fjzq:F*$!3;+++++++AF* 7$Fg`l$\"3;+++++++AF*-Fg_l6&Fi_lF]`lF]`l$\"*++++\"!\")-%*LINESTYLEG6# \"\"$-F$6%7$7$$\"3/+++Fjzq:F*Fi`l7$F[blF\\alF^alFcal-%+AXESLABELSG6%% \"xG%!G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#KF\\`l$\"#KF\\`l;$!#BF\\`l$\" #BF\\`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 53 " can be extended to a periodic function with period " } {XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "f := x -> piecewise(x<0,2*sin(x),sin(x)+1):\nf_ := x -> f(x-2*Pi* floor((x+Pi)/(2*Pi))):\n'f_(x)'='f(x-2*Pi*floor((x+Pi)/(2*Pi)))';\nplo t(f_(x),x=-Pi..4*Pi,color=COLOR(RGB,.5,0,1),thickness=2);" }}{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 614 180 180 {PLOTDATA 2 "6'-%'CURVESG6#7js7$$!3*****4tk#fTJ! #<$!3/UE[]'efD\"!#D7$$!3[HZ:*>)RqHF*$!3UM,!RG*=2M!#=7$$!3Sf%**4v.#*z#F *$!3v0pg\\dw9nF37$$!3c)HJqP[-l#F*$!3Ua%o#R7BO%*F37$$!3tPJ1.IH,DF*$!3s1 x#f_r[>\"F*7$$!3#>qzi5xPL#F*$!3CgkaT&ebW\"F*7$$!3mli\\47Em@F*$!3'*4S2x cxb;F*7$$!3q4/h3\\j(*>F*$!3po-7;#e0#=F*7$$!3w`Xs2'3!H=F*$!31Ho'G/'pL>F *7$$!3uF\"oH='4XF*7$$!3s,<@eP=h;F*$!3/DG)HnN=*>F*7$$!3 $))[LeaF#>;F*$!3_\\lYv[l(*>F*7$$!3#fFbMLrsd\"F*$!35G_Vu!e***>F*7$$!3\" G1x57:``\"F*$!3/;x5;7u)*>F*7$$!3!*\\))p3*eL\\\"F*$!3eEWBUk+%*>F*7$$!3x RV7!zjxL\"F*$!3wQ!Hw8Tf%>F*7$$!3iH)\\:no@=\"F*$!3)>-Lm()f3&=F*7$$!3xBp oQ*e5-\"F*$!3O`d38NK0Y ti*GHLpF3$!3qkK>-F?y7F*7$$!3aUnIMP4n_F3$!3@KO*o'GQ05F*7$$!3A^Q=%4Qig$F 3$!3?F3$!3sb:1@-FmQF37$$!3Q>Pdo`=\"4\"F3$!3 yWau_C/y@F37$$!3-*yk3$G))pB!#>$!3j()y*[+A$RZF_s7$$!3SQd+uk8-8F_s$!3aWU k`$*>/EF_s7$$!3#y(oYr6!RM#!#?$!3EZ+d]!)z(o%Fjs7$$\"3NGO7(RiNL)Fjs$\"3S ())QfYL$35F*7$$\"3X89d'f-6!>F_s$\"3Q*RG3))4!>5F*7$$\"3o9&*G5`fOSF_s$\" 34*>!)=*\\NS5F*7$$\"3#fh2S-)3shF_s$\"3sdoU+\\\"4[%F3$\"3Qt4VKiCL9F*7$$\"3%)RxL)G4*oyF3$ \"3d+s^P:;3F*7$$\"3m(*)e&Q)**4H\"F*$\"3%*yQ.g;6h>F*7$$\"35h%Q$o\"=\\ X\"F*$\"3))R\"=bj$H$*>F*7$$\"3*3q')=%\\MH:F*$\"3GJ)[V,T\"**>F*7$$\"3oS \\V:F*7$$\"3Y!=$)*)[)>y;F*$\"3W1C()HzB%*>F*7$$\"3E ?9`i_i_F*7$$\"3LbCXFliH>F*$\"34'e&Rt#3j$>F*7$$\"3S! \\tBzFm5#F*$\"3%z'=o;W%)f=F*7$$\"3D3!Q(pV^1CF*$\"31$z^pAX1n\"F*7$$\"3S )4p4v``v#F*$\"3Qc=Chqqw8F*7$$\"3LnIWJZu4HF*$\"3r^a&HRw(H7F*7$$\"3EOq\" >rNT1$F*$\"3;$z6M^zt2\"F*7$$\"3w&G@Qj4`3$F*$\"3S@)>\"4LDc5F*7$$\"3pNbs bN[1JF*$\"3@3XW&)=5N5F*7$$\"3mgwn;02X-\"F*7$$\"3j&yHwZ dw7$F*$\"3[RW&\\sMR,\"F*7$$\"3g5>eQWCQJF*$\"3$4+_(3#[L+\"F*7$$\"3eNS`* RJ)[JF*$!39G)R`aOxW\"F_s7$$\"3)\\`ULCz6>$F*$!3]$)R)QGaK\"**F_s7$$\"3VM 5:(3FNB$F*$!3y*p([%o*4O=F37$$\"3HL!oZxA#=LF*$!3I)Q/GeiU^$F37$$\"3fK]Qi %=HS$F*$!3SJt5y9Bn^F37$$\"3%ohy\"4\">Uc$F*$!3I\")ylGY9.#)F37$$\"34,A(f v>bs$F*$!3)*y/%R:4E5\"F*7$$\"3#HG_$f+#Q*QF*$!3CP$>:nLlO\"F*7$$\"3?lBti .7iSF*$!3!z=d@TU=f\"F*7$$\"3tsR!*yDn;UF*$!37RmMTjDfdJZ)=F*7$$\"3]/dGF*7$$\"3'y#e\\R-$z`%F*$ !3[s`ak7kp>F*7$$\"3)**)3g+mgzXF*$!3#\\P<#HZR#)>F*7$$\"3@^fqhHG@YF*$!3= (zHIV0<*>F*7$$\"3K85\"GKfHm%F*$!3iIUQ1sb(*>F*7$$\"3Xvg\"RoNYq%F*$!3A1] p'))R***>F*7$$\"3[Jj<5d#zu%F*$!3+F7ttst)*>F*7$$\"3[(eOkt:7z%F*$!3k(fD$ f&*y$*>F*7$$\"3SWopid]M[F*$!3Msu<9g5&)>F*7$$\"3U+r&*)y&zx[F*$!3Qo#Q*3H qs>F*7$$\"3O8wZTePk\\F*$!3=oCd/z$o$>F*7$$\"3SD\")*R*e&40&F*$!3E$y$=EKY ')=F*7$$\"3d. 5F*7$$\"3l\"zxMO8HK'F*$\"3=ra`1wrR5F*7$$\"3dyL\"Qs![VjF*$\"3u`>!*)))e- 1\"F*7$$\"3CUDC`0o-lF*$\"3!=>As+Px@\"F*7$$\"3!fqrEQ!)=m'F*$\"3q6vq;$3( p8F*7$$\"31G\"y\"=*3a,(F*$\"35$Q&)H'\\_o;F*7$$\"3Ctu9J+2LtF*$\"3EWY\"= 'eOn=F*7$$\"3mO$)*=bbE](F*$\"3HlQSwr\"*Q>F*7$$\"33+#\\E2TAn(F*$\"3Y)e! p!fIN)>F*7$$\"3G0:B;f2\\xF*$\"3M=[yRC]%*>F*7$$\"3Y5Q\")f2\"f#yF*$\"371 A))Qgg**>F*7$$\"3l:hR.cu-zF*$\"3a'pHKF6))*>F*7$$\"3%3UypW!ezzF*$\"3SWD \\KG7#*>F*7$$\"3W*pKX+#eZ\")F*$\"3IYE2L\"3s&>F*7$$\"3-yp3iNe:$)F*$\"33 LP;v,M&*=F*7$$\"37g!p!3jtJ')F*$\"38)p`h$3\\7`')p$ *F*$\"3)HT2x/&)[0\"F*7$$\"37ud<7g+\"R*F*$\"3`\"fC#HbwL5F*7$$\"3\\qYP/n 97%*F*$\"3)f490#4j75F*7$$\"3c>TZ]qrA%*F*$\"3dej$)341-5F*7$$\"3&octlR(G L%*F*$!3dh\"H=im=q\"F_s7$$\"399InUx&QW*F*$!3+#Qq$f[s:QF_s7$$\"3AjCx)3G WX*F*$!3(p)[[hn:HfF_s7$$\"3V]!ov&3**Q&*F*$!3QkyHu\\HzAF37$$\"3lPOOEObB '*F*$!3AR5Bf]Q\\RF37$$\"3!)))36?;W'y*F*$!3qnM*3<:m2(F37$$\"3=Q\"eQhH$ \\**F*$!3F&Q*)z(*\\;+\"F*7$$\"3I#*)*Q61f65!#;$!3')*=Y6s8[F\"F*7$$\"3hq RRh#[#G5F[bm$!3(\\'=4&*fo7:F*7$$\"3H\\dkxzwW5F[bm$!3i#4#\\(GYsq\"F*7$$ \"3)z_(*Qp(Gh5F[bm$!3O1&)H8:Kb=F*7$$\"3=`:Y(fnk2\"F[bm$!3kNs-9C#p%>F*7 $$\"3Ayb-,vk\"4\"F[bm$!31Ed$ecYP*>F*7$$\"3%4[f;(p*f4\"F[bm$!3?$HR%[Ct) *>F*7$$\"3$QQ$HUkM+6F[bm$!3I:o3Yx$***>F*7$$\"3t'GFH\"fp/6F[bm$!3=E$o*y ,O(*>F*7$$\"3X*=hNQX!46F[bm$!3Q-&HDi/5*>F*7$$\"31&**G[KWx6\"F[bm$!3w6R !4v9q'>F*7$$\"3o+o4mKWE6F[bm$!3H70\">b\\\"G>F*7$$\"3![JQ'RT+U6F[bm$!3h OOR^m^A=F*7$$\"3uG)zJ,lv:\"F[bm$!3a\"F[bm$!3J\")[J<9\\C7F*7$$\"39\"3j:7Fm?\" F[bm$!3#[5FlLb-f*F37$$\"3c'eCk&p]A7F[bm$!3'RkV@hnUp'F37$$\"3G$Hsw+s&R7 F[bm$!35$p[TYpkR$F37$$\"3+++#*eqjc7F[bm$!3AY2i/X$=.&F--%&COLORG6&%$RGB G$\"\"&!\"\"$\"\"!Fehm$\"\"\"Fehm-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q \"x6\"Q!F`im-%%VIEWG6$;$!+aEfTJ!\"*$\"+iqjc7!\")%(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 "" }}{PARA 0 "" 0 "" {TEXT 300 39 "Calculation of t he constant coefficient" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "f := x -> piecewise(x <0,2*sin(x),sin(x)+1):\nc=1/(2*Pi)*Int('f(x)',x=-Pi..Pi);\nsimplify(va lue(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*&#\"\"\"\"\"#F(,$ -%$IntG6$-%\"fG6#%\"xG/F1;,$%#PiG!\"\"F5*$F5F6F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"cG,$*(\"\"#!\"\"%#PiGF(,&F)\"\"\"F'F(F+F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 51 "Calculat ion of the coefficients of the cosine terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "f := x -> piecewise(x<0,2*sin(x),sin(x)+1):\nassume(k_,integer):\na[k]= 1/Pi*Int('f(x)'*cos(k*x),x=-Pi..Pi);\nsubs(k_=k,value(subs(k=k_,%))); \naa := unapply(rhs(%),k):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 #%\"kG,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#*&F'F1F0F1F1/F0;,$%#PiG !\"\"F9*$F9F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%\"kG*(%#PiG !\"\",&\"\"\"F,)F*F'F,F,,&*$)F'\"\"#F,F,F,F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The formula does not \"wo rk\" when " }{XPPEDIT 18 0 "k=1" "6#/%\"kG\"\"\"" }{TEXT -1 26 ", so w e need to calculate " }{XPPEDIT 18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT -1 13 " separately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "a[1]=1/Pi*Int('f(x)'*cos(x),x=-Pi..Pi);\n value(%);\naa(1) := rhs(%):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6#\"\"\",$-%$IntG6$*&-%\"fG6#%\"xGF'-%$cosGF/F'/F0;,$%#PiG!\"\"F6*$F 6F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "ma trix([[k,`|`,seq(k,k=1..12)],['a'[k],`|`,seq(aa(k),k=1..12)]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$70%\"kG%\"|grG\"\"\"\"\" #\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#770&%\"aG6#F(F)\"\"!,$* (F+F*F,!\"\"%#PiGF=F*F:,$*(F+F*\"#:F=F>F=F*F:,$*(F+F*\"#NF=F>F=F*F:,$* (F+F*\"#jF=F>F=F*F:,$*(F+F*\"#**F=F>F=F*F:,$*(F+F*\"$V\"F=F>F=F*Q)ppri nt136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "a[k]=0" "6#/&%\"aG6#%\"kG\"\"!" } {TEXT -1 6 " when " }{TEXT 306 1 "k" }{TEXT -1 4 " is " }{TEXT 262 3 " odd" }{TEXT -1 15 ", as expected. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 299 49 "Calculation of the coefficients of the s ine terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "f := x -> piecewise(x<0,2*sin(x),s in(x)+1):\nassume(k_,integer):\nb[k]=1/Pi*Int('f(x)'*sin(k*x),x=-Pi..P i);\nsubs(k_=k,value(subs(k=k_,%)));\nbb := unapply(rhs(%),k):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%\"kG,$-%$IntG6$*&-%\"fG6#%\" xG\"\"\"-%$sinG6#*&F'F1F0F1F1/F0;,$%#PiG!\"\"F9*$F9F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%\"kG,$*(%#PiG!\"\",&)F+F'\"\"\"F.F+F.F'F +F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "T he previous formula does not give the correct coefficient when " } {XPPEDIT 18 0 "k=1" "6#/%\"kG\"\"\"" }{TEXT -1 26 ", so we need to cal culate " }{XPPEDIT 18 0 "b[1];" "6#&%\"bG6#\"\"\"" }{TEXT -1 13 " sepa rately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "b[1]=1/Pi*Int('f(x)'*sin(x),x=-Pi..Pi);\nsimplify(val ue(%));\nbb(1) := rhs(%):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG 6#\"\"\",$-%$IntG6$*&-%\"fG6#%\"xGF'-%$sinGF/F'/F0;,$%#PiG!\"\"F6*$F6F 7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*(\"\"#!\"\",&*& \"\"$F'%#PiGF'F'\"\"%F'F'F/F+F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "matrix([[k,`|`,seq(k,k=1..12 )],['b'[k],`|`,seq(bb(k),k=1..12)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$70%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\" \")\"\"*\"#5\"#6\"#770&%\"bG6#F(F),$*(F+!\"\",&*&F,F*%#PiGF*F*F-F*F*F? F " 0 "" {MPLTEXT 1 0 126 "FS := (x,n)->(Pi-2)/(2*Pi)+(3*Pi+4)/(2*Pi)*sin(x)+\n sum((1+(-1)^k)/((k^2-1)*Pi)*cos(k*x)-((-1)^k-1)/(k*Pi)*sin(k*x),k=2 ..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)op eratorG%&arrowGF),(*(\"\"#!\"\",&%#PiG\"\"\"F/F0F3F2F0F3*&#F3F/F3*(,&* &\"\"$F3F2F3F3\"\"%F3F3F2F0-%$sinG6#9$F3F3F3-%$sumG6$,&**,&F3F3)F0%\"k GF3F3,&*$)FFF/F3F3F3F0F0F2F0-%$cosG6#*&FFF3F>F3F3F3**,&FEF3F3F0F3F2F0F FF0-F " 0 "" {MPLTEXT 1 0 9 "FS(x,11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,:*(\"\"# !\"\",&%#PiG\"\"\"F%F&F)F(F&F)*&#F)F%F)*(,&*&\"\"$F)F(F)F)\"\"%F)F)F(F &-%$sinG6#%\"xGF)F)F)*&#F%F/F)*&F(F&-%$cosG6#,$*&F%F)F4F)F)F)F)F)*&F6F )*&F(F&-F26#,$*&F/F)F4F)F)F)F)F)*&#F%\"#:F)*&F(F&-F96#,$*&F0F)F4F)F)F) F)F)*&#F%\"\"&F)*&F(F&-F26#,$*&FMF)F4F)F)F)F)F)*&#F%\"#NF)*&F(F&-F96#, $*&\"\"'F)F4F)F)F)F)F)*&#F%\"\"(F)*&F(F&-F26#,$*&FhnF)F4F)F)F)F)F)*&#F %\"#jF)*&F(F&-F96#,$*&\"\")F)F4F)F)F)F)F)*&#F%\"\"*F)*&F(F&-F26#,$*&Fi oF)F4F)F)F)F)F)*&#F%\"#**F)*&F(F&-F96#,$*&\"#5F)F4F)F)F)F)F)*&#F%\"#6F )*&F(F&-F26#,$*&FjpF)F4F)F)F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 301 39 "Graphs of some truncated Fourier series " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "FS := (x,n)->(Pi-2)/(2*Pi)+(3*Pi+4)/(2*Pi)*s in(x)+\n sum((1+(-1)^k)/((k^2-1)*Pi)*cos(k*x)-((-1)^k-1)/(k*Pi)*sin( k*x),k=2..n):\nf_ :=x-> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nplot([f_(x),F S(x,1),FS(x,3),FS(x,7),FS(x,11)],x=-Pi..2*Pi,\n color=[black,red,bl ue,brown,magenta],linestyle=[3,1$4]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 623 321 321 {PLOTDATA 2 "6)-%'CURVESG6%7\\q7$$!3*****4tk#fTJ!#<$!3/UE[ ]'efD\"!#D7$$!3?wlTyf()QIF*$!3![Ut1'Hs]?!#=7$$!3S_J_4$fh$HF*$!3-'eF4AG )zSF37$$!3OG?9&3'yYGF*$!3Cz`46eF37$$!3I/4wgGTdFF*$!3;afqvo(f\\(F37$ $!37p-i%y$RcDF*$!3**4iuT.t/6F*7$$!3mEnbBA/aBF*$!3$>\"Q>\\SD<9F*7$$!3u \"zSTS_E:#F*$!3YEztp!))3n\"F*7$$!3//4&=EQf'>F*$!3Ad')euQ)e%=F*7$$!3W9H 8ACFp=F*$!3D:.e[9d6>F*7$$!3kC\\T#e1Ex\"F*$!3sNz#)f1Tf>F*7$$!3%fzbvg?Es \"F*$!3A\\\\KJO*p(>F*7$$!3+nmpKYjs;F*$!3Yx)f&>zj*)>F*7$$!34Qv$yl[Ei\"F *$!3ar57Q>J(*>F*7$$!3Q4%yHoiEd\"F*$!3_2cr;l****>F*7$$!3o&fuP,PG_\"F*$! 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!3;-f<,YDvFF-7$$\"3Q%[sl7UcC\"F]bo$!3mYSjY+C\"R#F-7$$\"3#R#f6$\\;qC\"F ]bo$!3B9-%>r&3+DF-7$$\"3Gj$f'f3R[7F]bo$!3=Hf+y[c!\\#F-7$$\"3k-G?E_w\\7 F]bo$!3jn&*3w\"Q$=AF-7$$\"3=Uiu#fR6D\"F]bo$!3!o;H#>f`l:F-7$$\"35@J$eK) )QD\"F]bo$\"3c+JLss1\\5F-7$$\"3+++#*eqjc7F]bo$\"3q$*GZ,(fB#\\F--%'COLO URG6&%$RGBG$\"*++++\"!\")$\"\"!Fa^pF`^p-%*LINESTYLEG6#\"\"\"-%+AXESLAB ELSG6$Q\"x6\"Q!Fj^p-%%VIEWG6$;$!+aEfTJ!\"*$\"+iqjc7F_^p%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 42.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 74 "Construc ting the first few terms of the Fourier series using the procedure" } {TEXT -1 1 " " }{TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "f := x -> piecewise(x<0,2*sin(x),sin(x)+1):\nFourierSeries(f(x),x=-Pi..Pi, numterms=11,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coe fficient~-->G,$*(\"\"#!\"\",&%#PiG\"\"\"F&F'F*F)F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 55 " be the \"pseudo even\" function defined on the interval " } {XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 5 " by: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([-2* Pi-2*x, -Pi <= x and x < -3*Pi/4],[x/2, -3*Pi/4 <= x and x < -Pi/4],[2 *x, -Pi/4 <= x and x < Pi/2],[2*Pi-2*x, Pi/2 <= x and x <= Pi])" "6#/- %\"fG6#%\"xG-%*PIECEWISEG6&7$,&*&\"\"#\"\"\"%#PiGF/!\"\"*&F.F/F'F/F131 ,$F0F1F'2F',$*(\"\"$F/F0F/\"\"%F1F17$*&F'F/F.F131,$*(F9F/F0F/F:F1F1F'2 F',$*&F0F/F:F1F17$*&F.F/F'F/31,$*&F0F/F:F1F1F'2F'*&F0F/F.F17$,&*&F.F/F 0F/F/*&F.F/F'F/F131*&F0F/F.F1F'1F'F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 53 "and then extended to a periodic function with period \+ " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 310 "f := x -> piecewise(x<-3*Pi/4,-2*Pi-2*x,x<-Pi/4,-Pi/2,x2g9v#F*$!3g]\\-K;l-y F37$$!3)4577.flh#F*$!3E%QbZCn+0\"F*7$$!3u\"QM::*H#[#F*$!3vA36/qe=8F*7$ $!3w:L2/61?CF*$!3raH.*4jIW\"F*7$$!3y\\AhcI#yN#F*$!3o'3bR>Rvc\"F*7$$!3/ =ZVj\"zLH#F*$!3c'*[zEjzq:F*7$$!3&e=d-FN*GAF*FU7$$!3u%*GBP$Rc4#F*FU7$$! 3mCj&f)3xi>F*FU7$$!3(or4AN*4E=F*FU7$$!3QQQ*3**=dq\"F*FU7$$!3e!>jg@*>q: F*FU7$$!3`;m,%)H7M9F*FU7$$!3v(4F,K))HI\"F*FU7$$!3S*4!R#[0R=\"F*FU7$$!3 pY@QqWIU5F*FU7$$!3Wp3w$R)\\B#*F3FU7$$!3N]#[4j>e_)F3FU7$$!3EJc8o39GyF3$ !3EErit\"Gcc\"F*7$$!3cc&Hf%pd5sF3$!3J6f=*Q:@W\"F*7$$!3'=[BP-8If'F3$!3P 'pWZg-'=8F*7$$!3CB3<@?)yB&F3$!3mkTB/kdZ5F*7$$!3_*)R6;/\"F*7$$\"3q]1XIqOClF3$\"39I,41M([I\"F*7$$\"3 %HoBlmlzz(F3$\"3gOZILJff:F*7$$\"3^u'f#4)z?@*F3$\"3!\\$>&='fTU=F*7$$\"3 9Hvb,$F*7$$\"35+K?Bs#**p\"F*$\"3+'=t2'3L$)GF*7$$ \"3<#H$eMa;H=F*$\"3*=+8!QW&[i#F*7$$\"3%>>CZ'eYk>F*$\"3O-7txNDaBF*7$$\" 3kYuyfix%4#F*$\"3'Hp/wyKO4#F*7$$\"3T4q))fu.GAF*$\"3UnbS(Q5r#=F*7$$\"3y ?w'**=&>gBF*$\"3mWVCF\\zi:F*7$$\"3!*>3a=Wj\"[#F*$\"3WYz4qk\"*>8F*7$$\" 3A?c*)yu\"3i#F*$\"3zX$)Q\\.bT5F*7$$\"3-i-HnWIXFF*$\"3=A1*fsjd#zF37$$\" 3u=RWhN.yGF*$\"3E([ " 0 "" {MPLTEXT 1 0 221 "f := x -> piecewise(x<-3*Pi/4,-2*Pi-2*x,x<-Pi/4,-Pi/2,x f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n'f_(x)'='f(x-2*Pi*floor((x+P i)/(2*Pi)))';\nplot(f_(x),x=-Pi..4*Pi,color=COLOR(RGB,.5,0,1),thicknes s=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\" \"\"*(\"\"#F,%#PiGF,-%&floorG6#,$*(F.!\"\",&F/F,F'F,F,F/F5F,F,F5" }} {PARA 13 "" 1 "" {GLPLOT2D 614 180 180 {PLOTDATA 2 "6'-%'CURVESG6#7gr7 $$!3*****4tk#fTJ!#<$!3)fq*)fiefD\"!#D7$$!3[HZ:*>)RqHF*$!3&HF,()3*)QU$! #=7$$!3Sf%**4v.#*z#F*$!3Uum!=0yx%oF37$$!3c)HJqP[-l#F*$!30\"*)p6`&)o#)* F37$$!3tPJ1.IH,DF*$!3w5L0,$*f!G\"F*7$$!3#)>9na]`:\"[9F*7$ $!3#>qzi5xPL#F*$!3c'*[zEjzq:F*7$$!3e$)z)y:>+D#F*FK7$$!3mli\\47Em@F*FK7 $$!3q4/h3\\j(*>F*FK7$$!3w`Xs2'3!H=F*FK7$$!3!*\\))p3*eL\\\"F*FK7$$!3iH) \\:no@=\"F*FK7$$!3xBpoQ*e5-\"F*FK7$$!3C\"=S#e?\\*f)F3FK7$$!3!>Yti*GHLp F3$!3Q#pa#z&emQ\"F*7$$!3aUnIMP4n_F3$!3_[8'ou=M0\"F*7$$!3A^Q=%4Qig$F3$! 3V-xO)=wC@(F37$$!3!*f41aCQX>F3$!3y>>73\\w!*QF37$$!3-*yk3$G))pB!#>$!3.y &H\\\"4[%F3$\"3=AmaQ)H='*)F37$$\"3YDb!Q57\\<'F3$\"35 06w?C)\\B\"F*7$$\"3%)RxL)G4*oyF3$\"3'zanw&=yt:F*7$$\"3-Tb1)=i)p&*F3$\" 3?3JhPC(R\">F*7$$\"3AM$z(3:3F6F*$\"3Wo'ev,jTD#F*7$$\"3m(*)e&Q)**4H\"F* $\"3K&ze#F*7$$\"35h%Q$o\"=\\X\"F*$\"3@ApnOj$)4HF*7$$\"3)3e7^bJ@ \\\"F*$\"3yh^A5JE%)HF*7$$\"3*3q')=%\\MH:F*$\"3y,Mx$))*oeIF*7$$\"3+hPFN ;&za\"F*$\"3+AvaqK!f4$F*7$$\"3!4#3mG$elc\"F*$\"3!=k@tl;J8$F*7$$\"3y!)y /A];&e\"F*$\"3lCQ3j_&G6$F*7$$\"3oS\\V:y;F*$\"3HDK@H$)yEHF*7$$\"3E?9`i_i_F*$\"3dvYF_A$RU#F*7$$\"3S!\\tBzFm5#F*$\"3U0EVA(H*p?F*7$$\"3M\\d0\" 3rlD#F*$\"3e(3o]9V+x\"F*7$$\"3D3!Q(pV^1CF*$\"3upNqnl:q9F*7$$\"3K`NNgS$ 4e#F*$\"3ezCZ'=<87\"F*7$$\"3S)4p4v``v#F*$\"3E%*QT_!yZs(F37$$\"3LnIWJZu 4HF*$\"3k:X$HWepj%F37$$\"3EOq\">rNT1$F*$\"3/P^XL)Q\"\\:F37$$\"3VM5:(3F NB$F*$!3FE[Ar')oQ=F37$$\"3fK]Qi%=HS$F*$!3X)z/f<;lA&F37$$\"3%ohy\"4\">U c$F*$!3[ukx6\"HDX)F37$$\"34,A(fv>bs$F*$!3/:[w/U&y;\"F*7$$\"3yTAm2*p'4Q F*$!3L(*[93X:O8F*7$$\"3#HG_$f+#Q*QF*$!3iz\\_6[X/:F*7$$\"3@$zusfd[\"RF* $!3I***pt))Hla\"F*7$$\"3\\.t>N^*e$RF*FK7$$\"3y8)>JnKp&RF*FK7$$\"33CB/6 -(z(RF*FK7$$\"3kWt)oGX+-%F*FK7$$\"3?lBti.7iSF*FK7$$\"3tsR!*yDn;UF*FK7$ $\"3E!ev]zC7P%F*FK7$$\"3Xvg\"RoNYq%F*FK7$$\"3SD\")*R*e&40&F*FK7$$\"3d. 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H\"=^DI&G*F*$\"3)exh<>3&*y#F37$$\"3AjCx)3GWX*F*$!3(Gx;1gD+$fFfp7$$\"3l POOEObB'*F*$!3)f;&)=J8b(RF37$$\"3!)))36?;W'y*F*$!3;*=Io=tKB(F37$$\"3=Q \"eQhH$\\**F*$!3o\"F]jl$!3WIoo%\\&z<8F*7$$\"39 \"3j:7Fm?\"F]jl$!38AnX(z)>+5F*7$$\"3c'eCk&p]A7F]jl$!3GQhE-5-EoF37$$\"3 G$Hsw+s&R7F]jl$!3P%zCFv5IT$F37$$\"3+++#*eqjc7F]jl$!3)ou?Y]M=.&F--%&COL ORG6&%$RGBG$\"\"&!\"\"$\"\"!Fa^m$\"\"\"Fa^m-%*THICKNESSG6#\"\"#-%+AXES LABELSG6$Q\"x6\"Q!F\\_m-%%VIEWG6$;$!+aEfTJ!\"*$\"+iqjc7!\")%(DEFAULTG " 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 39 "Calcul ation of the constant coefficient" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "f := x -> p iecewise(x<-3*Pi/4,-2*Pi-2*x,x<-Pi/4,-Pi/2,x " 0 "" {MPLTEXT 1 0 260 "f := x -> piecewise(x<-3*Pi/4,-2*Pi-2*x,x<-P i/4,-Pi/2,xF:,$*&F*F**&F+F*F=F*F>F*F:,$*&F*F**&F2F*F=F*F>F>F:F:F:,$*&F*F**&\"#DF *F=F*F>F>F:,$*&F*F**&\"#=F*F=F*F>F*Q)pprint226\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "a[k]=0" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 288 1 " k" }{TEXT -1 4 " is " }{TEXT 262 3 "odd" }{TEXT -1 15 ", as expected. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 49 "Calc ulation of the coefficients of the sine terms" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "f := x -> piecewise(x<-3*Pi/4,-2*Pi-2*x,x<-Pi/4,-Pi/2,xF*F*FA,$**F+F*\"$@\"F=F " 0 "" {MPLTEXT 1 0 172 "FS := (x,n) ->Pi/16+sum(-2*(cos(1/4*Pi*k)-2*cos(1/2*Pi*k)+cos(3/4*Pi*k))/(k^2*Pi)* cos(k*x)+\n 2*(sin(1/4*Pi*k)+2*sin(1/2*Pi*k)+sin(3/4*Pi*k))/(k^2 *Pi)*sin(k*x),k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$% \"xG%\"nG6\"6$%)operatorG%&arrowGF),&*&\"#;!\"\"%#PiG\"\"\"F2-%$sumG6$ ,&*,\"\"#F2,(-%$cosG6#,$*(\"\"%F0F1F2%\"kGF2F2F2*&F8F2-F;6#,$*(F8F0F1F 2F@F2F2F2F0-F;6#,$**\"\"$F2F?F0F1F2F@F2F2F2F2F@!\"#F1F0-F;6#*&F@F29$F2 F2F0*,F8F2,(-%$sinGF " 0 "" {MPLTEXT 1 0 9 "FS(x,11) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8*&\"#;!\"\"%#PiG\"\"\"F(**\"\"# F(,&F*F(*$F*#F(F*F(F(F'F&-%$sinG6#%\"xGF(F(*&F'F&-%$cosG6#,$*&F*F(F1F( F(F(F&*&#F*\"\"*F(*(,&F,F(F*F&F(F'F&-F/6#,$*&\"\"$F(F1F(F(F(F(F(*&F-F( *&F'F&-F46#,$*&\"\"%F(F1F(F(F(F(F(*&#F*\"#DF(*(,&F*F(F,F&F(F'F&-F/6#,$ *&\"\"&F(F1F(F(F(F(F(*&#F(F:F(*&F'F&-F46#,$*&\"\"'F(F1F(F(F(F(F&*&#F* \"#\\F(*(,&F,F&F*F&F(F'F&-F/6#,$*&\"\"(F(F1F(F(F(F(F(*&#F*\"#\")F(*(F+ F(F'F&-F/6#,$*&F:F(F1F(F(F(F(F(*&#F(FKF(*&F'F&-F46#,$*&\"#5F(F1F(F(F(F (F&*&#F*\"$@\"F(*(F " 0 "" {MPLTEXT 1 0 333 "FS := (x,n)->Pi/16+sum( -2*(cos(1/4*Pi*k)-2*cos(1/2*Pi*k)+cos(3/4*Pi*k))/(k^2*Pi)*cos(k*x)+\n \+ 2*(sin(1/4*Pi*k)+2*sin(1/2*Pi*k)+sin(3/4*Pi*k))/(k^2*Pi)*sin(k*x ),k=1..n):\nf_ :=x-> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nplot([f_(x),FS(x ,1),FS(x,3),FS(x,5),FS(x,7)],x=-Pi..2*Pi,\n color=[black,red,blue,b rown,magenta],linestyle=[3,1$4]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 623 321 321 {PLOTDATA 2 "6)-%'CURVESG6%7gn7$$!3*****4tk#fTJ!#<$!3)fq*)fief D\"!#D7$$!3S_J_4$fh$HF*$!3@9GL\")om3T!#=7$$!3I/4wgGTdFF*$!3Awxdcef$o(F 37$$!37p-i%y$RcDF*$!3+[!Rzt(Rq6F*7$$!3)y\\)3/!=_X#F*$!3Y!f-!*H\\FP\"F* 7$$!3mEnbBA/aBF*$!3c'*[zEjzq:F*7$$!3?f([QJZLD#F*FF7$$!3u\"zSTS_E:#F*FF 7$$!3//4&=EQf'>F*FF7$$!3kC\\T#e1Ex\"F*FF7$$!3Q4%yHoiEd\"F*FF7$$!3yaJ;1 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!3R>&y-wIn_\"F*7$$\"3I(*G%=k=re&F*$!3Yb7\"QeqhR\"F*7$$\"3!fJ([#e-hl&F* $!3GCU1!R(4]7F*7$$\"3c;T**))>7\"y&F*$!3a5_H:&\\a+\"F*7$$\"3@<4]&RTh!fF *$!3WNh'y*y.\\vF37$$\"3f:awG1#G/'F*$!3cFM=@3#)*y%F37$$\"3'R\"*H?')*\\z hF*$!3Y]ud+X2x?F37$$\"33$3q)oV53jF*$\"3Y.]y'f)Q!)[F_s7$$\"3>_-rv)3nV'F *$\"3Afnt!*f`%3$F37$$\"3k/D1Tw8rlF*$\"3/=#=/vKhv&F37$$\"35dZT1kc0nF*$ \"3=2u_H[eN%)F37$$\"3]=y)e4&*p$oF*$\"31FnlAt)*36F*7$$\"3#*z3O&yB%opF*$ \"3CY>4!pW/P\"F*7$$\"34u@1Vu,1rF*$\"3A_>(QK!*Qk\"F*7$$\"3EoMw+6hVsF*$ \"3GTt'fs\"3A>F*7$$\"31VR_Y,8wtF*$\"3'oP1\"G.D(=#F*7$$\"3&yT%G#>\\'3vF *$\"3!f8c'\\NOZCF*7$$\"3&yPCn0pTk(F*$\"34C24/P\"Hs#F*7$$\"3'yLk6#*)ozx F*$\"3b*R`RZf^+$F*7$$\"3s6$\\SM)G8yF*$\"3Q)fl51?'pIF*7$$\"3e&GMpw()o%y F*$\"3]L**Qkkz+JF*7$$\"3]snPyuojyF*$\"3)y([!4?1**4$F*7$$\"3Wf#>)*=([!) yF*$\"3koU%fk*y(3$F*7$$\"3[X<)F*$\"3]Q)ood&\\.DF*7$$\"3#3*eGX$z aH)F*$\"3-p)y><^9E#F*7$$\"3+j3\\S9-P%)F*$\"3/#=ji?$)[(>F*7$$\"3'p$epNN cy&)F*$\"3Q6ciCEq!p\"F*7$$\"3mHF)HZg^q)F*$\"3/B%[L(3uS9F*7$$\"38C'p-Td <$))F*$\"3e!)>W&[*y&=\"F*7$$\"3'Q5>+:On'*)F*$\"3%y%f'[zl\"[\"*F37$$\"3 c$eo(*)[r,\"*F*$\"3#\\'oA*4sEZ'F37$$\"3%y%zgmn!4B*F*$\"3]?g*)HehyQF37$ $\"3!RJZMk)4g$*F*$\"3aQ5tC$\\1G\"F37$$\"39r&)f>'\\N]*F*$!3aXqL63(oc\"F 37$$\"3QG)\\df+qk*F*$!3V%>%pNb2JWF37$$\"3go)31g=7x*F*$!3Gh*fv:P%[pF37$ $\"3!)3zY0mV&*)*F*$!3FUB!y%G3,%*F37$$\"3nU[?:Dk.5!#;$!3^l5\"*>Y6?7F*7$ $\"3j%*G')p8u<5F[am$!30$Q'\\%*\\-3:F*7$$\"3jv]sjP&4-\"F[am$!3'eC$of5F] :F*7$$\"3icsedh;C5F[am$!3+#f-%G))3s:F*7$$\"3iP%\\9byt-\"F[am$!3asMQe5; x:F*7$$\"3h=;JX4fI5F[am$!3%z:Kn;9Nd\"F*7$$\"3h!)f.Ld,P5F[am$!3s05%>yGy c\"F*7$$\"3gU.w?0WV5F[am$!3so!*[4+!>d\"F*7$$\"3g?(4p$f\\d5F[am$!3O28%Q j5)p:F*7$$\"3g)4fIN^:2\"F[am$!3]Y=.(z#pq:F*7$$\"3S:uDZi!e4\"F[am$!315E yhUpq:F*7$$\"3/K#H0uHM7\"F[am$!3%=&)Gai6.d\"F*7$$\"3/EMJrE6]6F[am$!3Qw q\\9Z(=d\"F*7$$\"3SQ*[5UXM;\"F[am$!3OK_gNU'yc\"F*7$$\"3y]Wyq\"yn<\"F[a m$!39]XQ:m4i:F*7$$\"3_l\"R<(=1!>\"F[am$!3u1qj&f@*H8F*7$$\"3D!)QpsbM.7F [am$!3c5]'\\-P(o5F*7$$\"3&QQ'p]q5;7F[am$!3i#\\Wzmc_4)F37$$\"3j()))pG&o )G7F[am$!3!**f0q0taa&F37$$\"3!RW4Qz_FC\"F[am$!3KFo:iSb#z#F37$$\"3+++#* eqjc7F[am$\"3oOOGqQ`zPF--%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!FjgmFigm -%*LINESTYLEG6#\"\"\"-%+AXESLABELSG6$Q\"x6\"Q!Fchm-%%VIEWG6$;$!+aEfTJ! \"*$\"+iqjc7Fhgm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 286 74 "Constructing the first few terms of the Fourier s eries using the procedure" }{TEXT -1 1 " " }{TEXT 0 13 "FourierSeries " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "f := x -> piecewise(x<-3*Pi/4,-2*Pi-2*x,x<-P i/4,-Pi/2,xG,$*&\"#;!\"\"%#PiG\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 55 " be the \+ \"pseudo even\" function defined on the interval " }{XPPEDIT 18 0 "[-P i,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)= PIECEWISE([-x/2-Pi/2, -Pi<=x and \+ x<-Pi/2],[x/2,-Pi/2<=x and x<0],[x, 0<=x and x " 0 "" {MPLTEXT 1 0 302 "f := x -> piecewise(x<-Pi/2,-x/2-P i/2,x<0,x/2,x7$$!3?3E[*ysa )GF*$!3]Cf`?$*f!G\"!#=7$$!3gX&))>2g9v#F*$!3mPi+3Hm]>F97$$!3)4577.flh#F *$!3mg%))=6o^i#F97$$!3u\"QM::*H#[#F*$!3'o0x-^nkH$F97$$!3y\\AhcI#yN#F*$ !3o;x)[)z%)=RF97$$!3&e=d-FN*GAF*$!3KOIm;pGjXF97$$!3u%*GBP$Rc4#F*$!3$= \\%y\"em(H_F97$$!3mCj&f)3xi>F*$!3KUt;Q)3T*eF97$$!3(or4AN*4E=F*$!3E\"Q+ p]mud'F97$$!3QQQ*3**=dq\"F*$!3ot(zMJo$zrF97$$!3f9&yM5fzj\"F*$!3k#Rc0vn \"=vF97$$!3e!>jg@*>q:F*$!3'G&fJ!3'*4&yF97$$!3b.*R+5h@]\"F*$!3w<&*>+b!3 ^(F97$$!3`;m,%)H7M9F*$!3m#3$3?\\hqrF97$$!3v(4F,K))HI\"F*$!3x)[N1gT\\^' F97$$!3S*4!R#[0R=\"F*$!3.(\\]>TF&>fF97$$!3pY@QqWIU5F*$!3VL2\">NA:@&F97 $$!3Wp3w$R)\\B#*F9$!3sM/)o>\\F97$$!38TqX=W2,EF9$!3c?&G#4s`+8F97$$!3qTJY)ocYO\"F9 $!3\\3dJUMGBoF37$$!38,z7VKJ,J!#?$!3e]Rc@ml]:Fjs7$$\"39SFf3xEa8F9F^t7$$ \"3'R2()Hve,c#F9Fat7$$\"3IcwRsU78F*7$$\"3%>>CZ'eYk>F*$\"3=,c'))yEr<\"F*7$$\"3kYuyfix%4#F*$\" 3[YB!QR;o/\"F*7$$\"3T4q))fu.GAF*$\"33Py-P>bN\"*F97$$\"3y?w'**=&>gBF*$ \"3GB3a=Wj\"[#F*$\"3=K(*[]Be*f'F97$$\"3A?c*)yu\"3i #F*$\"3$*G<%pu^x?&F97$$\"3-i-HnWIXFF*$\"336`*H'=)G'RF97$$\"3u=RWhN.yGF *$\"3jV(e9#4fNEF97$$\"3ss(*RSA20IF*$\"3'R?+>8/_O\"F97$$\"3!)***\\/l#fT JF*$\"3\"[(HB9LzRJF--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F][lF\\[l-%*THIC KNESSG6#\"\"#-F$6%7$7$$!3/+++Fjzq:F*$F[[lF][l7$Ff[l$\"3%************** p\"F*-Ffz6&FhzF\\[lF\\[l$\"*++++\"!\")-%*LINESTYLEG6#\"\"$-F$6%7$7$$\" 3/+++Fjzq:F*Fh[l7$Fi\\lFj[lF\\\\lFa\\l-%+AXESLABELSG6%%\"xG%!G-%%FONTG 6#%(DEFAULTG-%%VIEWG6$;$!#KF[[l$\"#KF[[l;$!#6F[[l$\"# " 0 "" {MPLTEXT 1 0 207 "f := x -> piecewise(x<- Pi/2,-x/2-Pi/2,x<0,x/2,x f(x-2*Pi*floor((x+P i)/(2*Pi))):\n'f_(x)'='f(x-2*Pi*floor((x+Pi)/(2*Pi)))';\nplot(f_(x),x= -Pi..4*Pi,color=COLOR(RGB,.5,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floo rG6#,$*(F.!\"\",&F/F,F'F,F,F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 560 168 168 {PLOTDATA 2 "6'-%'CURVESG6#7aq7$$!3*****4tk#fTJ!#<$!3&\\Eu \\c'*)RJ!#E7$$!3Sf%**4v.#*z#F*$!3ho;&H^W>r\"!#=7$$!3tPJ1.IH,DF*$!3#pFL ED)\\,KF37$$!3mli\\47Em@F*$!3GPwY?slw[F37$$!3w`Xs2'3!H=F*$!3y'>E$H-#Hc 'F37$$!3uF\"oH='4X;F*$!3Y@:yQb#=h(F37$$!3#fFbMLrsd\"F*$!3&fes1g1;#yF37$$!3 \"G1x57:``\"F*$!3-9`Q0cdwwF37$$!3!*\\))p3*eL\\\"F*$!3a\\U\\VXzmuF37$$! 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3Si2Bp+\\*p'F37$$\"3Ayb-,vk\"4\"Fael$!3-9?VZ_[euF37$$\"3%4[f;(p*f4\"Fa el$!3'*\\r7z(efn(F37$$\"3$QQ$HUkM+6Fael$!3zqm7d4`9yF37$$\"3t'GFH\"fp/6 Fael$!3%f_JaUdqf(F37$$\"3X*=hNQX!46Fael$!3-!ROP*QeztF37$$\"31&**G[KWx6 \"Fael$!3C4hMIojWpF37$$\"3o+o4mKWE6Fael$!3YGe&pw*o4lF37$$\"3uG)zJ,lv: \"Fael$!3QDW!GT-O&\\F37$$\"3sv:q'GZ2>\"Fael$!33wqrO()[%H$F37$$\"3c'eCk &p]A7Fael$!3dMlc]_]1 " 0 "" {MPLTEXT 1 0 105 "f := x -> p iecewise(x<-Pi/2,-x/2-Pi/2,x<0,x/2,x " 0 "" {MPLTEXT 1 0 246 "f := x -> piecewise(x<-Pi/2,-x/2-Pi/2,x<0,x/2,xF*F?F?F:F:F:,$*&F*F**&\"#]F*F>F*F?F?F:F:Q(pprin t96\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 308 49 "Calculation of the coefficients of the sine terms" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "f := x -> piecewise(x<-Pi/2,-x/2-Pi/2,x<0,x/2,x,$*(F,F*\" #DF=F,$*(F,F*\"#\\F=F,$*&F*F**&\"#FF*F,$*(F,F* \"$@\"F=FQ(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 311 53 "A procedure for constructing truncated F ourier series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "FS := (x,n)->Pi/16+sum((-(- 1)^k-1+2*cos(Pi*k/2))/(2*Pi*k^2)*cos(k*x)+\n \+ 3*sin(Pi*k/2)/(Pi*k^2)*sin(k*x),k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&*&\"#;! \"\"%#PiG\"\"\"F2-%$sumG6$,&*&#F2\"\"#F2**,()F0%\"kGF0F2F0*&F9F2-%$cos G6#,$*(F9F0F1F2F=F2F2F2F2F2F1F0F=!\"#-F@6#*&F=F29$F2F2F2F2*,\"\"$F2-%$ sinGFAF2F1F0F=FD-FLFFF2F2/F=;F29%F2F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 312 44 "The first few terms of the Fou rier series of" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "FS(x,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, 6*&\"#;!\"\"%#PiG\"\"\"F(*(\"\"$F(F'F&-%$sinG6#%\"xGF(F(*&#F(\"\"#F(*& F'F&-%$cosG6#,$*&F1F(F.F(F(F(F(F&*&#F(F*F(*&F'F&-F,6#,$*&F*F(F.F(F(F(F (F&*&#F*\"#DF(*&F'F&-F,6#,$*&\"\"&F(F.F(F(F(F(F(*&#F(\"#=F(*&F'F&-F46# ,$*&\"\"'F(F.F(F(F(F(F&*&#F*\"#\\F(*&F'F&-F,6#,$*&\"\"(F(F.F(F(F(F(F&* &#F(\"#FF(*&F'F&-F,6#,$*&\"\"*F(F.F(F(F(F(F(*&#F(\"#]F(*&F'F&-F46#,$*& \"#5F(F.F(F(F(F(F&*&#F*\"$@\"F(*&F'F&-F,6#,$*&\"#6F(F.F(F(F(F(F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 310 39 "Graphs of some truncated Fourier series" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "FS := (x,n)->Pi/16+Sum((-(-1)^k-1+2*cos(Pi*k/2) )/(2*Pi*k^2)*cos(k*x)+\n 3*sin(Pi*k/2)/ (Pi*k^2)*sin(k*x),k=1..n);\nf_ :=x-> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n plot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,7)],x=-Pi..2*Pi,\n color=[ black,red,blue,brown,magenta],linestyle=[3,1$4]);\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&*&\"# ;!\"\"%#PiG\"\"\"F2-%$SumG6$,&*&#F2\"\"#F2**,()F0%\"kGF0F2F0*&F9F2-%$c osG6#,$*(F9F0F1F2F=F2F2F2F2F2F1F0F=!\"#-F@6#*&F=F29$F2F2F2F2*,\"\"$F2- %$sinGFAF2F1F0F=FD-FLFFF2F2/F=;F29%F2F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 623 321 321 {PLOTDATA 2 "6)-%'CURVESG6%7Y7$$!3*****4tk#fTJ!# <$!3&\\Eu\\c'*)RJ!#E7$$!3S_J_4$fh$HF*$!3b.KL?n;F5!#=7$$!3I/4wgGTdFF*$! 30WW9k*)*3#>F37$$!37p-i%y$RcDF*$!3+?w%[M%*f#HF37$$!3mEnbBA/aBF*$!3JK`; ]@vPRF37$$!3u\"zSTS_E:#F*$!3$p+XsC,Z%\\F37$$!3//4&=EQf'>F*$!3SXWpe>Fye F37$$!3kC\\T#e1Ex\"F*$!3RUV(eNI\\%oF37$$!3+nmpKYjs;F*$!3_IcY/,zWtF37$$ !3Q4%yHoiEd\"F*$!3k=p0`)\\Y%yF37$$!3?#yqXM6IZ\"F*$!3+6R&Gsc]O(F37$$!3y aJ;1+Ot8F*$!3%Rx:3.+o'oF37$$!3/&pXcq_$o6F*$!3Av%G#GNwTeF37$$!3sk6iP;#y ()*F3$!3O#e5)=3\"*Q\\F37$$!3$QLxh,D]%yF3$!3#pm)33D^ARF37$$!3#Rx)\\O:)Q !eF3$!3'pQ\\#o2%>!HF37$$!3%f+R\"z:'o$QF3$!3(H]p&*yI%=>F37$$!3++n(R,>10 #F3$!3+]$))p]4`-\"F37$$\"3IXD63lhRt!#?F_q7$$\"3U;GFHcrs=F3Fcq7$$\"3=)o km\">vlRF3Ffq7$$\"3pIqN#oV%=eF3Fiq7$$\"3w'=q^=S6&yF3F\\r7$$\"3')G\")pY zu'y*F3F_r7$$\"3)eUE(e^j!=\"F*Fbr7$$\"3sEG8=y4m8F*Fer7$$\"3=I_y%3@hY\" F*Fhr7$$\"3kLwV^V9m:F*F[s7$$\"3Q8DY9//q;F*$\"3uzs7RAbr9F*7$$\"39$R([xk $Rx\"F*$\"3)**R-hF*$\"3EYO`>Dx'=\"F*7$$\"3'Q5? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "f := x -> piecewise(x<-Pi/2,-x/2-Pi/2,x<0,x/2,x G,$*&\"#;!\" \"%#PiG\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "The Fourier series of a pseudo-odd functi on " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 36 " is a periodic function with period " }{XPPEDIT 18 0 "2*L " "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 40 " which satisfies the symmetry relation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(L-x) = -f(x);" "6#/-%\"fG6#,&%\"LG\"\"\"%\"xG!\"\",$-F%6#F*F+" }{TEXT -1 22 " for all real numbers " }{TEXT 352 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Then, because of the periodicity, if " }{TEXT 353 1 "n" }{TEXT -1 8 " is any " }{TEXT 262 11 "odd integer" }{TEXT -1 9 " we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f( n*L-x) = -f(x);" "6#/-%\"fG6#,&*&%\"nG\"\"\"%\"LGF*F*%\"xG!\"\",$-F%6# F,F-" }{TEXT -1 9 " for all " }{TEXT 354 1 "x" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 47 "Thus, i n the terminology of the first section, " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 6 " is a " }{TEXT 262 10 "pseudo-odd" }{TEXT -1 11 " function. 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" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 271 15 "____ ___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 270 1 "k" }{TEXT -1 23 " is even, th e function " }{XPPEDIT 18 0 "g(x)=cos(k*Pi*x/L)" "6#/-%\"gG6#%\"xG-%$c osG6#**%\"kG\"\"\"%#PiGF-F'F-%\"LG!\"\"" }{TEXT -1 25 " is periodic wi th period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 28 " and satisfies the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "g(L-x) = g(x);" "6#/-%\"gG6#,&%\"LG\"\"\"%\"xG!\"\"-F%6 #F*" }{TEXT -1 9 " for all " }{TEXT 269 1 "x" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "This rela tion follows since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(L-x)=cos(k*Pi*(L-x)/L)" "6#/-%\"gG6#,&%\"LG\"\"\"%\"xG!\"\"-%$cos G6#**%\"kGF)%#PiGF),&F(F)F*F+F)F(F+" }{XPPEDIT 18 0 "``=cos(k*Pi-k*Pi* x/L)" "6#/%!G-%$cosG6#,&*&%\"kG\"\"\"%#PiGF+F+**F*F+F,F+%\"xGF+%\"LG! \"\"F0" }{XPPEDIT 18 0 "`` = cos(-k*Pi*x/L);" "6#/%!G-%$cosG6#,$**%\"k G\"\"\"%#PiGF+%\"xGF+%\"LG!\"\"F/" }{XPPEDIT 18 0 "`` = cos(k*Pi*x/L); " "6#/%!G-%$cosG6#**%\"kG\"\"\"%#PiGF*%\"xGF*%\"LG!\"\"" }{XPPEDIT 18 0 "`` = g(x);" "6#/%!G-%\"gG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Thus, in the terminolog y of earlier sections, " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" } {TEXT -1 4 " is " }{TEXT 262 11 "pseudo-even" }{TEXT -1 30 " with resp ect to the interval " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\"\"F%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Since the product " }{XPPEDIT 18 0 "f(x)*cos(k*Pi*x/L)" " 6#*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF(%#PiGF(F'F(%\"LG!\"\"F(" } {TEXT -1 28 " of the pseudo-odd function " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 30 " and the pseudo-even function " }{XPPEDIT 18 0 "g(x)=cos(k*Pi*x/L)" "6#/-%\"gG6#%\"xG-%$cosG6#**%\"kG\"\"\"%#PiG F-F'F-%\"LG!\"\"" }{TEXT -1 32 " is pseudo-odd, it follows that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos(k*Pi*x/L ),x = -L .. L)=0" "6#/-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF, %#PiGF,F+F,%\"LG!\"\"F,/F+;,$F3F4F3\"\"!" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 5 "when " }{TEXT 275 1 "k" }{TEXT -1 16 " is even. Hen ce " }{XPPEDIT 18 0 "a[k]=0" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT -1 6 " whe n " }{TEXT 276 1 "k" }{TEXT -1 10 " is even. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 273 1 "k" } {TEXT -1 4 " is " }{TEXT 262 3 "odd" }{TEXT -1 15 ", the function " } {XPPEDIT 18 0 "h(x) = sin(k*Pi*x/L);" "6#/-%\"hG6#%\"xG-%$sinG6#**%\"k G\"\"\"%#PiGF-F'F-%\"LG!\"\"" }{TEXT -1 25 " is periodic with period \+ " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 28 " and sat isfies the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(L-x) = h(x);" "6#/-%\"hG6#,&%\"LG\"\"\"%\"xG!\"\"-F%6#F*" }{TEXT -1 9 " for all " }{TEXT 272 1 "x" }{TEXT -1 2 ". " }}{PARA 256 "" 0 " " {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "This relation follows \+ since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(L-x) = si n(k*Pi*(L-x)/L);" "6#/-%\"hG6#,&%\"LG\"\"\"%\"xG!\"\"-%$sinG6#**%\"kGF )%#PiGF),&F(F)F*F+F)F(F+" }{XPPEDIT 18 0 "`` = sin(k*Pi-k*Pi*x/L);" "6 #/%!G-%$sinG6#,&*&%\"kG\"\"\"%#PiGF+F+**F*F+F,F+%\"xGF+%\"LG!\"\"F0" } {XPPEDIT 18 0 "`` = sin(Pi-k*Pi*x/L);" "6#/%!G-%$sinG6#,&%#PiG\"\"\"** %\"kGF*F)F*%\"xGF*%\"LG!\"\"F/" }{XPPEDIT 18 0 "`` = sin(k*Pi*x/L);" " 6#/%!G-%$sinG6#**%\"kG\"\"\"%#PiGF*%\"xGF*%\"LG!\"\"" }{XPPEDIT 18 0 " `` = h(x);" "6#/%!G-%\"hG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "h(x );" "6#-%\"hG6#%\"xG" }{TEXT -1 9 " is also " }{TEXT 262 11 "pseudo-ev en" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Since the product \+ " }{XPPEDIT 18 0 "f(x)*sin(k*Pi*x/L);" "6#*&-%\"fG6#%\"xG\"\"\"-%$sinG 6#**%\"kGF(%#PiGF(F'F(%\"LG!\"\"F(" }{TEXT -1 28 " of the pseudo-odd f unction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 30 " and t he pseudo-even function " }{XPPEDIT 18 0 "h(x) = sin(k*Pi*x/L);" "6#/- %\"hG6#%\"xG-%$sinG6#**%\"kG\"\"\"%#PiGF-F'F-%\"LG!\"\"" }{TEXT -1 33 " is pseudo-odd, it follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(f(x)*sin(k*Pi*x/L),x = -L .. L) = 0;" "6#/-%$Int G6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#**%\"kGF,%#PiGF,F+F,%\"LG!\"\"F,/F+;, $F3F4F3\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "when " } {TEXT 274 1 "k" }{TEXT -1 15 " is odd. Hence " }{XPPEDIT 18 0 "b[k] = \+ 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 277 1 "k" } {TEXT -1 9 " is odd. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "Examples of Fourier series of pseudo-odd \+ functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 54 " be the \"pseudo odd\" function defined on the i nterval " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECE WISE([Pi*cos*x, -Pi <= x and x < 0],[Pi-2*x, 0 <= x and x < Pi]);" "6# /-%\"fG6#%\"xG-%*PIECEWISEG6$7$*(%#PiG\"\"\"%$cosGF.F'F.31,$F-!\"\"F'2 F'\"\"!7$,&F-F.*&\"\"#F.F'F.F331F5F'2F'F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 53 "and then extended to a periodic function with per iod " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "f := x -> piecewise(x<0,Pi*cos(x),Pi-2*x):\n'f(x)'=f(x);\npi := evalf(Pi): pi2 := pi/2:\np1 := plot(f(x),x=-Pi..Pi,thickness=2):\np2 \+ := plot([[[-pi2,-3.2],[-pi2,3.2]],[[pi2,-3.2],[pi2,3.2]]],color=blue,l inestyle=3):\nplots[display]([p1,p2],view=[-3.2..3.2,-3.3..3.3],labels =[`x`,``]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEW ISEG6$7$*&%#PiG\"\"\"-%$cosGF&F.2F'\"\"!7$,&F-F.*&\"\"#F.F'F.!\"\"%*ot herwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 440 282 282 {PLOTDATA 2 "6'-%'C URVESG6%7X7$$!3*****4tk#fTJ!#<$!37$z*e`EfTJF*7$$!35'\\8!o[6tIF*$!3=$fL @uHU8$F*7$$!3w\"*pr)3PY+$F*$!3d*o&=Bb<7JF*7$$!3)**z*4R\\0XHF*$!3/6kbYC 6\"3$F*7$$!3?3E[*ysa)GF*$!3(z!)Gfb9\"RIF*7$$!3gX&))>2g9v#F*$!3]\\Y1F\" Hb!HF*7$$!3)4577.flh#F*$!3=9qS%RU%=FF*7$$!3u\"QM::*H#[#F*$!39&4V=+%>$[ #F*7$$!3y\\AhcI#yN#F*$!3x*ovlGc]A#F*7$$!3&e=d-FN*GAF*$!3]1L&>BT:#>F*7$ $!3u%*GBP$Rc4#F*$!3gWDd)H!=u:F*7$$!3mCj&f)3xi>F*$!3PPSo^A8+7F*7$$!3(or 4AN*4E=F*$!32(4Lr%[tLz!#=7$$!3QQQ*3**=dq\"F*$!3!*\\oNP=(eA%Fco7$$!3e!> jg@*>q:F*$\"3A'*fjw&ye(=!#?7$$!3`;m,%)H7M9F*$\"3)eMx#zYO!G%Fco7$$!3v(4 F,K))HI\"F*$\"3\\GX4.lA8$)Fco7$$!3S*4!R#[0R=\"F*$\"3#*Q/*)QrN&=\"F*7$$ !3pY@QqWIU5F*$\"3)*=Y#)p+4%e\"F*7$$!3Wp3w$R)\\B#*Fco$\"3[CKq$eht*=F*7$ $!3EJc8o39GyFco$\"3#*fLz?WFco$\"3Oa$)zm[.!3$F*7$$! 3qTJY)ocYO\"Fco$\"3QH,Qt_Q7JF*7$$!3c.@Z/\"\\$yp!#>$\"34FdX%RYR8$F*7$$! 38,z7VKJ,JF^p$\"3Q+pXXvdTJF*7$$\"3p0t!3)GF;mF^t$\"3+ZO(f>n#4IF*7$$\"39 SFf3xEa8Fco$\"33X7(=6R2(GF*7$$\"3'R2()Hve,c#Fco$\"3LyB*H!4cHEF*7$$\"3I cwRn$=F*7$$\"3%HoBlmlzz(Fco$\"3_c]G?&**>e\"F*7$ $\"3^u'f#4)z?@*Fco$\"3?eyt\"pw\"*H\"F*7$$\"39Hv>CZ'eYk>F*$!3a2fee2RtyFco7$$\"3kYuyfix%4#F*$!3;+^)f' )fz/\"F*7$$\"3T4q))fu.GAF*$!3qDU=mA[98F*7$$\"3y?w'**=&>gBF*$!3Y[aMExzy :F*7$$\"3!*>3a=Wj\"[#F*$!3oY=\\$=w;#=F*7$$\"3A?c*)yu\"3i#F*$!3LZ9?/B/+ @F*7$$\"3-i-HnWIXFF*$!3!4t!*4G;!\\BF*7$$\"3u=RWhN.yGF*$!3RW!)HpWZ9EF*7 $$\"3ss(*RSA20IF*$!3K_(4s#=boGF*7$$\"3!)***\\/l#fTJF*$!3\\1-JZEfTJF*-% 'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F[]lFj\\l-%*THICKNESSG6#\"\"#-F$6%7$7$ $!3/+++Fjzq:F*$!3;+++++++KF*7$Fd]l$\"3;+++++++KF*-Fd\\l6&Ff\\lFj\\lFj \\l$\"*++++\"!\")-%*LINESTYLEG6#\"\"$-F$6%7$7$$\"3/+++Fjzq:F*Ff]l7$Fh^ lFi]lF[^lF`^l-%+AXESLABELSG6%%\"xG%!G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$! #KFi\\l$\"#KFi\\l;$!#LFi\\l$\"#LFi\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 53 " can be extended to a periodic function with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"# \"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "f := x -> piecewise(x<0,Pi* cos(x),Pi-2*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=-Pi..4*Pi,color=COLOR(R GB,.5,0,1),thickness=2);" }}{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 614 180 180 {PLOTDATA 2 "6' -%'CURVESG6#7er7$$!3*****4tk#fTJ!#<$!37$z*e`EfTJF*7$$!3C#=r_.%z)4$F*$! 3'H0$*4$erQJF*7$$!3'\\OKKU&*f0$F*$!3;LOiL14IJF*7$$!3@ZN>6o>8IF*$!3;%)o -eGt:JF*7$$!3[HZ:*>)RqHF*$!3[^/j*zoc4$F*7$$!3U%4x](4![)GF*$!3z^%pC?z&Q IF*7$$!3Sf%**4v.#*z#F*$!33Tnqv)R#fHF*7$$!3c)HJqP[-l#F*$!3mL*[)*QS*pFF* 7$$!3tPJ1.IH,DF*$!3^hGaReH>DF*7$$!3#>qzi5xPL#F*$!3W,3[w&)3r@F*7$$!3mli \\47Em@F*$!3W0W%G[*4iF*$!3zgqmuRg+8F*7$$!3w`Xs2'3!H =F*$!3E!f`(Qc8A!)!#=7$$!3s,<@eP=h;F*$!3%)Gq3U*Rd$GFco7$$!3!*\\))p3*eL \\\"F*$\"3k!3=gfP.V#Fco7$$!3xRV7!zjxL\"F*$\"3A>_m/K&[D(Fco7$$!3iH)\\:n o@=\"F*$\"3I6+X.%3/>\"F*7$$!3xBpoQ*e5-\"F*$\"3fqo[_pOT;F*7$$!3C\"=S#e? \\*f)Fco$\"3_.#)p'o8)\\?F*7$$!3!>Yti*GHLpFco$\"3`iFco$\"3EJ!>$4HL#3$F*7$$!3kRtJ6RG=:Fco$\"3Pdo`=\" 4\"Fco$\"3sw81gz!H7$F*7$$!3[\"*4Ie#o3k'!#>$\"3yxIg4ymMJF*7$$!3-*yk3$G) )pBFcs$\"3MQt_\"[529$F*7$$\"3#fh2S-)3shFcs$\"3!3k4J*3:=IF*7$$\"33-!))y )eSr9Fco$\"3q#>7gZ6t%GF*7$$\"3ec1e./;wHFco$\"3!=mtGdgja#F*7$$\"336LF> \\\"4[%Fco$\"3!48N(p'4aC#F*7$$\"3YDb!Q57\\<'Fco$\"3-)oGGB5m!>F*7$$\"3% )RxL)G4*oyFco$\"39XA#fz5yc\"F*7$$\"3-Tb1)=i)p&*Fco$\"3\"\\owf@?wA\"F*7 $$\"3AM$z(3:3F6F*$\"3!oC6.O'Hu))Fco7$$\"3m(*)e&Q)**4H\"F*$\"3%z(*>ZwHf f&Fco7$$\"35h%Q$o\"=\\X\"F*$\"334(G\"pJcF*$!3Yv6:8Sg wrFco7$$\"3S!\\tBzFm5#F*$!3q(=d6$Hmr5F*7$$\"3M\\d0\"3rlD#F*$!3a0<_3&\\ :P\"F*7$$\"3D3!Q(pV^1CF*$!3QBi)e3O9n\"F*7$$\"3K`NNgS$4e#F*$!3a8t6naF?? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "f := x -> piecewise(x <0,Pi*cos(x),Pi-2*x):\nc=1/(2*Pi)*Int('f(x)',x=-Pi..Pi);\nsimplify(val ue(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*&#\"\"\"\"\"#F(,$- %$IntG6$-%\"fG6#%\"xG/F1;,$%#PiG!\"\"F5*$F5F6F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 323 51 "Calculation of the coefficients of the cosine \+ terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "f := x -> piecewise(x<0,Pi*cos(x), Pi-2*x):\nassume(k_,integer):\na[k]=1/Pi*Int('f(x)'*cos(k*x),x=-Pi..Pi );\nsubs(k_=k,value(subs(k=k_,%)));\naa := unapply(rhs(%),k):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%\"kG,$-%$IntG6$*&-%\"fG6#%\" xG\"\"\"-%$cosG6#*&F'F1F0F1F1/F0;,$%#PiG!\"\"F9*$F9F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%\"kG,$**\"\"#\"\"\"%#PiG!\"\",&F+F-)F-F' F+F+F'!\"#F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The formula does not \"work\" when " }{XPPEDIT 18 0 "k=1 " "6#/%\"kG\"\"\"" }{TEXT -1 26 ", so we need to calculate " } {XPPEDIT 18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT -1 13 " separately. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "a[1]=1/Pi*Int('f(x)'*cos(x),x=-Pi..Pi);\nsimplify(value(%));\naa (1) := rhs(%):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\",$- %$IntG6$*&-%\"fG6#%\"xGF'-%$cosGF/F'/F0;,$%#PiG!\"\"F6*$F6F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\",$*(\"\"#!\"\",&*$)%#PiGF*F'F '\"\")F'F'F/F+F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "matrix([[k,`|`,seq(k,k=1..12)],['a'[k],`|`,se q(aa(k),k=1..12)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$ 70%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\" #770&%\"aG6#F(F),$*(F+!\"\",&*$)%#PiGF+F*F*F1F*F*F@F " 0 "" {MPLTEXT 1 0 161 "f := x -> piecewise(x<0,Pi*cos(x),Pi-2*x):\nassume(k_,integer):\n b[k]=1/Pi*Int('f(x)'*sin(k*x),x=-Pi..Pi);\nsubs(k_=k,value(subs(k=k_,% )));\nbb := unapply(rhs(%),k):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#%\"kG,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#*&F'F1F0F1F1/F0;,$ %#PiG!\"\"F9*$F9F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%\"kG,$ *(,&)!\"\"F'\"\"\"F-F-F-F'F,,&*$)F'\"\"#F-F-F-F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The previous formula d oes not give the coefficient when " }{XPPEDIT 18 0 "k=1" "6#/%\"kG\"\" \"" }{TEXT -1 26 ", so we need to calculate " }{XPPEDIT 18 0 "b[1];" " 6#&%\"bG6#\"\"\"" }{TEXT -1 13 " separately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "b[1]=1/Pi*In t('f(x)'*sin(x),x=-Pi..Pi);\nsimplify(value(%));\nbb(1) := rhs(%):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$-%$IntG6$*&-%\"fG6# %\"xGF'-%$sinGF/F'/F0;,$%#PiG!\"\"F6*$F6F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "matrix([[k,`|`,seq(k,k=1.. 12)],['b'[k],`|`,seq(bb(k),k=1..12)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$70%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"( \"\")\"\"*\"#5\"#6\"#770&%\"bG6#F(F)\"\"!#!\"\"F,F:#F<\"#IF:#F<\"$0\"F :#F<\"$_#F:#F<\"$&\\F:#F<\"$e)Q)pprint266\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "b[k ] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 330 1 "k" }{TEXT -1 4 " is " }{TEXT 262 3 "odd" }{TEXT -1 15 ", as expected. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 327 53 "A proce dure for constructing truncated Fourier series" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "FS := (x,n)->(Pi^2+8)/(2*Pi)*cos(x)+sum(-2*(-1+(-1)^k)/(k^2*Pi)*c os(k*x)-\n ((-1)^k+1)/(k*(k^2-1))*sin(k*x),k =2..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%) operatorG%&arrowGF),&*&#\"\"\"\"\"#F0*(,&*$)%#PiGF1F0F0\"\")F0F0F6!\" \"-%$cosG6#9$F0F0F0-%$sumG6$,&*,F1F0,&F0F8)F8%\"kGF0F0FD!\"#F6F8-F:6#* &FDF0F " 0 "" {MPLTEXT 1 0 9 "FS(x,11) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8*&#\"\"\"\"\"#F&*(,&*$)%#PiGF'F &F&\"\")F&F&F,!\"\"-%$cosG6#%\"xGF&F&F&*&#F&\"\"$F&-%$sinG6#,$*&F'F&F2 F&F&F&F.*&#\"\"%\"\"*F&*&F,F.-F06#,$*&F5F&F2F&F&F&F&F&*&#F&\"#IF&-F76# ,$*&F=F&F2F&F&F&F.*&#F=\"#DF&*&F,F.-F06#,$*&\"\"&F&F2F&F&F&F&F&*&#F&\" $0\"F&-F76#,$*&\"\"'F&F2F&F&F&F.*&#F=\"#\\F&*&F,F.-F06#,$*&\"\"(F&F2F& F&F&F&F&*&#F&\"$_#F&-F76#,$*&F-F&F2F&F&F&F.*&#F=\"#\")F&*&F,F.-F06#,$* &F>F&F2F&F&F&F&F&*&#F&\"$&\\F&-F76#,$*&\"#5F&F2F&F&F&F.*&#F=\"$@\"F&*& F,F.-F06#,$*&\"#6F&F2F&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 326 39 "Graphs of some truncated Fourier series " }{TEXT -1 2 ". 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!\"\"$\"\"!FdbmFcbm-%+AXESLABELSG6$Q\"x6\"Q!Fibm-%%VIEWG6$;$!+aEfTJ!\" *$\"+iqjc7!\")%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 329 74 "Constructing the first few terms of the Fourier s eries using the procedure" }{TEXT -1 1 " " }{TEXT 0 13 "FourierSeries " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "f := x -> piecewise(x<0,Pi*cos(x),Pi-2*x):\nF ourierSeries(f(x),x=-Pi..Pi,numterms=11,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%F&F2F&F&F&F&F&*&#F& \"$&\\F&-F76#,$*&\"#5F&F2F&F&F&F.*&#F=\"$@\"F&*&F,F.-F06#,$*&\"#6F&F2F &F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 54 " be the \"pseudo odd\" function defined on the interval " }{XPPEDIT 18 0 "[-Pi,Pi]" "6 #7$,$%#PiG!\"\"F%" }{TEXT -1 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([-Pi^2/8*(2*x+Pi), -Pi <= x and x < 0],[(x-Pi/2)^3, 0 <= x and x < Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISE G6$7$,$*(%#PiG\"\"#\"\")!\"\",&*&F/\"\"\"F'F4F4F.F4F4F131,$F.F1F'2F'\" \"!7$*$,&F'F4*&F.F4F/F1F1\"\"$31F9F'2F'F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 53 "and then extended to a periodic function with per iod " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "f := x -> piecewise(x<0,-Pi^2/8*(2*x+Pi),(x-Pi/2)^3):\n'f(x)'=f (x);\npi := evalf(Pi): pi2 := pi/2:\np1 := plot(f(x),x=-Pi..Pi,thickne ss=2):\np2 := plot([[[-pi2,-4.2],[-pi2,4.2]],[[pi2,-4.2],[pi2,4.2]]],c olor=blue,linestyle=3):\nplots[display]([p1,p2],view=[-3.2..3.2,-4.3.. 4.3],labels=[`x`,``]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\" xG-%*PIECEWISEG6$7$,$*(\"\")!\"\"%#PiG\"\"#,&*&F1\"\"\"F'F4F4F0F4F4F/2 F'\"\"!7$*$),&*&F1F/F0F4F/F'F4\"\"$F4%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 440 282 282 {PLOTDATA 2 "6'-%'CURVESG6%7X7$$!3*****4tk#fTJ!# <$\"3W3FapXyvQF*7$$!3w\"*pr)3PY+$F*$\"382>et-'y`$F*7$$!3?3E[*ysa)GF*$ \"3A2y`]T$QC$F*7$$!3gX&))>2g9v#F*$\"3&3ZUR-rJ\"HF*7$$!3)4577.flh#F*$\" 3w(y-v/;.e#F*7$$!3u\"QM::*H#[#F*$\"33^NmsI/\\AF*7$$!3y\\AhcI#yN#F*$\"3 cy]4>1\">%>F*7$$!3&e=d-FN*GAF*$\"3vF4B`F*Qi\"F*7$$!3u%*GBP$Rc4#F*$\"3y j\\U@#)*\\H\"F*7$$!3mCj&f)3xi>F*$\"3%)Gn`&e%er'*!#=7$$!3(or4AN*4E=F*$ \"3QYQ6e'\\$*H'FZ7$$!3QQQ*3**=dq\"F*$\"3GVS\\)H$3HLFZ7$$!3e!>jg@*>q:F* $!31qdRw;Jt9!#?7$$!3`;m,%)H7M9F*$!3[n-\\j&zAP$FZ7$$!3v(4F,K))HI\"F*$!3 j(oQJq(*yg'FZ7$$!3S*4!R#[0R=\"F*$!3-5pp^*[ha*FZ7$$!3pY@QqWIU5F*$!3J!z( *zQ,SI\"F*7$$!3Wp3w$R)\\B#*FZ$!3o/\\_wv(**f\"F*7$$!3EJc8o39GyFZ$!3p/]# fCoU%>F*7$$!3'=[BP-8If'FZ$!3qeH:EQ-\\AF*7$$!3CB3<@?)yB&FZ$!3W&*RB$F*7$ $!3qTJY)ocYO\"FZ$!3&4owN/p!RNF*7$$!3c.@Z/\"\\$yp!#>$!3_()GXA2g.PF*7$$! 38,z7VKJ,JFdo$!33%4H8SK\"oQF*7$$\"3p0t!3)GF;mF_s$!3TS=$>]tjS$F*7$$\"39 SFf3xEa8FZ$!3)4wb%f9FdHF*7$$\"312**yIK@d>FZ$!3k**p!*4[.+EF*7$$\"3'R2() Hve,c#FZ$!3a2+')=@zsAF*7$$\"3IcwRu:MlE9FZ7$$\"3Olj%G%3%R=\"F*$!3sy+3<&p&*y&F_s7$$\"3\\-)REfwoI\"F* $!3v-ZqZmHQ=F_s7$$\"3cM2vQyFT9F*$!3kjk;bwns@Fdo7$$\"3!y32s$*Qxc\"F*$!3 A7i-(*\\$z&G!#D7$$\"35+K?Bs#**p\"F*$\"3kw?Wu.B`@Fdo7$$\"3<#H$eMa;H=F*$ \"3twgHPDtC>CZ'eYk>F*$\"3_36W@9#45'F_s7$$\"3kYuyfix%4#F*$ \"3*p*4x[HhQ9FZ7$$\"3T4q))fu.GAF*$\"3S$zB>id!RGFZ7$$\"3y?w'**=&>gBF*$ \"3r6EPxM9>\\FZ7$$\"3!*>3a=Wj\"[#F*$\"3'y&**e%*)\\lb(FZ7$$\"3A?c*)yu\" 3i#F*$\"3y)\\#)>'\\pd6F*7$$\"3-i-HnWIXFF*$\"3u&f$32!)>?;F*7$$\"3u=RWhN .yGF*$\"3/(>W7V(*QB#F*7$$\"3tX=#4!HbTHF*$\"3O-=&\\R:cd#F*7$$\"3ss(*RSA 20IF*$\"3MKcJhp^]HF*7$$\"3/')[UXCLtIF*$\"3yeTggx9#R$F*7$$\"3!)***\\/l# fTJF*$\"3m)3'zhXyvQF*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F\\]lF[]l-%*THI CKNESSG6#\"\"#-F$6%7$7$$!3/+++Fjzq:F*$!3;+++++++UF*7$Fe]l$\"3;+++++++U F*-Fe\\l6&Fg\\lF[]lF[]l$\"*++++\"!\")-%*LINESTYLEG6#\"\"$-F$6%7$7$$\"3 /+++Fjzq:F*Fg]l7$Fi^lFj]lF\\^lFa^l-%+AXESLABELSG6%%\"xG%!G-%%FONTG6#%( DEFAULTG-%%VIEWG6$;$!#KFj\\l$\"#KFj\\l;$!#VFj\\l$\"#VFj\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 53 " can be e xtended to a periodic function with period " }{XPPEDIT 18 0 "2*Pi" "6# *&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "f := x -> piecewise(x<0 ,-Pi^2/8*(2*x+Pi),(x-Pi/2)^3):\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi ))):\n'f_(x)'='f(x-2*Pi*floor((x+Pi)/(2*Pi)))';\nplot(f_(x),x=-Pi..4*P i,color=COLOR(RGB,.5,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floorG6#,$*(F .!\"\",&F/F,F'F,F,F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 614 180 180 {PLOTDATA 2 "6'-%'CURVESG6#7ir7$$!3*****4tk#fTJ!#<$\"3W3FapXyvQF*7 $$!3[HZ:*>)RqHF*$\"3IyQ!)*>zLX$F*7$$!3Sf%**4v.#*z#F*$\"3h[]1IQ(4.$F*7$ $!3c)HJqP[-l#F*$\"3;JXWw6WjEF*7$$!3tPJ1.IH,DF*$\"3C8S#G_3fH#F*7$$!3#>q zi5xPL#F*$\"31cuy9'zD)=F*7$$!3mli\\47Em@F*$\"3w(*3v12Dp9F*7$$!3q4/h3\\ j(*>F*$\"3/#G5k)>=`5F*7$$!3w`Xs2'3!H=F*$\"3ilmpgE8rj!#=7$$!3s,<@eP=h;F *$\"3(p:'Gx/AIAFU7$$!3!*\\))p3*eL\\\"F*$!3pYV71FU7$$!3xRV7!zjxL\"F *$!3ug=!QPZ)\\dFU7$$!3iH)\\:no@=\"F*$!3'ePz9/.!*e*FU7$$!3xBpoQ*e5-\"F* $!3;a%yijAkN\"F*7$$!3C\"=S#e?\\*f)FU$!3Iq*3%o\\%Rv\"F*7$$!3!>Yti*GHLpF U$!3[B*fH8j];#F*7$$!3aUnIMP4n_FU$!3)o(3^(H\"=wDF*7$$!3A^Q=%4Qig$FU$!3? ^T*3+\")f)HF*7$$!3!*f41aCQX>FU$!3]DuF/2y&R$F*7$$!3Q>Pdo`=\"4\"FU$!3c*[ NrQXlg$F*7$$!3-*yk3$G))pB!#>$!30aN**p+J+\"Q1#F*7$$\"3%Q)pUhw`GPFU$!3!3D7B4F\">\\\"4[%FU$!3dI\\cO98:9F*7$$\"3YDb!Q57\\<'FU$!3a#41hrZNm)FU7$ $\"3%)RxL)G4*oyFU$!3S-1;>)er\"[FU7$$\"3-Tb1)=i)p&*FU$!31hIF[%3EJ#FU7$$ \"3AM$z(3:3F6F*$!39MZFjq7$$\" 3m(*)e&Q)**4H\"F*$!313!*\\\")oT!>#Fjq7$$\"3Qz'[M+fHP\"F*$!3Z=V_))\\FVx !#?7$$\"35h%Q$o\"=\\X\"F*$!3i*[Y\\m#)fb\"F^v7$$\"3oS\\V:*\\*e7T=F*$\"3#*> )zyX;b(>Fjq7$$\"3LbCXFliH>F*$\"3dX6)>$eE?YFjq7$$\"3(G(H\"*fr7=?F*$\"31 DpG6!38&*)Fjq7$$\"3S!\\tBzFm5#F*$\"3\"y3K(QWXQ:FU7$$\"3M\\d0\"3rlD#F*$ \"3Jh%4rNT1$F*$\"3.+=9\">S-L$F*7$$\"3pNbsbN[1JF*$\"3#[ %=:[`m@OF*7$$\"3eNS`*RJ)[JF*$\"3_h+L&pBz&QF*7$$\"3)\\`ULCz6>$F*$\"3u3G #*yXV`PF*7$$\"3VM5:(3FNB$F*$\"3'fb:DYX*[OF*7$$\"3HL!oZxA#=LF*$\"3%30,( Hs'*RMF*7$$\"3fK]Qi%=HS$F*$\"3SWl)o**))4B$F*7$$\"3%ohy\"4\">Uc$F*$\"3G 9M^.c*H$GF*7$$\"34,A(fv>bs$F*$\"3i%GS,@-]V#F*7$$\"3#HG_$f+#Q*QF*$\"3U- TScyt>?F*7$$\"3?lBti.7iSF*$\"3m=zm-NZ/;F*7$$\"3tsR!*yDn;UF*$\"3rllV%>J JA\"F*7$$\"3E!ev]zC7P%F*$\"3hG@0i)))yT)FU7$$\"3'y#e\\R-$z`%F*$\"3?YI@# *of/VFU7$$\"3Xvg\"RoNYq%F*$\"3$zjRPA\\I\">Fjq7$$\"3U+r&*)y&zx[F*$!3)\\ -TYT\\73%FU7$$\"3SD\")*R*e&40&F*$!3u8gl^P!QN)FU7$$\"3d.'[2t?\"F*7$$\"3u\"GX\\lGCN&F*$!3#\\vGsfM#z:F*7$$\"3HJ[Tv!G_^&F*$!3q?! >k1E4)>F*7$$\"3#3Q%)e\\F!ycF*$!3Y'G4c`gM.A2f7KF*7$$\"37Ijy*Rsm4'F*$!30k.7KCe:M F*7$$\"3gz'G6%=%*yhF*$!3#zE2A9u&=OF*7$$\"3Na)*zhl2?iF*$!3&)>2D(**p+s$F *7$$\"35H5Z#G67E'F*$!3yrTH_ec@QF*7$$\"3$Q?UJ+YBI'F*$!3Gb'*o`inNPF*7$$ \"3dyL\"Qs![VjF*$!3Inu_7&yjW$F*7$$\"3'*fz_Q13BkF*$!3U%H)3?ytHHF*7$$\"3 CUDC`0o-lF*$!3c1#*)p^&\\nCF*7$$\"3_Cr&zY!G#e'F*$!3_w5%HYDm0#F*7$$\"3!f qrEQ!)=m'F*$!3MvZl<:5%p\"F*7$$\"3)p\"\\U]YkQoF*$!3glTXF-sY5F*7$$\"31G \"y\"=*3a,(F*$!3+YW$[byo*eFU7$$\"3k+Gmu%RU<(F*$!3yw$H+fW29$FU7$$\"3Ctu 9J+2LtF*$!3PUeh$*y[89FU7$$\"3S0H_\"ziyT(F*$!3Cly#**)['\\H)Fjq7$$\"3mO$ )*=bbE](F*$!34&pC8R=kL%Fjq7$$\"3#zwtAJ[ue(F*$!39@E#f-`M*=Fjq7$$\"33+# \\E2TAn(F*$!3nZT&HzBG+'F^v7$$\"3Y5Q\")f2\"f#yF*$!3KJJiK=\">@#Fiv7$$\"3 %3UypW!ezzF*$\"3s1.mS!H8)>F^v7$$\"3CfbvD7ej!)F*$\"3C#f(GMp73#*F^v7$$\" 3W*pKX+#eZ\")F*$\"3t(Ha$o(p3`#Fjq7$$\"3iR)4Ly#eJ#)F*$\"3!>/F+UORQ&Fjq7 $$\"3-yp3iNe:$)F*$\"3X\"GJ'R\\kN)*Fjq7$$\"3==!y]$*fOZ)F*$\"37\"R5H4s&z BFU7$$\"37g!p!3jtJ')F*$\"3\\Wa?phl/ZFU7$$\"3M07[l;(pz)F*$\"3=P3jQ@N&Q) FU7$$\"3!)[L*G-2A'*)F*$\"3apiX')>3h8F*7$$\"3pM`'\\!4)H/*F*$\"3/q_:x6\" 4o\"F*7$$\"3d?t.(yaP7*F*$\"31+Vx^bGZ?F*7$$\"3n/$4\"p'GX?*F*$\"3AkarErO jCF*7$$\"3c!H\"=^DI&G*F*$\"3h&*3Q=zJKHF*7$$\"3wxo(*>`')p$*F*$\"3qL%*\\ *>aL[$F*7$$\"3AjCx)3GWX*F*$\"3O_%zl#ei-QF*7$$\"3V]!ov&3**Q&*F*$\"3uW6 \"ybvRf$F*7$$\"3lPOOEObB'*F*$\"3ePG/*GD`Q$F*7$$\"3!)))36?;W'y*F*$\"3=; (\\SF:M)HF*7$$\"3=Q\"eQhH$\\**F*$\"3A*fc!f_]\"e#F*7$$\"3I#*)*Q61f65!#; $\"3%4Xw&>U\\q@F*7$$\"3hqRRh#[#G5Fj]m$\"3c2j4!=$[fb(=N\"F*7$$\"3)z_(*Qp(Gh5Fj]m$\"3gizz*eyEW*FU7$$\"3=`:Y(f nk2\"Fj]m$\"3=\"e2U^(=(p&FU7$$\"3Ayb-,vk\"4\"Fj]m$\"3!RC<'Qkp^>FU7$$\" 3X*=hNQX!46Fj]m$!3Oj.Y7!f5M#FU7$$\"3o+o4mKWE6Fj]m$!3Lqz`jW\"Qj'FU7$$\" 3![JQ'RT+U6Fj]m$!3)Q5H8hIt/\"F*7$$\"3uG)zJ,lv:\"Fj]m$!311WIw(z7V\"F*7$ $\"3K-2%*\\h:u6Fj]m$!3a*))*Gn'\\1%=F*7$$\"3sv:q'GZ2>\"Fj]m$!3go`Fe&>+D #F*7$$\"39\"3j:7Fm?\"Fj]m$!3\"Qep$y(Q=k#F*7$$\"3c'eCk&p]A7Fj]m$!3e)zj% )*zlLIF*7$$\"3G$Hsw+s&R7Fj]m$!3y_ " 0 "" {MPLTEXT 1 0 107 "f := x -> piecewise(x<0,-Pi^2/8*(2*x+Pi),(x-Pi/2)^3):\nc=1/(2*Pi) *Int('f(x)',x=-Pi..Pi);\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*&#\"\"\"\"\"#F(,$-%$IntG6$-%\"fG6#%\"xG/F1;,$% #PiG!\"\"F5*$F5F6F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 51 "Calcu lation of the coefficients of the cosine terms" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "f := x -> piecewise(x<0,-Pi^2/8*(2*x+Pi),(x-Pi/2)^3):\nassume(k_, integer):\na[k]=1/Pi*Int('f(x)'*cos(k*x),x=-Pi..Pi);\nsubs(k_=k,value( subs(k=k_,%)));\naa := unapply(rhs(%),k):\nmatrix([[k,`|`,seq(k,k=1..9 )],['a'[k],`|`,seq(aa(k),k=1..9)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6#%\"kG,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#*&F'F1F0F1F1/ F0;,$%#PiG!\"\"F9*$F9F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#% \"kG*(%#PiG!\"\",**()F*F'\"\"\")F'\"\"#F.)F)F0F.F.\"\"'F.*&F1F.F/F.F** &F2F.)F*,&F.F.F'F.F.F.F.F'!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7$7-%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"* 7-&%\"aG6#F(F)*&%#PiG!\"\",&*&F+F*)F8F+F*F9\"#7F*F*\"\"!,$*(\"#\")F9F8 F9,&*&\"#=F*F,$*(\"$D'F9F8F9,&*&\"#]F*F, $*(\"%,CF9F8F9,&*&\"#)*F*F,$*(\"%hlF9F8F9,&*&\"$i\"F*F< F*F9F=F*F*F*Q)pprint296\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "a[k]=0" "6#/&%\"aG6#% \"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 322 1 "k" }{TEXT -1 4 " is " } {TEXT 262 4 "even" }{TEXT -1 15 ", as expected. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 49 "Calculation of the coeff icients of the sine terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "f := x -> piecewi se(x<0,-Pi^2/8*(2*x+Pi),(x-Pi/2)^3):\nassume(k_,integer):\nb[k]=1/Pi*I nt('f(x)'*sin(k*x),x=-Pi..Pi);\nsubs(k_=k,value(subs(k=k_,%)));\nbb := unapply(rhs(%),k):\nmatrix([[k,`|`,seq(k,k=1..9)],['b'[k],`|`,seq(bb( k),k=1..9)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%\"kG,$-%$I ntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#*&F'F1F0F1F1/F0;,$%#PiG!\"\"F9*$F9F :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%\"kG,$*(\"\"$\"\"\",&)! \"\"F'F+F+F+F+F'!\"$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7$7-%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*7-&%\"bG 6#F(F)\"\"!#F,F-F7#F,\"#KF7#F*\"#OF7#F,\"$c#F7Q)pprint306\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " } {XPPEDIT 18 0 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT -1 6 " when \+ " }{TEXT 321 1 "k" }{TEXT -1 4 " is " }{TEXT 262 3 "odd" }{TEXT -1 15 ", as expected. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 318 53 "A procedure for constructing truncated Fourier series" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "FS := (x,n)->sum(((-1)^k*k^2*Pi^2+6-Pi^2*k^2+6* (-1)^(1+k))/(k^4*Pi)*cos(k*x)+\n 3*((-1)^k+1 )/k^3*sin(k*x),k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$% \"xG%\"nG6\"6$%)operatorG%&arrowGF)-%$sumG6$,&**,**()!\"\"%\"kG\"\"\") F6\"\"#F7)%#PiGF9F7F7\"\"'F7*&F:F7F8F7F5*&F " 0 "" {MPLTEXT 1 0 8 "FS(x,9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4*(,&*&\"\"#\"\"\")%#P iGF'F(!\"\"\"#7F(F(F*F+-%$cosG6#%\"xGF(F(*&#\"\"$\"\"%F(-%$sinG6#,$*&F 'F(F0F(F(F(F(*&#F(\"#\")F(*(,&*&\"#=F(F)F(F+F,F(F(F*F+-F.6#,$*&F3F(F0F (F(F(F(F(*&#F3\"#KF(-F66#,$*&F4F(F0F(F(F(F(*&#F(\"$D'F(*(,&*&\"#]F(F)F (F+F,F(F(F*F+-F.6#,$*&\"\"&F(F0F(F(F(F(F(*&#F(\"#OF(-F66#,$*&\"\"'F(F0 F(F(F(F(*&#F(\"%,CF(*(,&*&\"#)*F(F)F(F+F,F(F(F*F+-F.6#,$*&\"\"(F(F0F(F (F(F(F(*&#F3\"$c#F(-F66#,$*&\"\")F(F0F(F(F(F(*&#F(\"%hlF(*(,&*&\"$i\"F (F)F(F+F,F(F(F*F+-F.6#,$*&\"\"*F(F0F(F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 317 39 "Graphs of some truncated Fourier series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "FS := (x,n)->sum(((-1)^k*k^ 2*Pi^2+6-Pi^2*k^2+6*(-1)^(1+k))/(k^4*Pi)*cos(k*x)+\n \+ 3*((-1)^k+1)/k^3*sin(k*x),k=1..n):\nf_ :=x-> f(x-2*Pi*floor(( x+Pi)/(2*Pi))):\nplot([f_(x),FS(x,1),FS(x,3),FS(x,5),FS(x,9)],x=-Pi..2 *Pi,\n color=[black,red,blue,magenta,brown],linestyle=[3,1$4]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 623 321 321 {PLOTDATA 2 "6)-%'CURVESG6%7[ o7$$!3*****4tk#fTJ!#<$\"3W3FapXyvQF*7$$!3S_J_4$fh$HF*$\"3%=#[&e7)*)oLF *7$$!3I/4wgGTdFF*$\"3Wx!4:%*ey#HF*7$$!37p-i%y$RcDF*$\"3%[%R'=Dk=V#F*7$ $!3mEnbBA/aBF*$\"3$[$RX(y\"eK>F*7$$!3u\"zSTS_E:#F*$\"3<4MWDBnN9F*7$$!3 //4&=EQf'>F*$\"37T$4KXO(\\(*!#=7$$!3kC\\T#e1Ex\"F*$\"3g$*4jn%o%z\\FK7$ $!3Q4%yHoiEd\"F*$\"3iR'*p+$\\]g%!#?7$$!3yaJ;1+Ot8F*$!3i?[gZfar[FK7$$!3 /&pXcq_$o6F*$!3jV7@O$)*)H**FK7$$!3sk6iP;#y()*FK$!3mQ9M(yH&Q9F*7$$!3$QL xh,D]%yFK$!31z%z$\\A5S>F*7$$!3#Rx)\\O:)Q!eFK$!3\"ztOh@MPW#F*7$$!3%f+R \"z:'o$QFK$!3?_9%o$p2HHF*7$$!3++n(R,>10#FK$!3-G\\!Ri9)pLF*7$$!3qsyZWU6 '))*!#>$!3S\\]R$\\a=j$F*7$$\"3IXD63lhRtFV$!3GjZO7(3<#QF*7$$\"3T4(p<(*e 0t*Fgp$!3e2yr!=1#*>$F*7$$\"3U;GFHcrs=FK$!35>d\"*f6E[EF*7$$\"3J_(oHxL#> HFK$!3QHNn(><;4#F*7$$\"3=)okm\">vlRFK$!3=W+'>b5!>;F*7$$\"3Wf3^*z(4#*[F K$!31%[8U!=Fl7F*7$$\"3pIqN#oV%=eFK$!3=lH@[q?s'*FK7$$\"3w'=q^=S6&yFK$!3 s4vIz3**\\[FK7$$\"3!y:Mf1W*=))FK$!3S!oFE2J%pKFK7$$\"3')G\")pYzu'y*FK$! 3/IZP@\\-w?FK7$$\"3Q>\")pw\\lz5F*$!35k=,4.t%=\"FK7$$\"3)eUE(e^j!=\"F*$ !3JEFgp7$$\"3sEG8=y4m8F*$!3 -+t$QNqrd)FV7$$\"3kLwV^V9m:F*$!3`nOGf\"Gn+\"!#C7$$\"39$R([xk$Rx\"F*$\" 3'z0CXPlFQ)FV7$$\"3+qF*$\" 3iP&\\BijLm&Fgp7$$\"3OD\"Qjy*\\_?F*$\"3lK\\MZht<6FK7$$\"3'Q5?'Q%z,:#F* $\"3Cp,y$)4!\\%>FK7$$\"3-T#)Qx?4^AF*$\"3Y(>XoVC%[JFK7$$\"3=yj:;Z+_BF*$ \"3!y=k!4(4ww%FK7$$\"3I>&y`P^%\\DF*$\"3]t`&pTAKP*FK7$$\"3ngi.t7(\\k#F* $\"3P,1\\6(Q%R7F*7$$\"31-Spq6\\SFF*$\"3R.TEU,O+;F*7$$\"3q)>Y8t\\l%GF*$ \"3uE#*Q1$[j2#F*7$$\"3O&R)*>H3E&HF*$\"3g1=2=xVQEF*7$$\"3%=W*)ei1z/$F*$ \"3Fu'G*[-%GA$F*7$$\"3J)[!yf\\?VJF*$\"3#*z&o*yj!=(QF*7$$\"3rl7Bsi&\\C$ F*$\"3.xLny]u?OF*7$$\"35V?o%e2nM$F*$\"3cu\"y$yPopLF*7$$\"3jV+G476JNF*$ \"3^g3:sgo9HF*7$$\"3m\"pX$yIrKPF*$\"3\\?8g)R`sT#F*7$$\"3WCa$fs/C#RF*$ \"3%3%fi=!3#\\>F*7$$\"3T-0$[:(o?TF*$\"3_`bSNd'*f9F*7$$\"3AgP!=ZWXJ%F*$ \"3=N*)4#y>k\")*FK7$$\"35'o8p6&\\F*7$$\"35euYH&=$G'F* $!3ISz/aXyvQF*-%'COLOURG6&%$RGBG\"\"!Fe_lFe_l-%*LINESTYLEG6#\"\"$-F$6% 7ap7$F($\"3;)4uHnmMY#F*7$$!3'yGjGJM-4$F*$\"3)fw*op%=-Y#F*7$$!3?wlTyf() QIF*$\"3m^!esU#[]CF*7$$!3_k)pRkL.!Q)Hl6F*7$F\\o$!3)y;1,aViN\"F*7$Fao$!3*3RxWZ$\\VPzvh]VG#F*7$$!3'H&yb'HSP%HFK$!3M7:'e1) \\dBF*7$F`p$!3$>j#=fI&=T#F*7$$!3jVA@>_h>:FK$!3115k0y2NCF*7$Fep$!3'R42u 4Q9X#F*7$$!334L$oHwgd%Fgp$!3Qev,:y)3Y#F*7$F[q$!3'f[L)>.SjCF*7$$\"3(>[! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "f : = x -> piecewise(x<0,-Pi^2*x/4-Pi^3/8,(x-Pi/2)^3):\nFourierSeries(f(x) ,x=-Pi..Pi,numterms=9,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9co nstant~coefficient~-->G\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 54 " be the \"pseudo odd\" function defined on the interval " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\" F%" }{TEXT -1 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([sin*2*x, -Pi <= x and x < 0],[2*x, 0 <= x and \+ x < Pi/2],[2*x-2*Pi, Pi/2 <= x and x <= Pi])" "6#/-%\"fG6#%\"xG-%*PIEC EWISEG6%7$*(%$sinG\"\"\"\"\"#F.F'F.31,$%#PiG!\"\"F'2F'\"\"!7$*&F/F.F'F .31F6F'2F'*&F3F.F/F47$,&*&F/F.F'F.F.*&F/F.F3F.F431*&F3F.F/F4F'1F'F3" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 53 "and then extended to a \+ periodic function with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\" %#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "f := x -> piecewise(x<0,sin(2*x),x 2g9v#F*$\"3?SJ%e-zY.(F37$$!3)4577.flh#F*$\"3w;@wEocu')F37$$!3OTKP\"4H %\\DF*$\"3;L?G-mXi#*F37$$!3u\"QM::*H#[#F*$\"3;K\\%HiMOo*F37$$!3w:L2/61 ?CF*$\"3,Q@!4(>`=**F37$$!3y\\AhcI#yN#F*$\"3;:_Zbp%*****F37$$!3/=ZVj\"z LH#F*$\"3DOLlL%)=@**F37$$!3&e=d-FN*GAF*$\"3#pS\\\"[l%yn*F37$$!3IS]u.tG i@F*$\"3+vPs,!ytD*F37$$!3u%*GBP$Rc4#F*$\"3gIH=f&pEn)F37$$!3mCj&f)3xi>F *$\"3ai_z&4931(F37$$!3(or4AN*4E=F*$\"3%f'GIZ31()[F37$$!3QQQ*3**=dq\"F* $\"3O-]<`O#em#F37$$!3e!>jg@*>q:F*$!37dEbz6A%>\"!#?7$$!3`;m,%)H7M9F*$!3 E3Lr]Pb*p#F37$$!3v(4F,K))HI\"F*$!3VMq2'z3P5&F37$$!3S*4!R#[0R=\"F*$!3i- \\L\"p^%))pF37$$!3pY@QqWIU5F*$!38CY`w$p(3()F37$$!3:o6z[:FB)*F3$!3J>s() )>aVB*F37$$!3Wp3w$R)\\B#*F3$!3^Z[h-TAF'*F37$$!3N]#[4j>e_)F3$!3)\\xr/[i )4**F37$$!3EJc8o39GyF3$!3qF9E^k')****F37$$!3cc&Hf%pd5sF3$!3@/\\ll-K<** F37$$!3'=[BP-8If'F3$!3is8@_Nn$o*F37$$!3b_rWAvW:fF3$!3!4\"e%zm%yd#*F37$ $!3CB3<@?)yB&F3$!3[6a[\"oZ@m)F37$$!3_*)R6;/\"F*7$$\"3q]1XIqOClF3$\"39I,41M([I\"F*7$$\"3%HoBlmlzz(F 3$\"3gOZILJff:F*7$$\"3^u'f#4)z?@*F3$\"3!\\$>&='fTU=F*7$$\"39Hv!*eJ,e \"F*$!3aR69H@#H7$F*7$$\"3s,m&HZiUe\"F*$!3z#Qm7OgY6$F*7$$\"3!*e8$3CCDf \"F*$!3Uoo^Do8)4$F*7$$\"3(e61(3gy+;F*$!3]atw*G8;3$F*7$$\"3-IcXW&4th\"F *$!3?E$o#=ic[IF*7$$\"3&R90-3LQj\"F*$!3L)Hpn9>b,$F*7$$\"3-sTq^,)om\"F*$ !3=U7x.]U\\HF*7$$\"35+K?Bs#**p\"F*$!3+'=t2'3L$)GF*7$$\"39YK*)Gjak>CZ'eYk>F*$!3O -7txNDaBF*7$$\"3kYuyfix%4#F*$!3'Hp/wyKO4#F*7$$\"3T4q))fu.GAF*$!3UnbS(Q 5r#=F*7$$\"3y?w'**=&>gBF*$!3mWVCF\\zi:F*7$$\"3!*>3a=Wj\"[#F*$!3WYz4qk \"*>8F*7$$\"3A?c*)yu\"3i#F*$!3zX$)Q\\.bT5F*7$$\"3-i-HnWIXFF*$!3=A1*fsj d#zF37$$\"3u=RWhN.yGF*$!3E([ " 0 "" {MPLTEXT 1 0 200 "f := x -> p iecewise(x<0,sin(2*x),x f(x-2*Pi*floor ((x+Pi)/(2*Pi))):\n'f_(x)'='f(x-2*Pi*floor((x+Pi)/(2*Pi)))';\nplot(f_( x),x=-Pi..5*Pi,color=COLOR(RGB,.5,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&f loorG6#,$*(F.!\"\",&F/F,F'F,F,F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 614 180 180 {PLOTDATA 2 "6'-%'CURVESG6#7eu7$$!3*****4tk#fTJ! #<$\"3/UE[]'efD\"!#D7$$!3I(pB&4$fh$HF*$\"3/4[5f./%*R!#=7$$!3i%RP<(fsIF F*$\"3aqNK#osKK(F37$$!3r2lNZFNTEF*$\"3quH=<8I<%)F37$$!3E@c(H_z>b#F*$\" 3nJ-k$**4JC*F37$$!3ex^y5HH2DF*$\"3#\\uuSt\\oa*F37$$!3OMZf)H1EY#F*$\"3) R>V!eSQu(*F37$$!36\"H/ko>zT#F*$\"3,tj*ys'*Q#**F37$$!3WZQ@uIBtBF*$\"3!y ,[\\H%>%***F37$$!3w@)y^IyHK#F*$\"3M[$G_\"=%z(**F37$$!33'zVh`BFF#F*$\"3 1Lf?F`(4')*F37$$!3Rq(3rwoCA#F*$\"3kH,]^aZW'*F37$$!3;XP2)*R@s@F*$\"3UYS YluiI$*F37$$!3C%p.+Y/<2#F*$\"3)H@+FH4XU)F37$$!35VO$>#\\>r>F*$\"3y*Q8rq 7\"zrF37$$!3KP1(3OV)o5l8F*$!3W=z5Ba#))*RF37$$!3;Io(4;7P;\"F*$!3_***o%> $\\:F(F37$$!3!R8K)*3b.2\"F*$!3HLz!zfu%>%)F37$$!3UxV(o=!)*p(*F3$!3I'GDb XIZF*F37$$!32'*3:J[>.$*F3$!3QB&>Dh))Ge*F37$$!3%eTFaZ4k$))F3$!3@z.#pL(e 2)*F37$$!3]MRq>Tip$)F3$!3-\\fLz'po%**F37$$!3:`/)RwQG!zF3$!3-x!o**fA&** **F37$$!3T'y\"Rl&4&>uF3$!37)H4MbqA'**F37$$!3y?J!oO!=OpF3$!3n&=#z\"o+?$ 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m$\"3)[4[cMEA,(F37$$\"339a:X5nz5F]`m$\"3%R$pmgQCtQF37$$\"3!pL9*4-k)4\" F]`m$\"3z)3\"f*zMV$=Fb]l7$$\"3<#z.GXo%=6F]`m$!3?5/ylIn#p$F37$$\"3VZKp& p'HQ6F]`m$!3suf/2qi&*pF37$$\"3u%)=ah&*)z9\"F]`m$!3&\\5]gZw6C)F37$$\"3) =_!RFCod6F]`m$!3Uwb+2e)z<*F37$$\"3aS[Jg)GD;\"F]`m$!3.$R&R>X:>&*F37$$\" 3-f\"RKHvt;\"F]`m$!3Wq%pY?f4x*F37$$\"3mxM;E9#=\"F]`m$!3C[>lj*)H n**F37$$\"3!eEV8p:s=\"F]`m$!3Gff%*\\#oT$)*F37$$\"3M+5Zd%*G#>\"F]`m$!3U Uls+'f)*f*F37$$\"33N()fBKO(>\"F]`m$!3Ec*z*yOym#*F37$$\"3Q/U&ev5v?\"F]` m$!3-fGJ.#=!>$)F37$$\"3&Qn4\")Gew@\"F]`m$!3z$3\"R#\\v(HqF37$$\"3lT$ft% [?P7F]`m$!3SnED)\\T$*y$F37$$\"3i4!4mS^nD\"F]`m$\"3'\\MZ=Y.pG#Fip7$$\"3 m,Rh'eSnF\"F]`m$\"3>$)ofN]q?SF37$$\"3s$z=mwHnH\"F]`m$\"3%HHvl.T&=!)F37 $$\"3'f#47EMb;8F]`m$\"3Ku,q$HF$)>\"F*7$$\"3AeIi&3xjL\"F]`m$\"3M>Gu$[+[ f\"F*7$$\"3M'))**)pHfa8F]`m$\"3w\"Qz#p\"=\"f>F*7$$\"3Y9nKJ'o$R \"F]`m$\"3]s0nN])4u#F*7$$\"3#)G,)zn0*)R\"F]`m$\"3OJU)3Ls`%GF*7$$\"3e63 uU]7/9F]`m$\"3m')y4E'f(\\HF*7$$\"3'H:@^sMnS\"F]`m$\"3K9ZqtK&>+$F*7$$\" 3N%\\,vSW$49F]`m$\"3)>a68#p9aIF*7$$\"3/l;p[#\\1T\"F]`m$\"3!e&\\6XPC!3$ F*7$$\"3tN=))*3a>T\"F]`m$\"3jp$=*o0M1JF*7$$\"3U1?2J*eKT\"F]`m$\"3Y$y@F RPC8$F*7$$\"3Hx@EsPc99F]`m$!3S&Qa14^Y7$F*7$$\"3<#*>^?oBL9F]`m$!3u(3ec7 !>^FF*7$$\"312=wo)4>X\"F]`m$!33!zh1;HxP#F*7$$\"33`[(GB>=Z\"F]`m$!3gp3S y=az>F*7$$\"35**y)pfG<\\\"F]`m$!35\\*Rhfa8e\"F*7$$\"3V.<#))R%y5:F]`m$! 3\\iQYf&Q-?\"F*7$$\"3x2bl+-%)H:F]`m$!3kex(yAD7>)F37$$\"3'Rv_@E=.b\"F]` m$!3aA&G%GHh&4%F37$$\"3'****\\OK'zq:F]`m$!3evPf>Kz*G'F--%&COLORG6&%$RG BG$\"\"&!\"\"$\"\"!F\\an$\"\"\"F\\an-%*THICKNESSG6#\"\"#-%+AXESLABELSG 6$Q\"x6\"Q!Fgan-%%VIEWG6$;$!+aEfTJ!\"*$\"+Fjzq:!\")%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 342 39 "Calculation of the constant coefficient" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "f := x -> piecewi se(x<0,sin(2*x),x " 0 "" {MPLTEXT 1 0 173 "f := x -> piecewise(x<0 ,sin(2*x),x " 0 "" {MPLTEXT 1 0 79 "a[2]=1/Pi*Int('f(x)'*co s(2*x),x=-Pi..Pi);\nsimplify(value(%));\naa(2) := rhs(%):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"#,$-%$IntG6$*&-%\"fG6#%\"xG\"\" \"-%$cosG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6#\"\"#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "matrix([[k,`|`,seq(k,k=1 ..10)],['a'[k],`|`,seq(aa(k),k=1..10)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7.%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"& \"\"'\"\"(\"\")\"\"*\"#57.&%\"aG6#F(F),$**F+F*F,!\"\"%#PiGF:,&F1F**&F, F*F;F*F:F*F:\"\"!,$**F+F*\"#XF:F;F:,&F1F**&\"#:F*F;F*F:F*F*F>,$**F+F* \"$D&F:F;F:,&F1F**&\"$0\"F*F;F*F*F*F*F>,$**F+F*\"%0AF:F;F:,&F1F**&\"$: $F*F;F*F:F*F*F>,$**F+F*\"%PiF:F;F:,&F1F**&\"$$pF*F;F*F*F*F*F>Q)pprint3 36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " Note that " }{XPPEDIT 18 0 "a[k]=0" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 348 1 "k" }{TEXT -1 4 " is " }{TEXT 262 4 "even" } {TEXT -1 15 ", as expected. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 341 49 "Calculation of the coefficients of the sine te rms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "f := x -> piecewise(x<0,sin(2*x),x " 0 "" {MPLTEXT 1 0 79 "b[2]=1/Pi*Int('f(x)'*sin(2*x),x=-Pi..Pi);\nsi mplify(value(%));\nbb(2) := rhs(%):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#,$*&F'F1F0F1 F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"\"##\"\"$F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "matrix([[k,`|`,seq(k,k=1..10)],['b'[k],`|`,se q(bb(k),k=1..10)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$ 7.%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\" bG6#F(F)\"\"!#F,F+F8#!\"\"F+F8#F*F,F8#F;F-F8#F*F.Q)pprint356\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT -1 6 " wh en " }{TEXT 347 1 "k" }{TEXT -1 4 " is " }{TEXT 262 3 "odd" }{TEXT -1 15 ", as expected. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 344 53 "A procedure for constructing truncated Fourier series " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "FS := (x,n)->2*(3*Pi-8)/(3*Pi)*cos(x)+3/2*si n(2*x)+\n sum(2*(4*(-1)^(1+k)+k^3*sin(Pi*k/2)*Pi+4-4*k*sin(Pi*k/2)*Pi) /(k^2*(k^2-4)*Pi)*cos(k*x)-\n 2*cos(Pi*k/2)/ k*sin(k*x),k=3..n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG %\"nG6\"6$%)operatorG%&arrowGF),(*&#\"\"#\"\"$\"\"\"*(,&\"\")!\"\"*&F1 F2%#PiGF2F2F2F8F6-%$cosG6#9$F2F2F2*&#F1F0F2-%$sinG6#,$*&F0F2F " 0 "" {MPLTEXT 1 0 9 "FS(x,11);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#,8*&#\"\"#\"\"$\"\"\"*(,&\"\")!\"\"*&F'F(%#PiGF( F(F(F.F,-%$cosG6#%\"xGF(F(F(*&#F'F&F(-%$sinG6#,$*&F&F(F2F(F(F(F(*&#F& \"#XF(*(,&F+F(*&\"#:F(F.F(F,F(F.F,-F06#,$*&F'F(F2F(F(F(F(F(*&#F(F&F(-F 66#,$*&\"\"%F(F2F(F(F(F,*&#F&\"$D&F(*(,&F+F(*&\"$0\"F(F.F(F(F(F.F,-F06 #,$*&\"\"&F(F2F(F(F(F(F(*&#F(F'F(-F66#,$*&\"\"'F(F2F(F(F(F(*&#F&\"%0AF (*(,&F+F(*&\"$:$F(F.F(F,F(F.F,-F06#,$*&\"\"(F(F2F(F(F(F(F(*&#F(FKF(-F6 6#,$*&F+F(F2F(F(F(F,*&#F&\"%PiF(*(,&F+F(*&\"$$pF(F.F(F(F(F.F,-F06#,$*& \"\"*F(F2F(F(F(F(F(*&#F(FWF(-F66#,$*&\"#5F(F2F(F(F(F(*&#F&\"&dT\"F(*(, &F+F(*&\"%(G\"F(F.F(F,F(F.F,-F06#,$*&\"#6F(F2F(F(F(F(F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 343 39 "Graphs of some tr uncated Fourier series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "FS := (x,n)->2*(3*Pi-8 )/(3*Pi)*cos(x)+3/2*sin(2*x)+\n sum(2*(4*(-1)^(1+k)+k^3*sin(Pi*k/2)*Pi +4-4*k*sin(Pi*k/2)*Pi)/(k^2*(k^2-4)*Pi)*cos(k*x)-\n \+ 2*cos(Pi*k/2)/k*sin(k*x),k=3..n):\nf_ :=x-> f(x-2*Pi*floor((x+ Pi)/(2*Pi))):\nplot([f_(x),FS(x,3),FS(x,5),FS(x,7),FS(x,17)],x=-Pi..2* Pi,\n color=[black,red,blue,brown,magenta],linestyle=[3,1$4]);\n" } }{PARA 13 "" 1 "" {GLPLOT2D 623 321 321 {PLOTDATA 2 "6)-%'CURVESG6%7hp 7$$!3*****4tk#fTJ!#<$\"3/UE[]'efD\"!#D7$$!3S_J_4$fh$HF*$\"3%**z9\"f./% *R!#=7$$!3I/4wgGTdFF*$\"3[E6h:jc\\pF37$$!3%pe!pAL!pl#F*$\"3sz$)Q>JMX#) F37$$!37p-i%y$RcDF*$\"3#=#fHIq04#*F37$$!3]$QaV*e!e]#F*$\"3+MrAk(ecb*F3 7$$!3)y\\)3/!=_X#F*$\"3/CdSUq_/)*F37$$!3F7E#Q6IYS#F*$\"3Cpq`([;J&**F37 $$!3mEnbBA/aBF*$\"37N'4^N2*****F37$$!3#Hu-(oZp.BF*$\"3hd6G6i#\\%**F37$ $!3?f([QJZLD#F*$\"3MR;66S>*y*F37$$!3YvZ**e)**H?#F*$\"3E<3+i%)GM&*F37$$ !3u\"zSTS_E:#F*$\"3A**Hm')>z#=*F37$$!3)y%e*HL&Hf?F*$\"3;5APgX@)G)F37$$ 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "f := x -> piecewise(x<0,sin(2*x),xG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 42 " be the \"pseudo odd\" function defined by " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=PIECEWISE([ln(1 +sin*x)*cos*x,x<>-Pi/2],[0,x=Pi/2])" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7 $*(-%#lnG6#,&\"\"\"F1*&%$sinGF1F'F1F1F1%$cosGF1F'F10F',$*&%#PiGF1\"\"# !\"\"F:7$\"\"!/F'*&F8F1F9F:" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 36 " is a p eriodic function with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"% #PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "f := x ->ln(1+sin(x))*cos(x):\n'f( x)'=f(x);\npi := evalf(Pi): pi2 := pi/2:\np1 := plot(f(x),x=-Pi..Pi,th ickness=2):\np2 := plot([[[-pi2,-1.1],[-pi2,1.1]],[[pi2,-1.1],[pi2,1.1 ]]],color=blue,linestyle=3):\nplots[display]([p1,p2],view=[-3.2..3.2,- 1.2..1.2],labels=[`x`,``]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG 6#%\"xG*&-%#lnG6#,&\"\"\"F--%$sinGF&F-F--%$cosGF&F-" }}{PARA 13 "" 1 " " {GLPLOT2D 440 282 282 {PLOTDATA 2 "6'-%'CURVESG6%7bp7$$!3*****4tk#fT J!#<$\"3mL()ogKzzi!#E7$$!3w\"*pr)3PY+$F*$\"3]>DvR1>a9!#=7$$!3?3E[*ysa) GF*$\"3:N^z=D,EGF37$$!3gX&))>2g9v#F*$\"3(f1FgF37$$!3u\"QM::*H#[#F*$\"3Qp\"36RPZ\\(F37$$!3y\\AhcI#yN#F*$ \"3Ek,7LB@p')F37$$!3&e=d-FN*GAF*$\"3kZWIC**oy&*F37$$!3IS]u.tGi@F*$\"3A waE6h.%))*F37$$!3u%*GBP$Rc4#F*$\"3HxTuTQ*[+\"F*7$$!3qBL#3yI!z?F*$\"3^< .h@(Hl+\"F*7$$!3A_PTCAUi?F*$\"3x_Zu>U425F*7$$!3u!=/!oO\"e/#F*$\"3Ex!zb nTl+\"F*7$$!3q4Yf6^?H?F*$\"3?xCB)4C[+\"F*7$$!3=nax)*z)f*>F*$\"3sW;#p/' )o(**F37$$!3mCj&f)3xi>F*$\"3m!)ze43Y_)*F37$$!3'3-$3>^V%*=F*$\"3$*eVsfl U2%*F37$$!3(or4AN*4E=F*$\"3mZ<&R5i\"F*$\"3#f[AAuH9 N$F37$$!3e_3xf\"zSg\"F*$\"3aU_Sjg@*>q:F*$!3ki;*)H#y!z#*!#?7$$!3#*ot0(o*=`:F*$!3nzVnwx RW:F37$$!31Z:0e,=O:F*$!3OQ,)e(o/oDF37$$!3?Dd/H1<>:F*$!3#==H(3r`;MF37$$ !3b.*R+5h@]\"F*$!37I\"zwJe,:%F37$$!3/g#G?/U\"o9F*$!3O$zR*=Ehw`F37$$!3` ;m,%)H7M9F*$!35n+oyF37$$!3v(4F,K))H I\"F*$!3>;j'*p\"HE#))F37$$!3Z)fe7!pWV7F*$!3;QUx=kwQ%*F37$$!3S*4!R#[0R= \"F*$!3Q;9)oeF%G)*F37$$!3G6\")QH_][6F*$!3yIS$3i.#o**F37$$!3%H7'Qw\\586 F*$!3Y%=0)))Gs/5F*7$$!3wG^))\\[S&4\"F*$!3DapWknc15F*7$$!3#[8%QBZqx5F*$ !3$*)=s0m)325F*7$$!3'39$)of/+1\"F*$!3-J#\\r]Yj+\"F*7$$!3pY@QqWIU5F*$!3 ww6!\\D&R/5F*7$$!3:o6z[:FB)*F3$!3%zDw#e%HZ*)*F37$$!3Wp3w$R)\\B#*F3$!3P mWmK\"[4j*F37$$!3N]#[4j>e_)F3$!3fz\"e&e$\\8?*F37$$!3EJc8o39GyF3$!3A/J- s)f6m)F37$$!3'=[BP-8If'F3$!3=5:Wpy\"[\\(F37$$!3CB3<@?)yB&F3$!3FBV,da,0 gF37$$!3_*)R>(R%=s[9F37$$!38,z7VKJ,JFfs$!3,/omcB61JFfs7$$\"39S Ff3xEa8F3$\"3wvWP*)o%[D\"F37$$\"3'R2()Hve,c#F3$\"3)4g'orxl$=#F37$$\"3I cwRmKF37$$\"3*4* =Jof03_F3$\"3c,m[&))oI]$F37$$\"3%3F\"Q*\\6i'eF3$\"3)e(\\7g)4*oOF37$$\" 3q]1XIqOClF3$\"3/Vj!p8%)*pPF37$$\"3#o;([[j;hrF3$\"3XL]@s$Hr!QF37$$\"3% HoBlmlzz(F3$\"3!pEe\")*o;'y$F37$$\"3ty;*ytA]])F3$\"3y27k&3Itp$F37$$\"3 ^u'f#4)z?@*F3$\"3m,\"4r3VHa$F37$$\"3%H[7$$\" 3!y32s$*Qxc\"F*$\"3!fW>i_5#>@Ffs7$$\"35+K?Bs#**p\"F*$!3_=*=cp)4s))Fi`l 7$$\"3<#H$eMa;H=F*$!3)=[!Gt.EG>CZ'eYk>F*$!3O07Y`#*=4DF37$$ \"3kYuyfix%4#F*$!3?!*fW=8g?JF37$$\"3-Gs$)foSh@F*$!3I-1IS<2nLF37$$\"3T4 q))fu.GAF*$!3GMeCOHpiNF37$$\"35:t#\\K;TH#F*$!3[Qm#GM-Dq$F37$$\"3y?w'** =&>gBF*$!3!pb1-&z!\\y$F37$$\"3M?UD/[\"4U#F*$!3I(*)RFSQv!QF37$$\"3!*>3a =Wj\"[#F*$!3?!>==!GXxPF37$$\"31?#=([fA^DF*$!39]\"4)GHRwOF37$$\"3A?c*)y u\"3i#F*$!30fnyjr(H]$F37$$\"35TH4t41$o#F*$!3KA:4a:9'G$F37$$\"3-i-HnWIX FF*$!3h(3FqW@7,$F37$$\"3u=RWhN.yGF*$!3\"HNX!fIGNAF37$$\"3ss(*RSA20IF*$ !3q*44$pB6k7F37$$\"3!)***\\/l#fTJF*$!3$yU[8`$zRJF--%'COLOURG6&%$RGBG$ \"#5!\"\"$\"\"!F\\glF[gl-%*THICKNESSG6#\"\"#-F$6%7$7$$!3/+++Fjzq:F*$!3 3+++++++6F*7$Fegl$\"33+++++++6F*-Fefl6&FgflF[glF[gl$\"*++++\"!\")-%*LI NESTYLEG6#\"\"$-F$6%7$7$$\"3/+++Fjzq:F*Fggl7$FihlFjglF\\hlFahl-%+AXESL ABELSG6%%\"xG%!G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#KFjfl$\"#KFjfl;$!#7F jfl$\"#7Fjfl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 53 " can be extended to a periodic function with period \+ " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "f := x -> ln(1+sin(x))*cos(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=-Pi ..5*Pi,color=COLOR(RGB,.5,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floo rG6#,$*(F.!\"\",&F/F,F'F,F,F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 614 180 180 {PLOTDATA 2 "6'-%'CURVESG6#7[z7$$!3*****4tk#fTJ!#<$\"3mL() ogKzzi!#E7$$!3()[oTyf()QIF*$\"3q`lYjB7w5!#=7$$!3I(pB&4$fh$HF*$\"3P\\a' H=![LAF37$$!3uX0jSEWLGF*$\"3zFR)*R.sVMF37$$!3i%RP<(fsIFF*$\"33#)pm;+-u YF37$$!3r2lNZFNTEF*$\"3w;q*)zK&=t&F37$$!3E@c(H_z>b#F*$\"3;-ZPa%\\\"\\n F37$$!3OMZf)H1EY#F*$\"3oo)QC(*4bp(F37$$!3WZQ@uIBtBF*$\"3b'R?Rs4t`)F37$ $!33'zVh`BFF#F*$\"3O21U64K7$*F37$$!3;XP2)*R@s@F*$\"3mLrL&*>*o%)*F37$$! 3KKib8m3Z@F*$\"3+cH:+<'[$**F37$$!3#*>(Q!H#f>7#F*$\"3zm\\3FJ@+5F*7$$!33 27_W=$o4#F*$\"3VhwEHjt/5F*7$$!3C%p.+Y/<2#F*$\"3(zBw'\\M\"p+\"F*7$$!3S 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pgw$e`+5F*7$$\"3KD(z]Mmb9\"Fg_n$!3dTMK#G_]+\"F*7$$\"3u%)=ah&*)z9\"Fg_n $!3d-87%f\"*p+\"F*7$$\"3)R//!yFT]6Fg_n$!3VN=]Z#3l+\"F*7$$\"3A.iY%*f$G: \"Fg_n$!3?n*G/iXP+\"F*7$$\"3ki$G4@f_:\"Fg_n$!3I%)\\zGcP))**F37$$\"3)=_ !RFCod6Fg_n$!3FOM20o4>**F37$$\"3-f\"RKHvt;\"Fg_n$!3?L;Y\\+yi%*F37$$\"3 K'z(3f\"oq<\"Fg_n$!35<'Rxj>#o()F37$$\"3!eEV8p:s=\"Fg_n$!3?\"prK10t%yF3 7$$\"33N()fBKO(>\"Fg_n$!3emdoX^i$y'F37$$\"3Q/U&ev5v?\"Fg_n$!3(H.\\%Gc& pi&F37$$\"3&Qn4\")Gew@\"Fg_n$!3O&>Z=%Rm@WF37$$\"3v2Xtn:VF7Fg_n$!3#)ylY ])o=D$F37$$\"3lT$ft%[?P7Fg_n$!3>c\")\\q8>0@F37$$\"3svT)p7ypC\"Fg_n$!3[ 2^5F@R45F37$$\"3i4!4mS^nD\"Fg_n$\"3as0$>\\(zU6Ff\\l7$$\"3m,Rh'eSnF\"Fg _n$\"3)4&G(>v9Ry\"F37$$\"3s$z=mwHnH\"Fg_n$\"3?[jzE+qLIF37$$\"3$*f)pjfT mI\"Fg_n$\"3F(zkG\"pBPMF37$$\"3'f#47EMb;8Fg_n$\"3+_Ls%=VJp$F37$$\"3)*e k*4M4:K\"Fg_n$\"3'p$zh'R*)fw$F37$$\"3+#*>(eDlkK\"Fg_n$\"3Xf*=M3gE!QF37 $$\"3?Dvuq6UJ8Fg_n$\"3\"Qa>N;LQ!QF37$$\"3AeIi&3xjL\"Fg_n$\"3c'*4t6KQqP F37$$\"3M'))**)pHfa8Fg_n$\"3GtS([Hb\"pLF37$$\"3Y9nKJ'o$R\"Fg_n$\"3Iyu.9S9f8F37$$\"3Hx@EsPc99Fg_n$!3[ ]BIAhFreFf\\l7$$\"3<#*>^?oBL9Fg_n$!3vF\")zq'\\fK\"F37$$\"312=wo)4>X\"F g_n$!3!G\\CxG*pYCF37$$\"33`[(GB>=Z\"Fg_n$!3cSl3XhkMLF37$$\"35**y)pfG< \\\"Fg_n$!3!*oWvSQ\"px$F37$$\"3C]jWZD\\'\\\"Fg_n$!3[__a^,D0QF37$$\"3=, [!z\\c7]\"Fg_n$!3h\"QZ**[1:!QF37$$\"3J_KO[/-1:Fg_n$!3Ykz=u#>\\w$F37$$ \"3V.<#))R%y5:Fg_n$!3U.Me+K*[p$F37$$\"3p0'Q(*H7._\"Fg_n$!3/.#)p3$=IX$F 37$$\"3x2bl+-%)H:Fg_n$!3_N$3X0*ouIF37$$\"3'Rv_@E=.b\"Fg_n$!3C0c7k`V7=F 37$$\"3'****\\OK'zq:Fg_n$!3rt%\\De'*[9$!#D-%&COLORG6&%$RGBG$\"\"&!\"\" $\"\"!F[go$\"\"\"F[go-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Ffgo -%%VIEWG6$;$!+aEfTJ!\"*$\"+Fjzq:!\")%(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 "" }}{PARA 0 "" 0 "" {TEXT 334 39 "Calculation of the const ant coefficient" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "f := x -> ln(1+sin(x))*cos(x ):\nc=1/(2*Pi)*Int('f(x)',x=-Pi..Pi);\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*&#\"\"\"\"\"#F(,$-%$IntG6$-%\"fG6#% \"xG/F1;,$%#PiG!\"\"F5*$F5F6F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"cG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 332 51 "Calculation of the coefficients of the cosine terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 129 "Maple cannot obtain a formul a for the cosine coefficients, but we can see a pattern if we calculat e the coefficients separately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "f := x -> ln(1+sin(x))*cos( x):\nfor k from 1 to 17 do\n print(a[k]=1/Pi*Int('f(x)'*cos(k*x),x=- Pi..Pi));\n aa(k) := simplify(1/Pi*int('f(x)'*cos(k*x),x=-Pi..Pi)); \n print(a[k]=aa(k));\nend do:\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\",$-%$IntG6$*&-%\"fG6#%\"xGF'-%$cosGF/F'/ F0;,$%#PiG!\"\"F6*$F6F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\" \"\",&-%#lnG6#\"\"#!\"\"#F'F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6#\"\"#,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/F 0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"$,$-%$IntG6$* &-%\"fG6#%\"xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"$#\"\"\"\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"%,$-%$IntG6$*&-%\"fG6#%\" xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"&,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6# ,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"&#!\"\"\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6#\"\"',$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/F 0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\" \"'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"(,$-%$IntG6$* &-%\"fG6#%\"xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"(#\"\"\"\"#C" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"),$-%$IntG6$*&-%\"fG6#%\" xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\")\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"*,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6# ,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"*#!\"\"\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6#\"#5,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/F0 ;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#5 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#6,$-%$IntG6$*&-% \"fG6#%\"xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#6#\"\"\"\"#g" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#7,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$ cosG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6#\"#8,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/ F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\" #8#!\"\"\"#%)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#9,$-%$Int G6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F :F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#9\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#:,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-% $cosG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#:#\"\"\"\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#;,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#, $*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#;\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6# \"#<,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#,$*&F'F1F0F1F1F1/F0;,$%#P iG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"#<#!\"\" \"$W\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "matrix([[k,`|`,seq(k,k=1..17)],['a'[k],`|`,seq(aa(k), k=1..17)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$75%\"kG% \"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\" #9\"#:\"#;\"#<75&%\"aG6#F(F),&-%#lnG6#F+!\"\"#F*F+F*\"\"!#F*F-FE#FCF5F E#F*\"#CFE#FC\"#SFE#F*\"#gFE#FC\"#%)FE#F*\"$7\"FE#FC\"$W\"Q)pprint386 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "It looks as though we can generate the coefficients " }{XPPEDIT 18 0 "a[ k]" "6#&%\"aG6#%\"kG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "k=2,3,` . . \+ . `" "6%/%\"kG\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 26 " by means of the for mula: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k]=-2*sin (k*Pi/2)/(k^2-1)" "6#/&%\"aG6#%\"kG,$*(\"\"#\"\"\"-%$sinG6#*(F'F+%#PiG F+F*!\"\"F+,&*$F'F*F+F+F1F1F1" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "aa := k -> -2*sin(k*Pi/2)/(k^2-1):\naa(1) := 1/ 2-ln(2):\nmatrix([[k,`|`,seq(k,k=1..17)],['a'[k],`|`,seq(aa(k),k=1..17 )]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$75%\"kG%\"|grG \"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#: \"#;\"#<75&%\"aG6#F(F),&-%#lnG6#F+!\"\"#F*F+F*\"\"!#F*F-FE#FCF5FE#F*\" #CFE#FC\"#SFE#F*\"#gFE#FC\"#%)FE#F*\"$7\"FE#FC\"$W\"Q)pprint416\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "a[k]=0" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT -1 6 " when \+ " }{TEXT 349 1 "k" }{TEXT -1 4 " is " }{TEXT 262 4 "even" }{TEXT -1 15 ", as expected. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 333 49 "Calculation of the coefficients of the sine terms" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "f := x -> ln(1+sin(x))*cos(x):\nfor k from 1 to 17 do\n print(b[k]=1/Pi*Int('f(x)'*sin(k*x),x=-Pi..Pi));\n bb(k) \+ := simplify(1/Pi*int('f(x)'*sin(k*x),x=-Pi..Pi));\n print(b[k]=bb(k) );\nend do:\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"\",$-%$IntG6$*&-%\"fG6#%\"xGF'-%$sinGF/F'/F0;,$%#PiG!\"\"F6*$F6F7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$s inG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"##F'\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#,$*&F'F1F0F1 F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%,$-%$In tG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$ F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%#!\"#\"#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$-%$IntG6$*&-%\"fG6#%\" xG\"\"\"-%$sinG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6# ,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"\"#\"#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"\"(,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#,$*&F'F1F0F1F1F1/F 0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"),$-%$IntG6$* &-%\"fG6#%\"xG\"\"\"-%$sinG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!\"#\"#j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*,$-%$IntG6$*&-%\"fG6#%\"xG\"\" \"-%$sinG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"#5,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#,$*&F'F1F0F1F1 F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6 #\"#5#\"\"#\"#**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6,$-%$ IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F: *$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7,$-%$IntG6$*&-%\"fG6#%\"xG\"\" \"-%$sinG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"#7#!\"#\"$V\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#, $*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"#9,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#,$*&F'F1F0F1F1F1/F0;,$%#P iG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#9#\"\"# \"$&>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#:,$-%$IntG6$*&-% \"fG6#%\"xG\"\"\"-%$sinG6#,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#:\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"#;,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6 #,$*&F'F1F0F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#;#!\"#\"$b#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#<,$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#,$*&F'F1F0 F1F1F1/F0;,$%#PiG!\"\"F:*$F:F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"#<\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 65 "matrix([[k,`|`,seq(k,k=1..17)],['b'[k],`|`,seq (bb(k),k=1..17)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7 5%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"# 7\"#8\"#9\"#:\"#;\"#<75&%\"bG6#F(F)\"\"!#F+F,F?#!\"#F8F?#F+\"#NF?#FB\" #jF?#F+\"#**F?#FB\"$V\"F?#F+\"$&>F?#FB\"$b#F?Q)pprint396\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "It looks as tho ugh we can generate the coefficients " }{XPPEDIT 18 0 "b[k];" "6#&%\"b G6#%\"kG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "k=2,3,` . . . `" "6%/%\" kG\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 26 " by means of the formula: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = -2*cos(k*Pi/2) /(k^2-1);" "6#/&%\"aG6#%\"kG,$*(\"\"#\"\"\"-%$cosG6#*(F'F+%#PiGF+F*!\" \"F+,&*$F'F*F+F+F1F1F1" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "bb := k -> -2*cos(k*Pi/2)/(k^2-1):\nbb(1) := 0:\nmat rix([[k,`|`,seq(k,k=1..17)],['b'[k],`|`,seq(bb(k),k=1..17)]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$75%\"kG%\"|grG\"\"\"\"\" #\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<75 &%\"bG6#F(F)\"\"!#F+F,F?#!\"#F8F?#F+\"#NF?#FB\"#jF?#F+\"#**F?#FB\"$V\" F?#F+\"$&>F?#FB\"$b#F?Q)pprint436\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 339 1 "k" }{TEXT -1 4 " is " }{TEXT 262 3 "odd" }{TEXT -1 15 ", as expected. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 336 53 "A procedure fo r constructing truncated Fourier series" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "FS : = (x,n)->(1/2-ln(2))*cos(x)+sum(-2*sin(k*Pi/2)/(k^2-1)*cos(k*x)-\n \+ 2*cos(k*Pi/2)/(k^2-1)*sin(k*x),k=2..n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operato rG%&arrowGF),&*&,&-%#lnG6#\"\"#!\"\"#\"\"\"F3F6F6-%$cosG6#9$F6F6-%$sum G6$,&**F3F6-%$sinG6#,$*(F3F4%#PiGF6%\"kGF6F6F6,&*$)FFF3F6F6F6F4F4-F86# *&FFF6F:F6F6F4**F3F6-F8FBF6FGF4-FAFKF6F4/FF;F39%F6F)F)F)" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 337 44 "The first few ter ms of the Fourier series of" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "FS(x,11);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#,8*&,&-%#lnG6#\"\"#!\"\"#\"\"\"F)F,F,-%$cosG6#%\"xGF, F,*&#F)\"\"$F,-%$sinG6#,$*&F)F,F0F,F,F,F,*&#F,\"\"%F,-F.6#,$*&F3F,F0F, F,F,F,*&#F)\"#:F,-F56#,$*&F;F,F0F,F,F,F**&#F,\"#7F,-F.6#,$*&\"\"&F,F0F ,F,F,F**&#F)\"#NF,-F56#,$*&\"\"'F,F0F,F,F,F,*&#F,\"#CF,-F.6#,$*&\"\"(F ,F0F,F,F,F,*&#F)\"#jF,-F56#,$*&\"\")F,F0F,F,F,F**&#F,\"#SF,-F.6#,$*&\" \"*F,F0F,F,F,F**&#F)\"#**F,-F56#,$*&\"#5F,F0F,F,F,F,*&#F,\"#gF,-F.6#,$ *&\"#6F,F0F,F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 335 39 "Graphs of some truncated Fourier series" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 304 "FS := (x,n)->(1/2-ln(2))*cos(x)+sum(-2*sin(k*Pi/2)/( k^2-1)*cos(k*x)-\n 2*cos(k*Pi/2)/(k^2 -1)*sin(k*x),k=2..n):\nf_ :=x-> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nplot( [f_(x),FS(x,3),FS(x,5),FS(x,7),FS(x,17)],x=-Pi..2*Pi,\n color=[blac k,red,blue,brown,magenta],linestyle=[3,1$4]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 660 244 244 {PLOTDATA 2 "6)-%'CURVESG6%7[t7$$!3*****4tk#fTJ! #<$\"3mL()ogKzzi!#E7$$!3?wlTyf()QIF*$\"3!)H&pMOAh2\"!#=7$$!3S_J_4$fh$H F*$\"3bc<(H=![LAF37$$!3OG?9&3'yYGF*$\"3m>T#)Qmi%G$F37$$!3I/4wgGTdFF*$ \"3ssP=v\"\\WN%F37$$!3%pe!pAL!pl#F*$\"3c&>R5I)*)\\bF37$$!37p-i%y$RcDF* $\"39@sL&H2.q'F37$$!3)y\\)3/!=_X#F*$\"3))=VZB'\\&pxF37$$!3mEnbBA/aBF*$ \"3*G#*GC(z!4q)F37$$!3?f([QJZLD#F*$\"3:#zV]5BhV*F37$$!3u\"zSTS_E:#F*$ \"3,(od[[:r\"**F37$$!3)ebaj88$H@F*$\"3](3^5mjY)**F37$$!3.?$o&oQ(f5#F*$ \"3'f$*yt[aL+\"F*7$$!3=%3#y+Yj#3#F*$\"3/#G%\\UUE15F*7$$!3)y%e*HL&Hf?F* $\"3&em5q'p225F*7$$!3f6'4_1cf.#F*$\"3D/\"\\F'fm05F*7$$!3uvLU(z;E,#F*$ \"3/!)H))Q_*=+\"F*7$$!3))RrjHvF*)>F*$\"3s*y,^uzh&**F37$$!3//4&=EQf'>F* $\"3_B7)>CYn')*F37$$!3W9H8ACFp=F*$\"3IG\\#G!p*4<*F37$$!3kC\\T#e1Ex\"F* $\"3slkhri17yF37$$!3%fzbvg?Es\"F*$\"3'of1rmRJv'F37$$!3+nmpKYjs;F*$\"3W \\I2TdB]`F37$$!3c-rEX;kZ;F*$\"3os&z'zgEsWF37$$!34Qv$yl[Ei\"F*$\"3'\\_D n]#)pU$F37$$!3itzSqcl(f\"F*$\"3$*p5Kke,H@F37$$!3Q4%yHoiEd\"F*$\"3]&Gzk ;9\\Z#!#>7$$!3-cunli?g:F*$!3Z4$zjDCm.\"F37$$!3U-lP[)\\xa\"F*$!3[Nvu(R/ u*=F37$$!30\\b2JMHN:F*$!3`x3#HX[eh#F37$$!3o&fuP,PG_\"F*$!3lD')o*)3rWKF 37$$!3%*)os\"zT#z\\\"F*$!3%>1m/AB(=VF37$$!3?#yqXM6IZ\"F*$!3>NwY%G?s@&F 37$$!3\\opOvc=B9F*$!3_Z@j:lt\\mF37$$!3yaJ;1+Ot8F*$!3/jG!>?f1t(F37$$!3! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "xx := Pi/4;\neval((1/2-ln(2))*cos( x)+Sum(-2*sin(k*Pi/2)/(k^2-1)*cos(k*x)-\n \+ 2*cos(k*Pi/2)/(k^2-1)*sin(k*x),k=2..10000),x=xx);\n``=evalf(%);\n 'f'(xx)=f(xx);\n``=evalf(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#xxG,$*&\"\"%!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& *&#\"\"\"\"\"#F&*&,&-%#lnG6#F'!\"\"F%F&F&F'F%F&F&-%$SumG6$,&**F'F&-%$s inG6#,$*(F'F-%#PiGF&%\"kGF&F&F&,&*$)F9F'F&F&F&F-F--%$cosG6#,$*(\"\"%F- F8F&F9F&F&F&F-**F'F&-F>F5F&F:F--F4F?F&F-/F9;F'\"&++\"F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G$\"+%32;y$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#,$*&\"\"%!\"\"%#PiG\"\"\"F,,$*&#F,\"\"#F,*&-%#lnG6#,&F, F,*&F0F*F0F/F,F,F0F/F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+Sq g\"y$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "xx := -Pi/4;\neval((1/2-ln(2))*cos(x)+Sum(-2*sin(k*P i/2)/(k^2-1)*cos(k*x)-\n 2*cos(k*Pi/2 )/(k^2-1)*sin(k*x),k=2..10000),x=xx);\n``=evalf(%);\n'f'(xx)=f(xx);\n` `=evalf(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$*&\"\"%!\" \"%#PiG\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&* &,&-%#lnG6#F'!\"\"F%F&F&F'F%F&F&-%$SumG6$,&**F'F&-%$sinG6#,$*(F'F-%#Pi GF&%\"kGF&F&F&,&*$)F9F'F&F&F&F-F--%$cosG6#,$*(\"\"%F-F8F&F9F&F&F&F-**F 'F&-F>F5F&F:F--F4F?F&F&/F9;F'\"&++\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$!+>v*Go)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#,$* &\"\"%!\"\"%#PiG\"\"\"F*,$*&#F,\"\"#F,*&-%#lnG6#,&F,F,*&F0F*F0F/F*F,F0 F/F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$!+bx*Go)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 338 74 "Constructing t he first few terms of the Fourier series using the procedure" }{TEXT -1 1 " " }{TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "f := x -> \+ ln(1+sin(x))*cos(x):\nFourierSeries(f(x),x=-Pi..Pi,numterms=11,mode=te rmbyterm);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8*&,&-%#lnG6#\"\"#!\"\" #\"\"\"F)F,F,-%$cosG6#%\"xGF,F,*&#F)\"\"$F,-%$sinG6#,$*&F)F,F0F,F,F,F, *&#F,\"\"%F,-F.6#,$*&F3F,F0F,F,F,F,*&#F)\"#:F,-F56#,$*&F;F,F0F,F,F,F** &#F,\"#7F,-F.6#,$*&\"\"&F,F0F,F,F,F**&#F)\"#NF,-F56#,$*&\"\"'F,F0F,F,F ,F,*&#F,\"#CF,-F.6#,$*&\"\"(F,F0F,F,F,F,*&#F)\"#jF,-F56#,$*&\"\")F,F0F ,F,F,F**&#F,\"#SF,-F.6#,$*&\"\"*F,F0F,F,F,F**&#F)\"#**F,-F56#,$*&\"#5F ,F0F,F,F,F,*&#F,\"#gF,-F.6#,$*&\"#6F,F0F,F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 111 "Simplification of the integral formulas for the Fourie r series coefficients and associated \"half-range\" series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " be a " }{TEXT 262 11 "pseu do-even" }{TEXT -1 31 " periodic function with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The following " }{TEXT 262 20 "alternative formulas" } {TEXT -1 77 " can be used for the Fourier series coefficients of the p seudo-even function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c=1/L" "6 #/%\"cG*&\"\"\"F&%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x), x=-L/2..L/2)" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$*&%\"LG\"\"\"\"\"#!\"\"F1 *&F.F/F0F1" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[k] = 2/L;" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos(k*Pi*x/L),x = -L/2 .. L/2 );" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"L G!\"\"F+/F*;,$*&F2F+\"\"#F3F3*&F2F+F8F3" }{TEXT -1 7 ", when " }{TEXT 355 1 "k" }{TEXT -1 10 " is even, " }{XPPEDIT 18 0 "a[k]=0" "6#/&%\"aG 6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 356 1 "k" }{TEXT -1 8 " is o dd," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k] = 2/L;" " 6#/&%\"bG6#%\"kG*&\"\"#\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin(k*Pi*x/L),x = -L/2 .. L/2);" "6#-%$IntG6$*&-%\"fG6#%\" xG\"\"\"-%$sinG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$*&F2F+\"\"#F3F3 *&F2F+F8F3" }{TEXT -1 7 ", when " }{TEXT 357 1 "k" }{TEXT -1 9 " is od d, " }{XPPEDIT 18 0 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT -1 6 " \+ when " }{TEXT 358 1 "k" }{TEXT -1 9 " is even." }}{PARA 0 "" 0 "" {TEXT -1 84 "The simplified integral formulas can be established from \+ the fact that the function " }{XPPEDIT 18 0 "f(x)*cos(k*Pi*x/L)" "6#*& -%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF(%#PiGF(F'F(%\"LG!\"\"F(" }{TEXT -1 21 " is pseudo-even when " }{TEXT 359 1 "k" }{TEXT -1 26 " is even \+ and the function " }{XPPEDIT 18 0 "f(x)*sin(k*Pi*x/L)" "6#*&-%\"fG6#% \"xG\"\"\"-%$sinG6#**%\"kGF(%#PiGF(F'F(%\"LG!\"\"F(" }{TEXT -1 21 " is pseudo-even when " }{TEXT 360 1 "k" }{TEXT -1 9 " is odd. " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 4 " is " }{TEXT 262 8 "also odd" }{TEXT -1 7 ", then " } {XPPEDIT 18 0 "c=0" "6#/%\"cG\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a [k] = 0;" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT -1 9 " for all " }{TEXT 377 1 "k" }{TEXT -1 5 " and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[k] = 4/L;" "6#/&%\"bG6#%\"kG*&\"\"%\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin(k*Pi*x/L),x = 0 .. L/2); " "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#**%\"kGF+%#PiGF+F*F+%\"LG! \"\"F+/F*;\"\"!*&F2F+\"\"#F3" }{TEXT -1 7 ", when " }{TEXT 375 1 "k" } {TEXT -1 9 " is odd, " }{XPPEDIT 18 0 "b[k] = 0;" "6#/&%\"bG6#%\"kG\" \"!" }{TEXT -1 6 " when " }{TEXT 376 1 "k" }{TEXT -1 10 " is even. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Similarl y, the following formulas can be used for the Fourier series coefficie nts of a " }{TEXT 262 10 "pseudo-odd" }{TEXT -1 10 " function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " c=0" "6#/%\"cG\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = 2/L;" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"%\"LG!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos(k*Pi*x/L),x = -L/2 .. L/2);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+ %\"LG!\"\"F+/F*;,$*&F2F+\"\"#F3F3*&F2F+F8F3" }{TEXT -1 7 ", when " } {TEXT 361 1 "k" }{TEXT -1 9 " is odd, " }{XPPEDIT 18 0 "a[k]=0" "6#/&% \"aG6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 362 1 "k" }{TEXT -1 9 " \+ is even," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k] = 2/ L;" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x)*sin(k*Pi*x/L),x = -L/2 .. L/2);" "6#-%$IntG6$* &-%\"fG6#%\"xG\"\"\"-%$sinG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$*&F 2F+\"\"#F3F3*&F2F+F8F3" }{TEXT -1 7 ", when " }{TEXT 363 1 "k" }{TEXT -1 10 " is even, " }{XPPEDIT 18 0 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 364 1 "k" }{TEXT -1 8 " is odd." }}{PARA 0 "" 0 "" {TEXT -1 84 "The simplified integral formulas can be establi shed from the fact that the function " }{XPPEDIT 18 0 "f(x)*cos(k*Pi*x /L)" "6#*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF(%#PiGF(F'F(%\"LG!\"\"F( " }{TEXT -1 21 " is pseudo-even when " }{TEXT 381 1 "k" }{TEXT -1 25 " is odd and the function " }{XPPEDIT 18 0 "f(x)*sin(k*Pi*x/L)" "6#*&-% \"fG6#%\"xG\"\"\"-%$sinG6#**%\"kGF(%#PiGF(F'F(%\"LG!\"\"F(" }{TEXT -1 21 " is pseudo-even when " }{TEXT 382 1 "k" }{TEXT -1 10 " is even. " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 4 " is " }{TEXT 262 9 "also even" }{TEXT -1 7 ", then \+ " }{XPPEDIT 18 0 "c=0" "6#/%\"cG\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT -1 9 " for all " }{TEXT 380 1 "k" }{TEXT -1 5 " and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[k] = 4/L;" "6#/&%\"aG6#%\"kG*&\"\"%\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos(k*Pi*x/L),x = 0 .. L/2); " "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG! \"\"F+/F*;\"\"!*&F2F+\"\"#F3" }{TEXT -1 7 ", when " }{TEXT 378 1 "k" } {TEXT -1 9 " is odd, " }{XPPEDIT 18 0 "a[k]=0" "6#/&%\"aG6#%\"kG\"\"! " }{TEXT -1 6 " when " }{TEXT 379 1 "k" }{TEXT -1 10 " is even. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Since th e integrals used to calculate any (possibly) non-zero coefficient are \+ all over the interval " }{XPPEDIT 18 0 "[-L/2,L/2]" "6#7$,$*&%\"LG\"\" \"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 47 ", this motivates the definitio n of associated \"" }{TEXT 262 10 "half-range" }{TEXT -1 104 "\" serie s which are pseudo-even and pseodo-odd periodic extensions of a functi on defined on the interval " }{XPPEDIT 18 0 "[-L/2,L/2]" "6#7$,$*&%\"L G\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 39 " is initially defined on the interval " }{XPPEDIT 18 0 "[-L/2,L/2]" "6#7$,$*&%\"LG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 58 ", Fourier se ries associated with the unique extensions of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 74 " to a pseudo-even and pseudo-odd func tion periodic functions with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"# \"\"\"%\"LGF%" }{TEXT -1 67 " can be constructed provided that the app ropriate integrals exist. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "In detail, let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 38 " be a function defined on an interval " } {XPPEDIT 18 0 "[-L/2, L/2];" "6#7$,$*&%\"LG\"\"\"\"\"#!\"\"F)*&F&F'F(F )" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "We can extend " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 15 " to a periodic " }{TEXT 368 11 "pseudo-even" }{TEXT -1 22 " function with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LG F%" }{TEXT -1 13 " as follows. " }}{PARA 0 "" 0 "" {TEXT -1 13 "First \+ extend " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 15 " to a f unction " }{XPPEDIT 18 0 "f_psev(x);" "6#-%'f_psevG6#%\"xG" }{TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "-L/2<=x" "6#1,$*&%\"LG \"\"\"\"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "``<3*L/2" "6#2%!G*(\"\"$\"\" \"%\"LGF'\"\"#!\"\"" }{TEXT -1 5 " by:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_psev(x) = PIECEWISE([f(x), -L/2 <= x and x < P i/2],[f(L-x), L/2 <= x and x < 3*L/2]);" "6#/-%'f_psevG6#%\"xG-%*PIECE WISEG6$7$-%\"fG6#F'31,$*&%\"LG\"\"\"\"\"#!\"\"F6F'2F'*&%#PiGF4F5F67$-F -6#,&F3F4F'F631*&F3F4F5F6F'2F'*(\"\"$F4F3F4F5F6" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "and then \+ extend " }{XPPEDIT 18 0 "f_psev(x);" "6#-%'f_psevG6#%\"xG" }{TEXT -1 56 " to a pseudo-even function defined for all real numbers " }{TEXT 372 1 "x" }{TEXT -1 17 " by periodicity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "We can also extend " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 15 " to a periodic " }{TEXT 368 10 "pseudo-odd" }{TEXT -1 22 " function with period " }{XPPEDIT 18 0 " 2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 13 " as follows. " }}{PARA 0 " " 0 "" {TEXT -1 13 "First extend " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 15 " to a function " }{XPPEDIT 18 0 "f_psod(x);" "6#-% 'f_psodG6#%\"xG" }{TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "-L/2<=x" "6#1,$*&%\"LG\"\"\"\"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "` `<3*L/2" "6#2%!G*(\"\"$\"\"\"%\"LGF'\"\"#!\"\"" }{TEXT -1 5 " by:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_psod(x) = PIECEWISE ([f(x), -L/2 <= x and x < Pi/2],[0, x = Pi/2],[-f(L-x), L/2 < x and x \+ < 3*L/2]);" "6#/-%'f_psodG6#%\"xG-%*PIECEWISEG6%7$-%\"fG6#F'31,$*&%\"L G\"\"\"\"\"#!\"\"F6F'2F'*&%#PiGF4F5F67$\"\"!/F'*&F9F4F5F67$,$-F-6#,&F3 F4F'F6F632*&F3F4F5F6F'2F'*(\"\"$F4F3F4F5F6" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "and then extend " }{XPPEDIT 18 0 "f_psod(x);" "6#-%'f_psodG6#%\"xG" }{TEXT -1 55 " to a pseudo-odd function defined for all real numbers " }{TEXT 373 1 "x " }{TEXT -1 17 " by periodicity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Examples of pseudo half range cos ine series " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"! \"\"F%" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = x^2-4*x+3;" "6#/-%\" fG6#%\"xG,(*$F'\"\"#\"\"\"*&\"\"%F+F'F+!\"\"\"\"$F+" }{TEXT -1 45 " ca n be extended to the psuedo-even function " }{XPPEDIT 18 0 "f_(x)" "6# -%#f_G6#%\"xG" }{TEXT -1 21 " with period 4 where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = PIECEWISE([x^2-4*x+3, -1 <= x and x < 1],[(2-x)^2-4*(2-x)+3, 1 <= x and x < 3]);" "6#/-%#f_G6#%\"xG-%*PIECEWISEG6$7$,(*$F'\"\"#\"\"\"*&\" \"%F/F'F/!\"\"\"\"$F/31,$F/F2F'2F'F/7$,(*$,&F.F/F'F2F.F/*&F1F/,&F.F/F' F2F/F2F3F/31F/F'2F'F3" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f_(x)" "6#-%# f_G6#%\"xG" }{TEXT -1 28 " is periodic with period 4, " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f_(x) = PIECEWISE([x^2-4*x+3, -1 <= x and x < 1],[x^2-1 , 1 <= x and x < 3]);" "6#/-%#f_G6#%\"xG-%*PIECEWISEG6$7$,(*$F'\"\"#\" \"\"*&\"\"%F/F'F/!\"\"\"\"$F/31,$F/F2F'2F'F/7$,&*$F'F.F/F/F231F/F'2F'F 3" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 5 " and " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 28 " is periodic with period 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "f := x -> x^2-4*x+3:\n'f(x )'=f(x);\nf_psev := unapply(simplify(f(piecewise(x<1,x,2-x))),x):\n'f_ psev(x)'=f_psev(x);\nf_ := x -> f_psev(x-4*floor(x/4+1/4)):\n'f_(x)'=' f_psev(x-4*floor(x/4+1/4))';\nplot(f_(x),x=-2..8,color=COLOR(RGB,.4,0, 1),thickness=2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,( *$)F'\"\"#\"\"\"F,*&\"\"%F,F'F,!\"\"\"\"$F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'f_psevG6#%\"xG-%*PIECEWISEG6$7$,(*$)F'\"\"#\"\"\"F0 *&\"\"%F0F'F0!\"\"\"\"$F02F'F07$,&F0F3F-F01F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%'f_psevG6#,&F'\"\"\"*&\"\"%F,-%&floorG6 #,&*&F.!\"\"F'F,F,#F,F.F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 686 223 223 {PLOTDATA 2 "6'-%'CURVESG6#7iq7$$!\"#\"\"!$\"\"$F*7$$!3smm;HU,\"*= !#<$\"3?pud\")4#yW$F07$$!3YLLLe%G?y\"F0$\"3p4KkfxR>RF07$$!3!***\\(=_+s o\"F0$\"3as_,E9/\\VF07$$!3gmmT&esBf\"F0$\"3qFKXn(pmz%F07$$!3!)**\\7`'G d[\"F0$\"3DFmr0/c@`F07$$!3ALL$3s%3z8F0$\"3A(Gp0!p>peF07$$!3;LLL$)Qtr7F 0$\"3>;9N.gVVkF07$$!31LL$e/$Qk6F0$\"32Xuv5NsSqF07$$!3Om\"zpti46\"F0$\" 3%)3H72j`YtF07$$!3))**\\7GCad5F0$\"3e>XGil0ewF07$$!3_;zptA$3.\"F0$\"3& Q)Q\"p)p&f\"yF07$$!3RL3F>@7/5F0$\"3k#GUiF%GvzF07$$!3O+vV['>Tx*!#=$\"3% *>ZR3?)\\'yF07$$!3#ommT5=q]*Fgo$\"3k%>'*Rf_(Fgo$\"3QNRnaCwwlF07$$!3(pmmT5j-]'Fgo$ \"3Vr0!fWRE-'F07$$!3-,+](o1YZ&Fgo$\"3?(>G)eeb*[&F07$$!3Kmm;a8(QT%Fgo$ \"3-(*z(\\9r.'\\F07$$!3iJL$3-OJN$Fgo$\"3cKa2?'*o`WF07$$!3Olm;zC!eH#Fgo $\"3+W,!>3G5(RF07$$!34****\\P*o%Q7Fgo$\"3bB:4Gcs5NF07$$!3A6L$3F9(3:!#> $\"3%QV3F>w01$F07$$\"3YoLLL3En$*Fjr$\"3ZN'=U7%3MEF07$$\"3)H++Dc#o%*=Fg o$\"3c9k7&>D!yAF07$$\"3;pmmT!RE&GFgo$\"3kTPNy)>.%>F07$$\"3_-++]K]4]Fgo $\"3%[g<\"G*\\rC\"F07$$\"3c,++]PAvrFgo$\"3U-kD'e)[ZkFgo7$$\"3@,++]-w=# )Fgo$\"3y0!)p/5wzQFgo7$$\"3'3+++v'Hi#*Fgo$\"3jcg]3r#)H:Fgo7$$\"3'[$ekG X?*\\*Fgo$\"3\"R@@Jbqm-\"Fgo7$$\"3xn;H2B6O(*Fgo$\"3](fH&o0RZ`Fjr7$$\"3 y%e9m>mX&)*Fgo$\"3Bya()eq#)HHFjr7$$\"3m+v$f3?I(**Fgo$\"3_\"3G,(>E.a!#? 7$$\"3)=/EvRZ\"45F0$\"3O;$eL*p%y$=Fjr7$$\"3ZL$eky#*4-\"F0$\"3m-_****pi UUFjr7$$\"3/+v=UVPo5F0$\"31NdW]tB99Fgo7$$\"3&om;z*ev:6F0$\"3Av#etB7\" \\CFgo7$$\"3>+]7.%Q%G7F0$\"3#\\+BF5414&Fgo7$$\"3_LLL347T8F0$\"3#pD)o2H 0')zFgo7$$\"3nLLLLY.K:F0$\"37-Gt<,8Z8F07$$\"33++D\"o7Tv\"F0$\"3Ok?_)H6 p2#F07$$\"3km;HK5S_=F0$\"3,K_V%e*QJCF07$$\"3?LLL$Q*o]>F0$\"3ez$\\-2*=0 GF07$$\"39n;H#GF&e?F0$\"3a<\"R>dMvB$F07$$\"3m++D\"=lj;#F0$\"3'yMK&)4QJ p$F07$$\"3a++]iB0pAF0$\"3_LowB')f[TF07$$\"3S++vV&R&fM(F07$$\"3emm;/T1&*HF0$\"3]ww1))*3/(zF07$$ \"3)omT5D,`5$F0$\"3OQ4AZ3GztF07$$\"3=nm\"zRQb@$F0$\"3C\"[wMTEKv'F07$$ \"3nLLLe,]6LF0$\"3(**oT')RJ!GiF07$$\"3:++v=>Y2MF0$\"3>J9t-PD@dF07$$\"3 \"QLe*[K56NF0$\"3chdMPqg%>&F07$$\"3Znm;zXu9OF0$\"3'*)z9EUV%*o%F07$$\"3 yLLe9i\"=s$F0$\"3!3TN,O@,>%F07$$\"34+++]y))GQF0$\"30Aw(y'zs8PF07$$\"3> ++DcljLRF0$\"3]fjm\")y&)pKF07$$\"3H++]i_QQSF0$\"3/\">x$yB$z%GF07$$\"3U +](=-N(RTF0$\"3S/%Q))y&egCF07$$\"3b++D\"y%3TUF0$\"3oc.vYFy$4#F07$$\"3+ ++]P![hY%F0$\"3'=5lG%=q_8F07$$\"3iKLL$Qx$oYF0$\"3y3==Kz=KxFgo7$$\"3cmm ;z)Qjx%F0$\"3w\"*3R'QlM(\\Fgo7$$\"3Y+++v.I%)[F0$\"35+9DKl&yW#Fgo7$$\"3 Ym\"Hd&\\@L\\F0$\"3u[#*z1LI!Q\"Fgo7$$\"3ML$ek`H@)\\F0$\"3G!f_]0Ggg$Fjr 7$$\"3SC1k\"=eV*\\F0$\"3%4K:LoY:8\"Fjr7$$\"3M;H#o#oe1]F0$\"3+rFM$*Rq@8 Fjr7$$\"3G3_+sa\")=]F0$\"3OO&>(*f'\\)z$Fjr7$$\"3A+v=0jFjr7$$\"3A$3_vS,b0&F0$\"3V*GWM@K39\"Fgo7$$\"3?mm\"zpe*z]F0$\"3m5'e ?*y5j;Fgo7$$\"3oL$e9\"=\"p=&F0$\"3#*HIz\"['f(3%Fgo7$$\"3;,++D\\'QH&F0$ \"3M&e_9Wk3u'Fgo7$$\"3%HL$e9S8&\\&F0$\"3/uSc@dUN7F07$$\"3s++D1#=bq&F0$ \"37?-&=N#z3>F07$$\"3#om\"H2FO3eF0$\"3uYK7\"ov,F#F07$$\"3\"HLL$3s?6fF0 $\"3)>H&=$*Hr_EF07$$\"3yl;zpe()=gF0$\"3kaqix(fe2$F07$$\"3a***\\7`Wl7'F 0$\"3#*z#*QV;>ANF07$$\"3cL$e*[ACIiF0$\"39D(Q!*[!)R(RF07$$\"3enmmm*RRL' F0$\"3g0S+odFZWF07$$\"3wnmTge)*RkF0$\"3amK.**4``\\F07$$\"3%zmmTvJga'F0 $\"3#***\\A$pxA[&F07$$\"3A,]PM&*>^mF0$\"3&HAq5(*e)GgF07$$\"3]MLe9tOcnF 0$\"3uUL!Hxgvf'F07$$\"3yn;H#e0I&oF0$\"3aGG%=%))R+xF07$$\"3-oT5SMLxpF0$\"37>Dc8W^kyF07$$\"3%\\L3-.B]+(F0$\" 3IkCe\\q))pzF07$$\"3)=]7.i7F.(F0$\"36*p_OR%z/yF07$$\"3%zm;/@-/1(F0$\"3 $G@BX;N7k(F07$$\"3$4+D1R\"y:rF0$\"36_]m')pr=tF07$$\"3![LL3dg6<(F0$\"3n &)z+;DL-qF07$$\"3K,+voTAqsF0$\"3q?m())3w;X'F07$$\"3%ymmmw(GptF0$\"3oY' 4Y&ok?fF07$$\"3/M$eRA5\\Z(F0$\"3!3:qqPygP&F07$$\"3C++D\"oK0e(F0$\"3o6$ *\\1@#Q&[F07$$\"3m+++]oi\"o(F0$\"3#4#4kiS&[P%F07$$\"35,+v=5s#y(F0$\"3X KIp!3Ej\"RF07$$\"3a+]P40O\"*yF0$\"3+dKU,0OYMF07$$\"\")F*F+-%*THICKNESS G6#\"\"#-%&COLORG6&%$RGBG$\"\"%!\"\"$F*F*$\"\"\"F*-%+AXESLABELSG6$Q\"x 6\"Q!Fg^m-%%VIEWG6$;F(Fc]m%(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 "" }} {PARA 0 "" 0 "" {TEXT 384 39 "Calculation of the constant coefficient " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "f := x -> x^2-4*x+3:\nc=1/2*Int('f(x)',x=-1.. 1);\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*& #\"\"\"\"\"#F(-%$IntG6$-%\"fG6#%\"xG/F0;!\"\"F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"cG#\"#5\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 383 62 "Calculation of the (non-zero) coefficien ts of the cosine terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "f := x -> x^2-4*x+3:\n assume(k_,integer):\na[2*k]=Int('f(x)'*cos(k*Pi*x),x=-1..1);\nsubs(k_= k,value(subs(k=k_,%)));\naa := unapply(eval(rhs(%),k=k/2),k):\nmatrix( [[k,`|`,seq(2*k,k=1..8)],['a'[2*k],`|`,seq(aa(2*k),k=1..8)]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#,$*&\"\"#\"\"\"%\"kGF*F*-%$In tG6$*&-%\"fG6#%\"xGF*-%$cosG6#*(F+F*%#PiGF*F3F*F*/F3;!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#,$*&\"\"#\"\"\"%\"kGF*F*,$**\"\"%F* F+!\"#%#PiGF/)!\"\"F+F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7$7,%\"kG%\"|grG\"\"#\"\"%\"\"'\"\")\"#5\"#7\"#9\"#;7,&%\"aG6#,$*&F *\"\"\"F(F8F8F),$*&F+F8%#PiG!\"#!\"\"*&F8F8*$)F;F*F8F=,$*(F+F8\"\"*F=F ;F " 0 "" {MPLTEXT 1 0 247 "f : = x -> x^2-4*x+3:\nassume(k_,integer):\nb[2*k-1]=Int('f(x)'*sin((2*k-1 )*Pi*x/2),x=-1..1);\nfactor(subs(k_=k,value(subs(k=k_,%))));\nbb := un apply(eval(rhs(%),k=(k+1)/2),k):\nmatrix([[k,`|`,seq(2*k-1,k=1..8)],[' b'[2*k-1],`|`,seq(bb(2*k-1),k=1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#,&*&\"\"#\"\"\"%\"kGF*F*F*!\"\"-%$IntG6$*&-%\"fG6#%\"xG F*-%$sinG6#,$**F)F,F'F*%#PiGF*F4F*F*F*/F4;F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#,&*&\"\"#\"\"\"%\"kGF*F*F*!\"\",$**\"#KF*)F,F+ F*%#PiG!\"#F'F2F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,% \"kG%\"|grG\"\"\"\"\"$\"\"&\"\"(\"\"*\"#6\"#8\"#:7,&%\"bG6#,&*&\"\"#F* F(F*F*F*!\"\"F),$*&\"#KF*%#PiG!\"#F9,$*(FF*,$*(FF9,$*(FF*,$*(FF9,$*(FF*, $*(FF9,$*(FF*Q)pprint106\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 388 53 "A procedure for c onstructing truncated Fourier series" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "FS := (x, n) -> 10/3+sum(4*(-1)^k/(k^2*Pi^2)*cos(k*Pi*x)+\n 32*(-1)^k/(Pi^2*( 2*k-1)^2)*sin((2*k-1)*Pi*x/2),k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&#\"#5\"\"$\"\"\" -%$sumG6$,&*,\"\"%F1%\"kG!\"#%#PiGF9)!\"\"F8F1-%$cosG6#*(F8F1F:F19$F1F 1F1*,\"#KF1F;F1F:F9,&*&\"\"#F1F8F1F1F1F " 0 "" {MPLTEXT 1 0 8 "FS(x,5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8#\"#5\" \"$\"\"\"*(\"\"%F'%#PiG!\"#-%$cosG6#*&F*F'%\"xGF'F'!\"\"*(\"#KF'F*F+-% $sinG6#,$*(\"\"#F1F*F'F0F'F'F'F1*&F*F+-F-6#,$*(F9F'F*F'F0F'F'F'F'*&#F3 \"\"*F'*&F*F+-F56#,$**F&F'F9F1F*F'F0F'F'F'F'F'*&#F)FAF'*&F*F+-F-6#,$*( F&F'F*F'F0F'F'F'F'F1*&#F3\"#DF'*&F*F+-F56#,$**\"\"&F'F9F1F*F'F0F'F'F'F 'F1*&#F'F)F'*&F*F+-F-6#,$*(F)F'F*F'F0F'F'F'F'F'*&#F3\"#\\F'*&F*F+-F56# ,$**\"\"(F'F9F1F*F'F0F'F'F'F'F'*&#F)FPF'*&F*F+-F-6#,$*(FVF'F*F'F0F'F'F 'F'F1*&#F3\"#\")F'*&F*F+-F56#,$**FAF'F9F1F*F'F0F'F'F'F'F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 394 39 "Graphs of some t runcated Fourier series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 334 "f := x -> x^2-4*x+3: \nf_psev := unapply(simplify(f(piecewise(x<1,x,2-x))),x):\nf_ := x -> \+ f_psev(x-4*floor(x/4+1/4)):\nFS := (x,n) -> 10/3+sum(4*(-1)^k/(k^2*Pi^ 2)*cos(k*Pi*x)+\n 32*(-1)^k/(Pi^2*(4*k^2-4*k+1))*sin((2*k-1)*Pi*x/2 ),k=1..n):\nplot([f_(x),FS(x,1),FS(x,2),FS(x,3)],x=-2..6,\n color= [black,red,magenta,coral],numpoints=75);" }}{PARA 13 "" 1 "" {GLPLOT2D 617 283 283 {PLOTDATA 2 "6(-%'CURVESG6$7`q7$$!\"#\"\"!$\"\"$ F*7$$!3=VKCV/*o)=!#<$\"31Pn&))*>BlMF07$$!3u-FqAWZ)y\"F0$\"3#H9S!z`%3*Q F07$$!3_%f%f9`zx;F0$\"3'z\")=4gMER%F07$$!3S8N^`DQm:F0$\"3p3M%e(Q\\A\\F 07$$!3#z$y$yJ*\\b9F0$\"3c3P_@P[uaF07$$!3QCVKkjp_8F0$\"3Y];YRl@3gF07$$! 3aS0ag)\\iC\"F0$\"3!4*HdI)RJe'F07$$!35aS0MF;O6F0$\"3C]2#e)Qc,sF07$$!37 ;i@'p&H\"3\"F0$\"3b#)f#H\"[$)=vF07$$!3Qy$y$e'Gk-\"F0$\"3,4MrBl7UyF07$$ !3Eq-F&G>B,\"F0$\"3V\\z;`gBEzF07$$!3]@;i@\"*4#)**!#=$\"3YSuQxEE*)zF07$ $!3OS0a!R05%)*Fbo$\"3v&p#=Fg&[!zF07$$!3Af%f%f;\"**p*Fbo$\"3TfpzDv%3#yF 07$$!3$pH(H(>CxT*Fbo$\"354D[p\\-awF07$$!3lM^8Nn`N\"*Fbo$\"3k)3W%3]z)[( F07$$!3$G0aSDL89)Fbo$\"3i%GdJPY$>pF07$$!3853\"3h&4AqFbo$\"3(z+)*>^O>I' F07$$!3;p-FqAE)*eFbo$\"3mXq>'))*>2dF07$$!3A/aS01C:[Fbo$\"3?6;/j;'z:&F0 7$$!3qy$y$yJvJQFbo$\"3/z9b&fC&zYF07$$!3a+Fq-:HiEFbo$\"3/5Z`hXzNTF07$$! 3Yk['[1/;n\"Fbo$\"3AtEWFUe'p$F07$$!3c'e%f%f!*>>&!#>$\"3_>sT+`P5KF07$$ \"3W\"Qy$yju3]F_s$\"3scJ*))*)e@!GF07$$\"3)yvcn:d+i\"Fbo$\"3k%))3#*)HAy BF07$$\"3K#Qy$yNy&o#Fbo$\"3!p^ZH!4#y*>F07$$\"3[*H(H(Hdxz$Fbo$\"31HG(fo E^i\"F07$$\"3G%*=*=H'*)=[Fbo$\"3zOL'z4fYI\"F07$$\"3h\\'['[EL?fFbo$\"38 ;4ZsKqB)*Fbo7$$\"31r-FqaTkqFbo$\"3A6$yDrMHt'Fbo7$$\"35H(H(HHMg!)Fbo$\" 35([lb4TbD%Fbo7$$\"38u-Fq=(f8*Fbo$\"3lveN?2r-=Fbo7$$\"3ffnvc$zPT*Fbo$ \"3vMHu]n!o?\"Fbo7$$\"3/XKCVoe\"p*Fbo$\"3-&y&R5=QjiF_s7$$\"3x(['['e!\\ I)*Fbo$\"3x-%*QhK6F07$$\"3_3\"3\"3PRs:F0$\"3wnzHt>Us9F07 $$\"3Y8N^8X#Rn\"F0$\"3f_9%pFB?!=F07$$\"3?A;i,X#\\y\"F0$\"3S'>*[wa&f=#F 07$$\"3m******RoO*)=F0$\"3Y%e\"4cqqpDF07$$\"3!yH(Hx\"R&)*>F0$\"3uf@>V) eT*HF07$$\"35#*=*=4v_5#F0$\"3\"[TSD@$=KMF07$$\"3\\-Fq-x,*=*evpMCF0$\"3#Gu0 M?_x#\\F07$$\"3PT0agY%Qa#F0$\"3%=igplX6Z&F07$$\"3]^8N6&RTk#F0$\"3cj$*[ bPZ\"*fF07$$\"3=\\'['G()3fFF0$\"33>gb71d7mF07$$\"3u=*=*y1!>'GF0$\"3anl *f\\v/>(F07$$\"3Yq-Fq,r;HF0$\"3&4.et@)>2vF07$$\"3=A;ih'>:(HF0$\"3g!)QS *4H*HyF07$$\"3C63\"3eMY)HF0$\"3)osr4eV!3zF07$$\"3G++++&\\x*HF0$\"3;^-v k?]')zF07$$\"3z*=*=>W'3,$F0$\"3\\z\"p/_J\\$zF07$$\"3&)y$y$Q$zR-$F0$\"3 1\"z)Ry*)pcyF07$$\"3Sdnvw\"4-0$F0$\"3PY+*3!fE,xF07$$\"3_N^8:!Rk2$F0$\" 3=$F0$\"3/R1[8WfzoF07$$\"3!RKCVW=QH$F0$\"33B #zAh=MK'F07$$\"3m8N^8\"=$3MF0$\"3V`@u+]\"or&F07$$\"3J#*=*=8sE^$F0$\"3G BsYB**z'=&F07$$\"3(*3\"3\"G!>oi$F0$\"3ZS(\\`#z)>j%F07$$\"3tM^8&\\3`s$F 0$\"3='Q/BW@U<%F07$$\"37A;i,lZPQF0$\"3+e'Q5)y]wOF07$$\"3Gz$y$e_#e%RF0$ \"3r_i6pQj>KF07$$\"3)Q^8NB.T0%F0$\"3/hD#eA9ly#F07$$\"3/9N^tG)>;%F0$\"3 uQwDdpIyBF07$$\"3K&f%fM9ilUF0$\"3%*4x83!p!3?F07$$\"3%Ry$yjulxVF0$\"3\\ jk\"[I&*>j\"F07$$\"3/WKCBfV%[%F0$\"3AfbvqW$pH\"F07$$\"3#Qy$yB_\"of%F0$ \"3AL&\\?;v#*o*Fbo7$$\"3UKCV_ba)p%F0$\"3\\NS$4zOy$pFbo7$$\"3'*H(H(4m(4 \"[F0$\"3be!)4EiwPTFbo7$$\"3+r-F!H@['[F0$\"37IlBIqI')GFbo7$$\"3963\"3( fm=\\F0$\"3sX_#o#G$Gp\"Fbo7$$\"3/_8N\"yAb%\\F0$\"3*Q()*[Y?A>6Fbo7$$\"3 1#*=*=fzB(\\F0$\"3VOv-;.P+cF_s7$$\"31i@;(*z!e)\\F0$\"3aMC-i=aeGF_s7$$ \"31KCV-kB**\\F0$\"3W#R\"Q;#yx_\"Fcw7$$\"31-Fq2[m7]F0$\"3+a]&fF,!\\DF_ s7$$\"3'H(H(H@$4E]F0$\"3Y&))>4#3&F0$\"3&phuh#*4 :r\"Fbo7$$\"3a%y$y$Q,$Q^F0$\"3#zySKS+t&HFbo7$$\"3C1aS0#\\;C&F0$\"3%ywQ G\\FpT&Fbo7$$\"39_8N6:\\Z`F0$\"396&R;tLt:)Fbo7$$\"399N^$*RNkaF0$\"3\"o (*=$=ELW6F07$$\"3?>*=*)Rj,d&F0$\"3%Gh/$*48aY\"F07$$\"3?O^8NLPycF0$\"3B SE4_q$p\"=F07$$\"3!R(H(H0?%)y&F0$\"3mae(eGY%)>#F07$$\"3w\"3\"3,Wd*)eF0 $\"3+H4A<9\\qDF07$$\"\"'F*F+-%'COLOURG6&%$RGBGF*F*F*-F$6$7bq7$F($\"3#G )Rw)f[!GHF07$F.$\"3\"Q^npD/k_$F07$F4$\"3ioU]Dy'=2%F07$F9$\"3'zsb!*fb.p %F07$F>$\"3e@QNh'z4H&F07$FC$\"3s'pNY/vp$eF07$FH$\"3ca$G#G'f:F'F07$FM$ \"3R_c6//+EmF07$$!3?(H(H(H17>\"F0$\"3Iu)eTE0[w'F07$FR$\"3u3K[&[)\\qoF0 7$$!37N^8:#H(36F0$\"3[*4*yKvI5pF07$FW$\"3!3wf0g\\8%pF07$$!3O(H(Hx@'Q0 \"F0$\"3!en;=*)4N'pF07$Ffn$\"3iY)3TI1n(pF07$F`o$\"3]Eg*zrw3)pF07$F[p$ \"3Qv_BkZ\\vpF07$F`p$\"3G+aGx:egpF07$Fep$\"3())RA\"zc>OpF07$$!3u$f%f%* \\VQ')Fbo$\"3aghF;&30(oF07$Fjp$\"3Cd_X\">Plx'F07$F_q$\"3_6BS^h!yY'F07$ Fdq$\"3`voG!3g%QgF07$Fiq$\"3zxH\\_g'\\`&F07$$!3Y\"*=*=*o\\BVFbo$\"3qi? 2o6Q%G&F07$F^r$\"3A4v(*)e(oB]F07$$!3iR0aSB-ZKFbo$\"3h3XF%*>$F07$Fcs$\"3OqnG)\\D#yEF07$Fhs$\"3e)*GyL^?j@F07$F ]t$\"3\\vp*)\\pJOF07$Fgz$\"3;CiBrUm\"R#F07$F\\[l$\"37&yfV 'Hh?HF07$Fa[l$\"3w2fdEVt$[$F07$Ff[l$\"3t4FX-\"HE5%F07$F[\\l$\"3#3#RIuH #Qq%F07$F`\\l$\"3A`,&[:'f'H&F07$Fe\\l$\"3B9E6tZ&R$eF07$Fj\\l$\"3wl'*o? 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The serie s can also be obtained as the standard Fourier sries of the function \+ " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 13 " defined by: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)=PIECEWISE([x ^2-4*x+3,-1<=x and x<1],[x^2-1,1<=x and x<3])" "6#/-%\"gG6#%\"xG-%*PIE CEWISEG6$7$,(*$F'\"\"#\"\"\"*&\"\"%F/F'F/!\"\"\"\"$F/31,$F/F2F'2F'F/7$ ,&*$F'F.F/F/F231F/F'2F'F3" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "g(x)" "6 #-%\"gG6#%\"xG" }{TEXT -1 27 " is periodic with period 4." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "g := x -> piecewise(x<1,x^2-4*x+3,x^2-1):\n'g(x)'=g(x);\nFourierSeries(g(x ),x=-1..3,numterms=9,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" gG6#%\"xG-%*PIECEWISEG6$7$,(*$)F'\"\"#\"\"\"F0*&\"\"%F0F'F0!\"\"\"\"$F 02F'F07$,&F0F3F-F0%*otherwiseG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6# \"#5\"\"$\"\"\"*(\"#KF'%#PiG!\"#-%$sinG6#,$*(\"\"#!\"\"F*F'%\"xGF'F'F' F2*(\"\"%F'F*F+-%$cosG6#*&F*F'F3F'F'F2*&#F)\"\"*F'*&F*F+-F-6#,$**F&F'F 1F2F*F'F3F'F'F'F'F'*&F*F+-F76#,$*(F1F'F*F'F3F'F'F'F'*&#F)\"#DF'*&F*F+- F-6#,$**\"\"&F'F1F2F*F'F3F'F'F'F'F2*&#F5F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " defined on the i nterval " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\" \"F)*&F&F'F(F)" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = sin*2*x;" "6 #/-%\"fG6#%\"xG*(%$sinG\"\"\"\"\"#F*F'F*" }{TEXT -1 45 " can be extend ed to the psuedo-even function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\" xG" }{TEXT -1 13 " with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\" \"%#PiGF%" }{TEXT -1 7 " where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = PIECEWISE([si n*2*x, -Pi/2 <= x and x < Pi/2],[sin(Pi-2*x), Pi/2 <= x and x < 3*Pi/2 ]);" "6#/-%#f_G6#%\"xG-%*PIECEWISEG6$7$*(%$sinG\"\"\"\"\"#F.F'F.31,$*& %#PiGF.F/!\"\"F5F'2F'*&F4F.F/F57$-F-6#,&F4F.*&F/F.F'F.F531*&F4F.F/F5F' 2F'*(\"\"$F.F4F.F/F5" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f_(x)" "6#-%# f_G6#%\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2 *Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = PIECEWISE([sin*2*x, -Pi/2 <= x and x < Pi/2],[-sin*2*x, \+ Pi/2 <= x and x < 3*Pi/2]);" "6#/-%#f_G6#%\"xG-%*PIECEWISEG6$7$*(%$sin G\"\"\"\"\"#F.F'F.31,$*&%#PiGF.F/!\"\"F5F'2F'*&F4F.F/F57$,$*(F-F.F/F.F 'F.F531*&F4F.F/F5F'2F'*(\"\"$F.F4F.F/F5" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 25 " is periodic with period \+ " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(x)=sin*2*x" "6#/-%\"fG6#%\"xG*(%$sinG\"\"\"\"\"#F *F'F*" }{TEXT -1 4 " is " }{TEXT 262 3 "odd" }{TEXT -1 27 ", so the Fo urier series of " }{XPPEDIT 18 0 "f_(x);" "6#-%#f_G6#%\"xG" }{TEXT -1 6 " is a " }{TEXT 262 11 "sine series" }{TEXT -1 24 " and the coeffici ent of " }{XPPEDIT 18 0 "sin*k*x" "6#*(%$sinG\"\"\"%\"kGF%%\"xGF%" } {TEXT -1 14 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[k] = 4/Pi;" "6#/&%\"bG6#%\"kG*&\"\"%\"\"\"%#PiG!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin*k*x,x = 0 .. Pi/2);" "6 #-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$sinGF+%\"kGF+F*F+/F*;\"\"!*&%#PiGF+\" \"#!\"\"" }{TEXT -1 7 ", when " }{TEXT 386 1 "k" }{TEXT -1 9 " is odd, " }{XPPEDIT 18 0 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT -1 6 " wh en " }{TEXT 387 1 "k" }{TEXT -1 10 " is even. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "f := x -> s in(2*x):\n'f(x)'=f(x);\nf_psev := unapply(simplify(f(piecewise(x f_psev(x-2*Pi*floor( x/(2*Pi)+1/4)):\n'f_(x)'='f_psev(x-2*Pi*floor(x/(2*Pi)+1/4))';\nplot(f _(x),x=-2..8,color=COLOR(RGB,.4,0,1),thickness=2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinG6#,$*&\"\"#\"\"\"F'F.F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'f_psevG6#%\"xG-%*PIECEWISEG6$7$-%$ sinG6#,$*&\"\"#\"\"\"F'F2F22F',$*&F1!\"\"%#PiGF2F27$,$F,F61F4F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%'f_psevG6#,&F'\"\"\"*( \"\"#F,%#PiGF,-%&floorG6#,&*(F.!\"\"F'F,F/F5F,#F,\"\"%F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 686 223 223 {PLOTDATA 2 "6'-%'CURVESG6#7cu7$$!\"# \"\"!$!3Y'GzI&\\-ov!#=7$$!3smm;HU,\"*=!#<$!3=P)fxT\\a(fF-7$$!3YLLLe%G? y\"F1$!3q&fG1/$4+TF-7$$!3!***\\(=_+so\"F1$!357tGg062BF-7$$!3gmmT&esBf \"F1$!3$fP'3#e7RJ%!#>7$$!3K3-)Q4U!z:F1$!3I>3^k'y!\\;FC7$$!3%)\\PM-;rl: F1$!3Lf7(>OFp,\"FC7$$!3e\"H236\"Q_:F1$!3y(*f9\"f5Ao$FC7$$!3JL3F>10R:F1 $!3x$>R0jw[M'FC7$$!3b;z>O'*Q7:F1$!3#\\!yi@!za;\"F-7$$!3!)**\\7`'Gd[\"F 1$!3+OJ$zVdJp\"F-7$$!3^m\"zpo1CV\"F1$!3&4$4n(H*eKFF-7$$!3ALL$3s%3z8F1$ !3!*)**ew0t4u$F-7$$!3;LLL$)Qtr7F1$!36$)eUS)Q4j&F-7$$!31LL$e/$Qk6F1$!3+ !pYY)=LisF-7$$!3))**\\7GCad5F1$!3CI@!)F-$!3Kj#Q\\ ^3W***F-7$$!3A++vVobtxF-$!3#3^9&)R1()***F-7$$!3$HLL3_>f_(F-$!3gIT$4x#[ y**F-7$$!3(pmmT5j-]'F-$!3\">TW4gAdj*F-7$$!3-,+](o1YZ&F-$!3C`y/]@#*)))) F-7$$!3Kmm;a8(QT%F-$!3>X*HWSN]s(F-7$$!3iJL$3-OJN$F-$!3_\")yv5[x9iF-7$$ !3Olm;zC!eH#F-$!3N1i33r&>V%F-7$$!34****\\P*o%Q7F-$!3#zwQP&yo^CF-7$$!31 ^m\"H#=qYpFC$!37!)*QT,v[Q\"F-7$$!3A6L$3F9(3:FC$!3t)*Q'\\U)F-7$$\"3/-+++N O#4'F-$\"3+^8#36MdQ*F-7$$\"3c,++]PAvrF-$\"3nfW&R'*)*z!**F-7$$\"3.-++vy 5OuF-$\"3g1>KFm4l**F-7$$\"3Q,+++?*pp(F-$\"3!=W$*eDr]***F-7$$\"3u+++Dh( y&zF-$\"3;b.ik7%y***F-7$$\"3@,++]-w=#)F-$\"3A@K39\"*Rt**F-7$$\"3.,+++& G0u)F-$\"3NRp@F$=K%)*F-7$$\"3'3+++v'Hi#*F-$\"3$RWopBXfg*F-7$$\"3ZL$eky #*4-\"F1$\"32(G&)ek!H5*)F-7$$\"3&om;z*ev:6F1$\"3/!eL@jLb*yF-7$$\"3>+]7 .%Q%G7F1$\"3]r7XF$HXK'F-7$$\"3_LLL347T8F1$\"3s%etEHjOV%F-7$$\"3gLL$3xx lV\"F1$\"3I$yw\\X[Al#F-7$$\"3nLLLLY.K:F1$\"3U@3#p#fdWxFC7$$\"3SD\"G8^9 fa\"F1$\"3'*GE+@%4V(\\FC7$$\"3#p\"HK*Q%zf:F1$\"3CRvBQ*4-?#FC7$$\"3c7.K GVtm:F1$\"3Mz&3ke2R7)!#?7$$\"3W3xJnUnt:F1$\"3MJ*fME\\ev&Fc\\l7$$\"3I/^ J1Uh!e\"F1$\"39EQ4E(\\M'>FC7$$\"3;+DJXTb(e\"F1$\"3e3`Lri$4N$FC7$$\"3j$ 3-8!RJ:;F1$\"3m)\\PN:c<*))FC7$$\"3)om\"HdO2V;F1$\"3'zA(>)\\<0W\"F-7$$ \"3eL3FpJf)p\"F1$\"3qZK%G^)>GDF-7$$\"33++D\"o7Tv\"F1$\"39$z2AqRZe$F-7$ $\"3km;HK5S_=F1$\"3=E9Huk-R`F-7$$\"3?LLL$Q*o]>F1$\"3kxO4kRm()oF-7$$\"3 9n;H#GF&e?F1$\"3k=B^&45'z#)F-7$$\"3m++D\"=lj;#F1$\"3G9Tuq?\"zG*F-7$$\" 3P+](=x3x@#F1$\"3!)fEL,*z)='*F-7$$\"3a++]iB0pAF1$\"3W[JBO!4&[)*F-7$$\" 3Q+D\"y:CZH#F1$\"3*\\GTB5BX#**F-7$$\"3p+]7`fR?BF1$\"3/l,*=C!Qu**F-7$$ \"3a+vV[x1YBF1$\"3G+*Q@0\\z***F-7$$\"3S++vV&R6`U#F1$\"3#z9nlm4Y!**F- 7$$\"3e]7`>q4_CF1$\"3w^Q9Tth;)*F-7$$\"3Kn;zWG))yCF1$\"3RL0SH)e/q*F-7$$ \"3y+DJ&\\aC`#F1$\"3sr)o0Dc]Q*F-7$$\"3CML$e9Ege#F1$\"3G_w`!>>?'*)F-7$$ \"3'QL3F9D(FC7 $$\"3!o;zW#)>/;$F1$!3CP-#yVWXw$FC7$$\"3=nm\"zRQb@$F1$!3wY2MF1$!3e--=FfKq]F-7$$\"3\"QL e*[K56NF1$!3R)Q7()oZct'F-7$$\"3Znm;zXu9OF1$!3u:Xmh'*f7\")F-7$$\"3yLLe9 i\"=s$F1$!33E(pgq:)p\"*F-7$$\"34+++]y))GQF1$!3oFE,mG83)*F-7$$\"3++DcE] 2bQF1$!35lm<\"eSn*)*F-7$$\"3#***\\7.AE\")QF1$!3'R)p&3-2#e**F-7$$\"3%)* \\(oz$\\u!RF1$!3%)3!pmhjB***F-7$$\"3>++DcljLRF1$!3=,r*fp;\"****F-7$$\" 3c+D\"GtB)fRF1$!3;Um))QxWy**F-7$$\"3[+]P44,')RF1$!37:S$)GMTI**F-7$$\"3 Q+v$f3)>7SF1$!3Uq^L(\\X^&)*F-7$$\"3H++]i_QQSF1$!3IM\">EO]Gv*F-7$$\"3U+ ](=-N(RTF1$!3W$G,!GCP3\"*F-7$$\"3b++D\"y%3TUF1$!3_()o'GF-7$$\"3iKLL$Qx$oYF1$!3=$GLK'p&4z)FC7$$\"3)em\"HdxO &p%F1$!3g\"=$eLre.MFC7$$\"3:***\\78eBs%F1$!3ahp^s!)p$*>FC7$$\"3IL$3_][ $\\ZF1$!3G_c?5_<&Q(FC7$$\"3cmm;z)Qjx%F1$!3%3^wax8bF\"F-7$$\"3'RL$3F'>. $[F1$!3F>HD#>0oL#F-7$$\"3Y+++v.I%)[F1$!3GRN0!*f)3P$F-7$$\"3ML$ek`H@)\\ F1$!3t'*)H%y'3p8&F-7$$\"3?mm\"zpe*z]F1$!3`zGPje!pq'F-7$$\"3oL$e9\"=\"p =&F1$!3_clpA,gG\")F-7$$\"3;,++D\\'QH&F1$!3%*f@o(=z(z\"*F-7$$\"3*Q$eR(> #=W`F1$!3%pP`F35=`*F-7$$\"3gm;zp%*\\%R&F1$!3ap#)f13R(y*F-7$$\"3U$e*)f5 e'>aF1$!3%HV@5Fl\"y)*F-7$$\"3A+v=Un\"[W&F1$!3gwSE(>NR%**F-7$$\"3/Ac5'[3$F-7$$\"3%ym \"H2)3I\\'F1$\"3+jtpu5QuSF-7$$\"3%zmmTvJga'F1$\"3D#yR0iC\"=]F-7$$\"3A, ]PM&*>^mF1$\"3Q0$G&*R(\\8nF-7$$\"3]MLe9tOcnF1$\"3z<-@:9&H6)F-7$$\"3yn; H#e0I&oF1$\"3eBnoJL$[3*F-7$$\"31,++]Qk\\pF1$\"3hG/p+!*R=(*F-7$$\"3-oT5 SMLxpF1$\"3t?))\\P.$R$)*F-7$$\"3%\\L3-.B]+(F1$\"3IOMoc,J>**F-7$$\"3)=] 7.i7F.(F1$\"3G'>_xnwU(**F-7$$\"3%zm;/@-/1(F1$\"3+h.^p8m)***F-7$$\"3)R$ 3_+=4)3(F1$\"3)y'*)ym%*Q#***F-7$$\"3$4+D1R\"y:rF1$\"3#o/!o*>![b**F-7$$ \"3'y;H2)4ZVrF1$\"3=FedMn/)))*F-7$$\"3![LL3dg6<(F1$\"3AjE>FeH!z*F-7$$ \"3K,+voTAqsF1$\"3O(oVe,#y(>*F-7$$\"3%ymmmw(GptF1$\"3/!\\LTX%RX#)F-7$$ \"3/M$eRA5\\Z(F1$\"3G6B!)o5uvoF-7$$\"3C++D\"oK0e(F1$\"37a`#f1,/?&F-7$$ \"3m+++]oi\"o(F1$\"3#)y<9XMBzLF-7$$\"35,+v=5s#y(F1$\"3\"GU,@J#R?9F-7$$ \"3v]iS\"*3))4yF1$\"3lQRp+pr3))FC7$$\"3Q+D1k2/PyF1$\"3]*=GYwDvQ$FC7$$ \"3dilAK2$Q%yF1$\"3,'H6:*3/I?FC7$$\"3kD1R+2i]yF1$\"3#>*)\\#[l\"=s'Fc\\ l7$$\"3s)oa&o1TdyF1$\"3?B$[6S:!eoFc\\l7$$\"3#4v=nj+U'yF1$\"3k+H8l#eO/# FC7$$\"3Gvo/t0yxyF1$\"3-?V\\b2*zv%FC7$$\"3a+]P40O\"*yF1$\"3w?)3%)o8)ou FC7$$\"3s+voa-oXzF1$\"3Wt2N%))3P#=F-7$$\"\")F*$\"3+e1lmJ.zGF--%*THICKN ESSG6#\"\"#-%&COLORG6&%$RGBG$\"\"%!\"\"$F*F*$\"\"\"F*-%+AXESLABELSG6$Q \"x6\"Q!F[an-%%VIEWG6$;F(Fe_n%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 374 60 "Calculation of the (non-zero) \+ coefficients of the sine terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "f := x -> s in(2*x):\n'f(x)'=f(x);\nassume(k_,integer):\nb[2*k-1]=4/Pi*Int('f(x)'* sin((2*k-1)*x),x=0..Pi/2);\nfactor(subs(k_=k,value(subs(k=k_,%))));\nb b := unapply(eval(rhs(%),k=(k+1)/2),k):\nmatrix([[k,`|`,seq(2*k-1,k=1. .8)],['b'[2*k-1],`|`,seq(bb(2*k-1),k=1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinG6#,$*&\"\"#\"\"\"F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#,&*&\"\"#\"\"\"%\"kGF*F*F*!\"\",$,$ -%$IntG6$*&-%\"fG6#%\"xGF*-%$sinG6#*&F'F*F6F*F*/F6;\"\"!,$*&F)F,%#PiGF *F**&\"\"%F*F@F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#,&*&\" \"#\"\"\"%\"kGF*F*F*!\"\",$*,\"\")F*)F,F+F*%#PiGF,,&*&F)F*F+F*F*F*F*F, ,&*&F)F*F+F*F*\"\"$F,F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7$7,%\"kG%\"|grG\"\"\"\"\"$\"\"&\"\"(\"\"*\"#6\"#8\"#:7,&%\"bG6#,&* &\"\"#F*F(F*F*F*!\"\"F),$*(\"\")F*F+F9%#PiGF9F*,$*(F " 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 4 " is " }{XPPEDIT 18 0 "Sum(8*(-1)^k/(Pi*(2*k+1)*(2*k-3))*sin((2*k-1)*x),k=1..infinity) " "6#-%$SumG6$**\"\")\"\"\"),$F(!\"\"%\"kGF(*(%#PiGF(,&*&\"\"#F(F,F(F( F(F(F(,&*&F1F(F,F(F(\"\"$F+F(F+-%$sinG6#*&,&*&F1F(F,F(F(F(F+F(%\"xGF(F (/F,;F(%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 389 53 "A procedure for constructing truncated Fourier series" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "FS := (x,n) -> sum(8*(-1)^k/(Pi*(2*k+1)*(2*k-3))*sin((2*k-1)*x), k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$% )operatorG%&arrowGF)-%$sumG6$,$*.\"\")\"\"\")!\"\"%\"kGF3%#PiGF5,&*&\" \"#F3F6F3F3F3F3F5,&*&F:F3F6F3F3\"\"$F5F5-%$sinG6#*&,&*&F:F3F6F3F3F3F5F 39$F3F3F3/F6;F39%F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 392 44 "The first few terms of the Fourier series of" } {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "FS(x,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&#\"\")\"\"$\"\" \"*&%#PiG!\"\"-%$sinG6#%\"xGF(F(F(*&#F&\"\"&F(*&F*F+-F-6#,$*&F'F(F/F(F (F(F(F(*&#F&\"#@F(*&F*F+-F-6#,$*&F2F(F/F(F(F(F(F+*&#F&\"#XF(*&F*F+-F-6 #,$*&\"\"(F(F/F(F(F(F(F(*&#F&\"#xF(*&F*F+-F-6#,$*&\"\"*F(F/F(F(F(F(F+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 395 39 "Gra phs of some truncated Fourier series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 310 "f := x - > sin(2*x):\nf_psev := unapply(simplify(f(piecewise(x f_psev(x-2*Pi*floor(x/(2*Pi)+1/4)):\nFS := (x,n) -> sum (8*(-1)^k/(Pi*(2*k+1)*(2*k-3))*sin((2*k-1)*x),k=1..n):\nplot([f_(x),FS (x,1),FS(x,2),FS(x,3),FS(x,4)],x=-2..6,\n color=[black,red,blue,ma genta,brown],numpoints=80);" }}{PARA 13 "" 1 "" {GLPLOT2D 617 283 283 {PLOTDATA 2 "6)-%'CURVESG6$7[t7$$!\"#\"\"!$!3Y'GzI&\\-ov!#=7$$!3+n$\\f F\\S*=!#<$!3VcGp$>6S-'F-7$$!3Xa$ou6i=!=F1$!3qC6Rd5deWF-7$$!3!HWNo1D+v \"F1$!3PM8g_7$$!3Y;`-KN%3e\"F1 $!3/(>W$=eI4?FM7$$!3KIWNo.'yc\"F1$!3]f7iPJ)=(e!#?7$$!3=WNo/s([b\"F1$!3 #oN,jb'G$=$FM7$$!3EeE,TS*=a\"F1$!3q*))yzeQsx&FM7$$!3>')3n8x#f^\"F1$!3E (>pZ6r^4\"F-7$$!379\"HjQh**[\"F1$!3Kt;Md`m4;F-7$$!3_f2')[\\m$R\"F1$!39 $)=kk**)*oMF-7$$!3W?'pbcbRH\"F1$!3G1ijyEAe_F-7$$!37xTtyf$3>\"F1$!3'\\* GP1'R'))oF-7$$!3')=:s(4Z!)3\"F1$!3AicbZkKB#)F-7$$!3Ci]S#*4;B)*F-$!3:M( *\\\\!RWB*F-7$$!3ur5[y#=yH')F-$!3#4N(>8\"o)z)*F-7$$!3i@xTt=on$)F-$!3u1'H 5#)os%**F-7$$!3ETt)*=>e0\")F-$!3+!**y08Ut)**F-7$$!3,ipbk>[VyF-$!3+FlR^ z(*****F-7$$!35Y;`-eI!e(F-$!3c+jLZS-&)**F-7$$!33Hj]S'HrJ(F-$!32T'RGM8C %**F-7$$!3175[yM&R0(F-$!3?'HMA'QEs)*F-7$$!39'pbkJx2z'F-$!3s4Y_2*pZx*F- 7$$!3=()*=:KRNG'F-$!3Lcr+KRy5&*F-7$$!3CyAeE8IwdF-$!3!4D+Va+!\\\"*F-7$$ !3%p]S#R*f]&[F-$!3I#zh5ntXD)F-7$$!3?bkJD[hfPF-$!3.4b35^WIoF-7$$!3%>j]S 7H;$GF-$!3GV'Ri@^`O&F-7$$!3w)z.V9h@v\"F-$!3i:#*4%*z.LMF-7$$!33h2')3^Su 7F-$!3a^?&)fFI@DF-7$$!3)RBxTt!\\mzFM$!3otpbcac'e\"F-7$$!39A[yA)\\Zs#FM $!3)G2y>\"G!oW&FM7$$\"3p*e2')3\"*p^#FM$\"3\\Z-=Yj&=.&FM7$$\"3q4+++!o$3 vFM$\"3Yjf26h.'\\\"F-7$$\"3)HCR6\\u*\\7F-$\"3')H*33;!*RZ#F-7$$\"3w!G#e E,d\"H#F-$\"35Y\"=*[vOCWF-7$$\"33zg)32!3[KF-$\"33BFNin!)[gF-7$$\"3w%yA eE0)zUF-$\"3'zfLWG7?b(F-7$$\"3/#[yAyx9N&F-$\"3kCsu(pBMx)F-7$$\"3EYyAe= P%G'F-$\"3=EF-C#)H6&*F-7$$\"37$ouzVZ\")y'F-$\"3y?`wD&eOx*F-7$$\"3'*>:s H(F-$\"3qmf1FX)o$**F-7$$\"3cM()*=vZ@b(F-$\"3)HI4Tz%y\")**F-7$$\"39 \\f2'[sB\"yF-$\"37U0wPPl****F-7$$\"3sjJD?sfs!)F-$\"3ItjvnHW!***F-7$$\" 3Ky.Va>#GL)F-$\"3[:>YFu%))F-$\"3'[z\\f;1a!)*F-7$$ \"3AR#R6\\Q6N*F-$\"3OQp**eP/b&*F-7$$\"3<#eE,@TO.\"F1$\"3Yd*G(pzh#z)F-7 $$\"3?\\f2Y!RI9\"F1$\"3c4Xk4[3\\vF-7$$\"3(=xTt'yLT7F1$\"3SRvxo'3E7'F-7 $$\"3-Hj]SGHY8F1$\"3X*>A8^62M%F-7$$\"3Suz.V_%QR\"F1$\"3IRQ-sxglMF-7$$ \"3!)>'pbk(RT9F1$\"3oql%Q\\\"=fDF-7$$\"3%=:s<)\\Q$\\\"F1$\"3*))>de7\\? a\"F-7$$\"3)QouzJs`a\"F1$\"3EF.v.4h#3&FM7$$\"3Gn$\\fK,wb\"F1$\"3#3$>FX7$$\"3KM()*=Mf?e\"F1$\"3eo&) >ADT_AFM7$$\"3sF1$\"3 %\\]hcRs$zoF-7$$\"3KBeE\"yF40#F1$\"3?b@?NKU$>)F-7$$\"3O[yA)\\?S:#F1$\" 3#GN\\M3\"f$>*F-7$$\"3IVa$o-S^?#F1$\"3U!>imM6ra*F-7$$\"3CQIWb&fiD#F1$ \"39tf#[AC4!)*F-7$$\"3U(z.VKY(zAF1$\"3aokh)3TL))*F-7$$\"3/dX;$4LKI#F1$ \"3\"Go.!RU&R%**F-7$$\"3n;`-i)>nK#F1$\"3cg)eo&*HE)**F-7$$\"3%e2')3j1-N #F1$\"3k1))Q>HG****F-7$$\"3[75[=^7xBF1$\"31knSb%R7***F-7$$\"39\\f21O// CF1$\"3ea]2yTCa**F-7$$\"3y&)3n$4i4V#F1$\"3sQ[M*G/%)))*F-7$$\"3UAeE\"e! )yX#F1$\"3a,;xu0\"Rz*F-7$$\"3r&)3n8I.1DF1$\"3y4x`%y=Vb*F-7$$\"3+\\f2Ya =aDF1$\"3o2E\"yr#=E#*F-7$$\"3u75[=l'ol#F1$\"3G&)fG#efdC)F-7$$\"3qyAe1a 9bFF1$\"3s,_?xI5#)pF-7$$\"3!y)*=:[rU'GF1$\"3W8eNYHRm_F-7$$\"3**4[yi!>: \"HF1$\"3Dtu0Y)*zSWF-7$$\"3;K10WmweHF1$\"3ab#G2&GevNF-7$$\"3v/CRrIR7IF 1$\"3a&[0m\">LbDF-7$$\"3MxTt)\\>g1$F1$\"3EJTJ>Zr0:F-7$$\"3H?'pb;%*[6$F 1$\"3$zM![30;P`FM7$$\"3Cj]SK)oP;$F1$!3u_]k!z\"yLWFM7$$\"3DD?'p.Is@$F1$ !3%RM&*>j%)p]\"F-7$$\"3G()*=:C\"pqKF1$!3_!e[(Q!zLb#F-7$$\"39Nou>#>oJ$F 1$!3l6joBVBLMF-7$$\"3+$ouz>ZHO$F1$!3WIv>;&*)QG%F-7$$\"3?/Vajf,oMF1$!3% *3a!o)4]ugF-7$$\"3/L10/s]pNF1$!3mdM%4GZ6b(F-7$$\"3G5[yU@$4n$F1$!3&=R`x 5*3<()F-7$$\"3#\\Nou&R)>x$F1$!3Po8&o%QHB&*F-7$$\"3-h)3n`B0#QF1$!33cexh '\\Tx*F-7$$\"39n$\\f6j!pQF1$!3`a(4LmiH$**F-7$$\"3%G#eE,%*H&*QF1$!3ehzA 2)>*z**F-7$$\"3eyAe'oN:#RF1$!3qS!\\p$[S****F-7$$\"3IM()*=(>xZRF1$!3s)H s_6k8***F-7$$\"3-!>:sD3S(RF1$!3;ic?w(>e&**F-7$$\"3>_@x@%=S-%F1$!3cn^M/ II7)*F-7$$\"3O9\"HjeGS2%F1$!3aT*><602d*F-7$$\"3!GR6HF&HzTF1$!3')*\\+&) =OOv)F-7$$\"3;dX;tpeuUF1$!3M[@eN`IzwF-7$$\"3O$\\f29-*zVF1$!3kBou-UQqhF -7$$\"3ErO\\>dx![%F1$!3gS7/S#p$oWF-7$$\"3'pz.Vu*3JXF1$!3)p9P4V\\qa$F-7 $$\"3cBR6pPS\"e%F1$!3?V(4>[U)*e#F-7$$\"3Auz.jp&Rj%F1$!3UI1*y<:Ac\"F-7$ $\"3wD?'p:5lo%F1$!3=?4/TTXt^FM7$$\"3ErO\\45h)p%F1$!3%*odDdaCbFFM7$$\"3 y;`-i=r5ZF1$!3+,00H/BaLFX7$$\"3Gipb9F\"Gs%F1$!3'[/*oEef%3#FM7$$\"3y2') 3nN\"\\t%F1$!3UK:9hPR.XFM7$$\"3y)*=:s_6fZF1$!32&)>Ho;mJ$*FM7$$\"3y*=:s (pJ$y%F1$!3#e3#eyq!QT\"F-7$$\"3cuz../Y#)[F1$!3/%3s\"ov@OLF-7$$\"39?'pb _E>*\\F1$!3i?d,@'poF-7$$\"3#p]S#R.S #>&F1$!3mtE^rf/#>)F-7$$\"334n$\\2#[&H&F1$!33rDx.%fD>*F-7$$\"3F<`-i\"eG M&F1$!3=(fgt*)oP_*F-7$$\"3cCR6\\UB!R&F1$!3et9****F-7$$\"3Un$\\ fZ&H;bF1$!3#z@!3:'\\J***F-7$$\"3j@xT`RNTbF1$!3W?egtE1i**F-7$$\"3%e2')3 V7kc&F1$!3<0Ozk\"ff!**F-7$$\"31IWN34Z\"f&F1$!3Rv([[(*z\\#)*F-7$$\"3UM( )*=Z!3XcF1$!3)[7h3OA#p&*F-7$$\"3uQIWN+p)p&F1$!3mQG:vGc.#*F-7$$\"3WY(z. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "g := x -> piecewise(xG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Examples of pseudo half range si ne series " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 25 " defined on the interval " } {XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 4 " by " } {XPPEDIT 18 0 "f(x) = x^2-4*x+3;" "6#/-%\"fG6#%\"xG,(*$F'\"\"#\"\"\"*& \"\"%F+F'F+!\"\"\"\"$F+" }{TEXT -1 44 " can be extended to the psuedo- odd function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 21 " with period 4 where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = PIECEWISE([x^2-4*x+3, -1 < = x and x < 1],[(2-x)^2-4*(2-x)+3, 1 <= x and x < 3]);" "6#/-%#f_G6#% \"xG-%*PIECEWISEG6$7$,(*$F'\"\"#\"\"\"*&\"\"%F/F'F/!\"\"\"\"$F/31,$F/F 2F'2F'F/7$,(*$,&F.F/F'F2F.F/*&F1F/,&F.F/F'F2F/F2F3F/31F/F'2F'F3" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 28 " is periodic with period 4, " }}{PARA 0 "" 0 "" {TEXT -1 9 "that i s, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = PIECEW ISE([x^2-4*x+3, -1 <= x and x < 1],[x^2-1, 1 <= x and x < 3]);" "6#/-% #f_G6#%\"xG-%*PIECEWISEG6$7$,(*$F'\"\"#\"\"\"*&\"\"%F/F'F/!\"\"\"\"$F/ 31,$F/F2F'2F'F/7$,&*$F'F.F/F/F231F/F'2F'F3" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " } {XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 28 " is periodic wit h period 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "f := x -> x^2-4*x+3:\n'f(x)'=f(x);\nf_psod := una pply(simplify(f(piecewise(x<1,x,2-x))*signum(1-x)),x):\n'f_psod(x)'=f_ psod(x);\nf_ := x -> f_psod(x-4*floor(x/4+1/4)):\n'f_(x)'='f_psod(x-4* floor(x/4+1/4))';\nplot(f_(x),x=-2..8,color=COLOR(RGB,.4,0,1),thicknes s=2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*$)F'\"\"# \"\"\"F,*&\"\"%F,F'F,!\"\"\"\"$F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%'f_psodG6#%\"xG-%*PIECEWISEG6$7$,(*$)F'\"\"#\"\"\"F0*&\"\"%F0F'F0!\" \"\"\"$F01F'F07$,&F0F0F-F32F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% #f_G6#%\"xG-%'f_psodG6#,&F'\"\"\"*&\"\"%F,-%&floorG6#,&*&F.!\"\"F'F,F, #F,F.F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 686 223 223 {PLOTDATA 2 "6'- %'CURVESG6#7^p7$$!\"#\"\"!$!\"$F*7$$!3YLLLe%G?y\"!#<$!3p4KkfxR>RF07$$! 3gmmT&esBf\"F0$!3qFKXn(pmz%F07$$!3ALL$3s%3z8F0$!3A(Gp0!p>peF07$$!3;LLL $)Qtr7F0$!3>;9N.gVVkF07$$!31LL$e/$Qk6F0$!32Xuv5NsSqF07$$!3Om\"zpti46\" F0$!3%)3H72j`YtF07$$!3))**\\7GCad5F0$!3e>XGil0ewF07$$!3_;zptA$3.\"F0$! 3&Q)Q\"p)p&f\"yF07$$!3RL3F>@7/5F0$!3k#GUiF%GvzF07$$!33E1k1eWu**!#=$\"3 YeB:9Sn%)zF07$$!39Tx*Fhn$\"3%*>ZR3?)\\'yF07$$!3f$3-j()o0k*Fhn$\" 3ax!3JBLcy(F07$$!3#ommT5=q]*Fhn$\"3k%>'*Rf_(Fhn$\"3QNRnaCwwlF07$$!3-,+](o1YZ&Fhn$ \"3?(>G)eeb*[&F07$$!3iJL$3-OJN$Fhn$\"3cKa2?'*o`WF07$$!34****\\P*o%Q7Fh n$\"3bB:4Gcs5NF07$$\"3YoLLL3En$*!#>$\"3ZN'=U7%3MEF07$$\"3;pmmT!RE&GFhn $\"3kTPNy)>.%>F07$$\"3_-++]K]4]Fhn$\"3%[g<\"G*\\rC\"F07$$\"3c,++]PAvrF hn$\"3U-kD'e)[ZkFhn7$$\"3'3+++v'Hi#*Fhn$\"3jcg]3r#)H:Fhn7$$\"3&om;z*ev :6F0$!3Av#etB7\"\\CFhn7$$\"3_LLL347T8F0$!3#pD)o2H0')zFhn7$$\"3nLLLLY.K :F0$!37-Gt<,8Z8F07$$\"33++D\"o7Tv\"F0$!3Ok?_)H6p2#F07$$\"3?LLL$Q*o]>F0 $!3ez$\\-2*=0GF07$$\"3m++D\"=lj;#F0$!3'yMK&)4QJp$F07$$\"3S++vV&R&fM(F07$$\"3emm;/T1&*HF0$!3]ww1) )*3/(zF07$$\"37aj%e#R&>+$F0$\"3y(3[FE!G))zF07$$\"3nTg_ZP%)3IF0$\"3maSr Ed,ZzF07$$\"3@Hd?pNt:IF0$\"3Xwb'*Hh%e!zF07$$\"3w;a)3RBE-$F0$\"3=`E]s9x kyF07$$\"3&=zWU..k.$F0$\"3wwMVvp!Hy(F07$$\"3%p;/wn#=]IF0$\"3$=_1bBA9q( F07$$\"39uxIF0$\"3b$Gzq-#fRvF07$$\"3)omT5D,`5$F0$\"3OQ4AZ3GztF07$ $\"3!o;zW#)>/;$F0$\"3UD4@tb@jqF07$$\"3=nm\"zRQb@$F0$\"3C\"[wMTEKv'F07$ $\"3nLLLe,]6LF0$\"3(**oT')RJ!GiF07$$\"3:++v=>Y2MF0$\"3>J9t-PD@dF07$$\" 3Znm;zXu9OF0$\"3'*)z9EUV%*o%F07$$\"34+++]y))GQF0$\"30Aw(y'zs8PF07$$\"3 H++]i_QQSF0$\"3/\">x$yB$z%GF07$$\"3b++D\"y%3TUF0$\"3oc.vYFy$4#F07$$\"3 +++]P![hY%F0$\"3'=5lG%=q_8F07$$\"3iKLL$Qx$oYF0$\"3y3==Kz=KxFhn7$$\"3Y+ ++v.I%)[F0$\"35+9DKl&yW#Fhn7$$\"3?mm\"zpe*z]F0$!3m5'e?*y5j;Fhn7$$\"3;, ++D\\'QH&F0$!3M&e_9Wk3u'Fhn7$$\"3%HL$e9S8&\\&F0$!3/uSc@dUN7F07$$\"3s++ D1#=bq&F0$!37?-&=N#z3>F07$$\"3\"HLL$3s?6fF0$!3)>H&=$*Hr_EF07$$\"3a*** \\7`Wl7'F0$!3#*z#*QV;>ANF07$$\"3enmmm*RRL'F0$!3g0S+odFZWF07$$\"3%zmmTv Jga'F0$!3#***\\A$pxA[&F07$$\"3]MLe9tOcnF0$!3uUL!Hxgvf'F07$$\"3yn;H#e0I &oF0$!3aGG%=%))R+xF07$$\"3a%3_]k)[jpF0 $!3eSEN\\\\E#y(F07$$\"3-oT5SMLxpF0$!37>Dc8W^kyF07$$\"3u4-jPeD%)pF0$!3W &)*HW!Hy0zF07$$\"3[^i:N#y6*pF0$!3SnCZMs9ZzF07$$\"3?$H#oK15)*pF0$!3!o'* *o.ug))zF07$$\"3%\\L3-.B]+(F0$\"3IkCe\\q))pzF07$$\"3)=]7.i7F.(F0$\"36* p_OR%z/yF07$$\"3%zm;/@-/1(F0$\"3$G@BX;N7k(F07$$\"3$4+D1R\"y:rF0$\"36_] m')pr=tF07$$\"3![LL3dg6<(F0$\"3n&)z+;DL-qF07$$\"3K,+voTAqsF0$\"3q?m()) 3w;X'F07$$\"3%ymmmw(GptF0$\"3oY'4Y&ok?fF07$$\"3C++D\"oK0e(F0$\"3o6$*\\ 1@#Q&[F07$$\"35,+v=5s#y(F0$\"3XKIp!3Ej\"RF07$$\"\")F*$\"\"$F*-%*THICKN ESSG6#\"\"#-%&COLORG6&%$RGBG$\"\"%!\"\"$F*F*$\"\"\"F*-%+AXESLABELSG6$Q \"x6\"Q!Fafl-%%VIEWG6$;F(F[el%(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 "" }}{PARA 0 "" 0 "" {TEXT 370 62 "Calculation of the (non-zero) \+ coefficients of the cosine terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "f := x -> x ^2-4*x+3:\nassume(k_,integer):\na[2*k-1]=Int('f(x)'*cos((2*k-1)*Pi*x/2 ),x=-1..1);\nfactor(subs(k_=k,value(subs(k=k_,%))));\naa := unapply(ev al(rhs(%),k=(k+1)/2),k):\nmatrix([[k,`|`,seq(2*k-1,k=1..5)],['a'[2*k-1 ],`|`,seq(aa(2*k-1),k=1..5)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6#,&*&\"\"#\"\"\"%\"kGF*F*F*!\"\"-%$IntG6$*&-%\"fG6#%\"xGF*-%$cosG 6#,$**F)F,F'F*%#PiGF*F4F*F*F*/F4;F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#,&*&\"\"#\"\"\"%\"kGF*F*F*!\"\",$*,\"#;F*)F,,&F*F*F+F*F*,* F)F,*(\"\"%F*)%#PiGF)F*)F+F)F*F**(F4F*F5F*F+F*F,*$F5F*F*F*F6!\"$F'F:F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7)%\"kG%\"|grG\"\"\" \"\"$\"\"&\"\"(\"\"*7)&%\"aG6#,&*&\"\"#F*F(F*F*F*!\"\"F),$*(\"#;F*,&F5 F6*$)%#PiGF5F*F*F*F=!\"$F*,$**F9F*\"#FF6,&F5F6*&F.F*FF6,$** F9F*\"$D\"F6,&F5F6*&\"#DF*FF*,$**F9F*\"$V$F6,&F5F6*&\"#\\F* FF6,$**F9F*\"$H(F6,&F5F6*&\"#\")F*FF*Q)pprint146 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 371 60 "C alculation of the (non-zero) coefficients of the sine terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "f := x -> x^2-4*x+3:\nassume(k_,integer):\nb[2*k]=In t('f(x)'*sin(k*Pi*x),x=-1..1);\nsubs(k_=k,value(subs(k=k_,%)));\nbb := unapply(eval(rhs(%),k=k/2),k):\nmatrix([[k,`|`,seq(2*k,k=1..8)],['b'[ 2*k-1],`|`,seq(bb(2*k),k=1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#,$*&\"\"#\"\"\"%\"kGF*F*-%$IntG6$*&-%\"fG6#%\"xGF*-%$sinG6#*( F+F*%#PiGF*F3F*F*/F3;!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#,$*&\"\"#\"\"\"%\"kGF*F*,$**\"\")F*F+!\"\"%#PiGF/)F/F+F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,%\"kG%\"|grG\"\"#\"\"% \"\"'\"\")\"#5\"#7\"#9\"#;7,&%\"bG6#,&*&F*\"\"\"F(F8F8F8!\"\"F),$*&F-F 8%#PiGF9F9,$*&F+F8F " 0 "" {MPLTEXT 1 0 126 "FS := (x,n) -> sum(16*(-1)^(1+k)*(Pi^2*(2*k-1)^2-2)/(Pi^3*(2* k-1)^3)*cos((2*k-1)*Pi*x/2)+\n8*(-1)^k/(k*Pi)*sin(k*Pi*x),k=1..n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%& arrowGF)-%$sumG6$,&*.\"#;\"\"\")!\"\",&F3F3%\"kGF3F3,&*&)%#PiG\"\"#F3) ,&*&FF@-%$cosG6#,$*&#F3FF3F;F3 9$F3F3F3F3F3*,\"\")F3)F5F7F3F7F5F;F5-%$sinG6#*(F7F3F;F3FHF3F3F3/F7;F39 %F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 391 44 "The first few terms of the Fourier series of" }{TEXT -1 1 " " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "FS(x,5); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6**\"#;\"\"\",&\"\"#!\"\"*$)%#PiG F(F&F&F&F,!\"$-%$cosG6#,$*(F(F)F,F&%\"xGF&F&F&F&*(\"\")F&F,F)-%$sinG6# *&F,F&F3F&F&F)*&#F%\"#FF&*(,&F(F)*&\"\"*F&F+F&F&F&F,F--F/6#,$**\"\"$F& F(F)F,F&F3F&F&F&F&F)*(\"\"%F&F,F)-F76#,$*(F(F&F,F&F3F&F&F&F&*&#F%\"$D \"F&*(,&F(F)*&\"#DF&F+F&F&F&F,F--F/6#,$**\"\"&F&F(F)F,F&F3F&F&F&F&F&*& #F5FEF&*&F,F)-F76#,$*(FEF&F,F&F3F&F&F&F&F)*&#F%\"$V$F&*(,&F(F)*&\"#\\F &F+F&F&F&F,F--F/6#,$**\"\"(F&F(F)F,F&F3F&F&F&F&F)*(F(F&F,F)-F76#,$*(FG F&F,F&F3F&F&F&F&*&#F%\"$H(F&*(,&F(F)*&\"#\")F&F+F&F&F&F,F--F/6#,$**F@F &F(F)F,F&F3F&F&F&F&F&*&#F5FWF&*&F,F)-F76#,$*(FWF&F,F&F3F&F&F&F&F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 396 39 "Graphs o f some truncated Fourier series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 365 "f := x -> x ^2-4*x+3:\nf_psod := unapply(simplify(f(piecewise(x<1,x,2-x))*signum(1 -x)),x):\nf_ := x -> f_psod(x-4*floor(x/4+1/4)):\nFS := (x,n) -> sum(1 6*(-1)^(1+k)*(Pi^2*(2*k-1)^2-2)/(Pi^3*(2*k-1)^3)*cos((2*k-1)*Pi*x/2)+ \n8*(-1)^k/(k*Pi)*sin(k*Pi*x),k=1..n):\nplot([f_(x),FS(x,1),FS(x,3),FS (x,5),FS(x,9)],x=-2..6,\n color=[black,red,blue,magenta,brown],num points=80);" }}{PARA 13 "" 1 "" {GLPLOT2D 617 283 283 {PLOTDATA 2 "6)- %'CURVESG6$7bq7$$!\"#\"\"!$!\"$F*7$$!3+n$\\fF\\S*=!#<$!3d%QH@XG]V$F07$ $!3Xa$ou6i=!=F0$!3212i],\"=$QF07$$!3cJD?;!)=)p\"F0$!3[OOv3%Q$)H%F07$$! 3Q-ip&pEQf\"F0$!3cbUMq*p'*y%F07$$!379\"HjQh**[\"F0$!3ci)=@%QH+`F07$$!3 _f2')[\\m$R\"F0$!3%o)pz)R#)Hz&F07$$!3W?'pbcbRH\"F0$!3sK;6\"[wEK'F07$$! 37xTtyf$3>\"F0$!3ooVP)\\-9*oF07$$!3+[yAQ:WR6F0$!3Qii@8Zz#=(F07$$!3')=: s(4Z!)3\"F0$!3kp!)30(p%zuF07$$!3kl75[Vhh5F0$!3H]VGR-6MwF07$$!3V75[)f\" 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The series can also be obtained as the \+ standard Fourier sries of the function " }{XPPEDIT 18 0 "h(x);" "6#-% \"hG6#%\"xG" }{TEXT -1 13 " defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = PIECEWISE([x^2-4*x+3, -1 <= x and x < \+ 1],[1-x^2, 1 <= x and x < 3]);" "6#/-%\"hG6#%\"xG-%*PIECEWISEG6$7$,(*$ F'\"\"#\"\"\"*&\"\"%F/F'F/!\"\"\"\"$F/31,$F/F2F'2F'F/7$,&F/F/*$F'F.F23 1F/F'2F'F3" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG " }{TEXT -1 27 " is periodic with period 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "h := x -> piecewis e(x<1,x^2-4*x+3,1-x^2):\n'h(x)'=h(x);\nFourierSeries(h(x),x=-1..3,numt erms=9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%*PIECEWISE G6$7$,(*$)F'\"\"#\"\"\"F0*&\"\"%F0F'F0!\"\"\"\"$F02F'F07$,&F0F0F-F3%*o therwiseG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4**\"#;\"\"\",&\"\"#!\" \"*$)%#PiGF(F&F&F&F,!\"$-%$cosG6#,$*(F(F)F,F&%\"xGF&F&F&F&*(\"\")F&F,F )-%$sinG6#*&F,F&F3F&F&F)*&#F%\"#FF&*(,&F(F)*&\"\"*F&F+F&F&F&F,F--F/6#, $**\"\"$F&F(F)F,F&F3F&F&F&F&F)*(\"\"%F&F,F)-F76#,$*(F(F&F,F&F3F&F&F&F& *&#F%\"$D\"F&*(,&F(F)*&\"#DF&F+F&F&F&F,F--F/6#,$**\"\"&F&F(F)F,F&F3F&F &F&F&F&*&#F5FEF&*&F,F)-F76#,$*(FEF&F,F&F3F&F&F&F&F)*&#F%\"$V$F&*(,&F(F )*&\"#\\F&F+F&F&F&F,F--F/6#,$**\"\"(F&F(F)F,F&F3F&F&F&F&F)*(F(F&F,F)-F 76#,$*(FGF&F,F&F3F&F&F&F&*&#F%\"$H(F&*(,&F(F)*&\"#\")F&F+F&F&F&F,F--F/ 6#,$**F@F&F(F)F,F&F3F&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " defined on t he interval " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\" #!\"\"F)*&F&F'F(F)" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = cos*2*x; " "6#/-%\"fG6#%\"xG*(%$cosG\"\"\"\"\"#F*F'F*" }{TEXT -1 45 " can be ex tended to the psuedo-even function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6 #%\"xG" }{TEXT -1 13 " with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"# \"\"\"%#PiGF%" }{TEXT -1 7 " where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = PIECEWISE([s in*2*x, -Pi/2 <= x and x < Pi/2],[sin(Pi-2*x), Pi/2 <= x and x < 3*Pi/ 2]);" "6#/-%#f_G6#%\"xG-%*PIECEWISEG6$7$*(%$sinG\"\"\"\"\"#F.F'F.31,$* &%#PiGF.F/!\"\"F5F'2F'*&F4F.F/F57$-F-6#,&F4F.*&F/F.F'F.F531*&F4F.F/F5F '2F'*(\"\"$F.F4F.F/F5" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f_(x)" "6#-%# f_G6#%\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2 *Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = PIECEWISE([sin*2*x, -Pi/2 <= x and x < Pi/2],[-sin*2*x, \+ Pi/2 <= x and x < 3*Pi/2]);" "6#/-%#f_G6#%\"xG-%*PIECEWISEG6$7$*(%$sin G\"\"\"\"\"#F.F'F.31,$*&%#PiGF.F/!\"\"F5F'2F'*&F4F.F/F57$,$*(F-F.F/F.F 'F.F531*&F4F.F/F5F'2F'*(\"\"$F.F4F.F/F5" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"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 10 "Note that " }{XPPEDIT 18 0 "f(x) = sin^2*2*x;" "6#/-%\"fG6#%\"xG*(%$sinG\"\"#F* \"\"\"F'F+" }{TEXT -1 4 " is " }{TEXT 262 4 "even" }{TEXT -1 27 ", so \+ the Fourier series of " }{XPPEDIT 18 0 "f_(x);" "6#-%#f_G6#%\"xG" } {TEXT -1 6 " is a " }{TEXT 262 13 "cosine series" }{TEXT -1 24 " and t he coefficient of " }{XPPEDIT 18 0 "cos*k*x;" "6#*(%$cosG\"\"\"%\"kGF% %\"xGF%" }{TEXT -1 14 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = 4/Pi;" "6#/&%\"aG6#%\"kG*&\"\"%\"\"\"%#PiG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*k*x,x = 0 .. Pi/2); " "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF+%\"kGF+F*F+/F*;\"\"!*&%#PiG F+\"\"#!\"\"" }{TEXT -1 7 ", when " }{TEXT 401 1 "k" }{TEXT -1 9 " is \+ odd, " }{XPPEDIT 18 0 "a[k] = 0;" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 402 1 "k" }{TEXT -1 10 " is even. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "f := x -> s in(2*x)^2:\n'f(x)'=f(x);\nf_psod := unapply(simplify(f(piecewise(x f_p sod(x-2*Pi*floor(x/(2*Pi)+1/4)):\n'f_(x)'='f_psod(x-2*Pi*floor(x/(2*Pi )+1/4))';\nplot(f_(x),x=-2..6,color=COLOR(RGB,.4,0,1),thickness=2);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$)-%$sinG6#,$*&\"\" #\"\"\"F'F0F0F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'f_psodG6#%\"x G-%*PIECEWISEG6$7$*$)-%$sinG6#,$*&\"\"#\"\"\"F'F4F4F3F41F',$*&F3!\"\"% #PiGF4F47$,$F,F82F6F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"x G-%'f_psodG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floorG6#,&*(F.!\"\"F'F,F/F5F ,#F,\"\"%F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 686 223 223 {PLOTDATA 2 "6'-%'CURVESG6#7is7$$!\"#\"\"!$!3!Q2V!p,]Fd!#=7$$!3MLLL$Q6G\">!#<$!3%) e/,x#\\K*RF-7$$!3ommmmFiD=F1$!3-0[/NI@!Q#F-7$$!37++]%!#>7$$!3$om;a\\S7j\"F1$!3]-cK3RHa 9FC7$$!35++]AHe)e\"F1$!3oq.iZ#=\\E\"!#?7$$!3um;/OTDn:F1$\"3'>1Z2kOeF-$\"3MZ4#)[Dde%)F-7$$!3%>+++:v2*\\F-$ \"3I_V!3zVR1(F-7$$!37nmmTrp?TF-$\"3yCKKC^-(Q&F-7$$!3IKLLL\">1D$F-$\"33 sH')R!*pjOF-7$$!3Q******\\RD%[#F-$\"3#=\\(*=aU?F#F-7$$!3Ymmmm())yr\"F- $\"3LS#4!4CtM6F-7$$!3A+++Df^'G\"F-$\"3ScT*=VtcZ'FC7$$!3)*RLLL3V^&)FC$ \"3=!*)He[pm*GFC7$$!3%)3++Dmc%R'FC$\"3OZZ5;7sE;FC7$$!3pxmm;CqPUFC$\"3G .eQ'[lg;(FN7$$!3aYLL3#Q33#FC$\"3rvU8,a&4t\"FN7$$\"3fg%)******f-w!#@$\" 3jcuk)Gz>J#!#B7$$\"3)e)*****\\Y3]d#F-$\"3!=\\n[$)>f U#F-7$$\"3!)*******RP)4MF-$\"3;gV%4,aI(RF-7$$\"3Ummm\"HUz;%F-$\"3S>'3A @d6[&F-7$$\"3.LLL$=Zg#\\F-$\"3Ay)zXG``%pF-7$$\"3H*****\\A2v#eF-$\"3KHt IudNX%)F-7$$\"3clmmms'*GnF-$\"3*yf@J,BA]*F-7$$\"3=lmm;Zz5rF-$\"3]'p/*4 :p!y*F-7$$\"3![mmm;AE\\(F-$\"3'\\ok6hey%**F-7$$\"31kmm\"*e`$o(F-$\"3M3 -B+QQ))**F-7$$\"3Wkmm;'\\W(yF-$\"3__)4cUK)****F-7$$\"3![mm;Mj`1)F-$\"3 ]48$\\vP@)**F-7$$\"31kmmmqFc#)F-$\"3nVl&y#HSN**F-7$$\"3.JLLe#*eW\"*F-$ \"3u3*3Q-*R[$*F-7$$\"3!)*****\\9!H.5F1$\"3z!oIzrO\"=#)F-7$$\"3/LL$e#3# >3\"F1$\"3+Gn=7\"F-7$$\"3jmmm6lCE=F1$!3w'Ga^gY3R#F-7$$\"3%*****\\(*)[ 6\">F1$!3o!H!*o..2'RF-7$$\"3ELLL$G^g*>F1$!3%[2QTwy#\\cF-7$$\"3'HLL3+TU 3#F1$!3J[[BAB6AtF-7$$\"3oKLL=2Vs@F1$!3279:N!\\&*F-7$$\"3f*****\\`pfK#F1$!3i%)H0sD]j**F-7$$\"39m;/,ypYBF1 $!33CUe\\HR'***F-7$$\"3qKL3ngUnBF1$!3ig\"ze\"\\&\\***F-7$$\"3q**\\7LV: )Q#F1$!3#)fz9^$)>f**F-7$$\"3Dmm;*f#))3CF1$!3y[9$e#*o$*))*F-7$$\"3!)*** \\78R.X#F1$!3SI&\\/RT'\\'*F-7$$\"3!HLLLm&z\"\\#F1$!3YYBX=$\\BG*F-7$$\" 3KmmmrHXxDF1$!3!y0=oA;j;)F-7$$\"3s******z-6jEF1$!3U%)H&QH'>#o'F-7$$\"3 W*****\\C4pu#F1$!3iF%367'oR]F-7$$\"3<******4#32$GF1$!3o(>ivbeFR$F-7$$ \"3F****\\F-7$$\"3O*****\\#y'G*HF1$!3OfWv3Nv*e)FC7 $$\"3N***\\iZ!)y.$F1$!3yM@ybX8TUFC7$$\"3K****\\FJ*G3$F1$!3+GBBDM$>P\"F C7$$\"35**\\7`%*R0JF1$!30xAdFckI_FN7$$\"3J***\\(yd!z7$F1$!3#\\L+)*G]8 \\(F^w7$$\"3_**\\P/@T]JF1$!3^8%4[#e)46$F^w7$$\"3G******H%=H<$F1$!39Bgd \")R3?RFN7$$\"3qKLLo,\"QD$F1$!3sfP-f$*4`\\FC7$$\"35mmm1>qMLF1$!3g:b\\` (R*=9F-7$$\"3(HLLL5r5U$F1$!3tOndjN?7GF-7$$\"3%)*******HSu]$F1$!3Ua\"=a A.SY%F-7$$\"3Smm;HOq&e$F1$!3<>,[lc+@gF-7$$\"3'HLL$ep'Rm$F1$!3oM*\\4kh( yuF-7$$\"3Umm;\\%H&\\PF1$!3aR>')pHJ#z)F-7$$\"3')******R>4NQF1$!31?$)=b 2(fm*F-7$$\"3AL$e*[)=_&QF1$!3cOL*Hqi`z*F-7$$\"3emm\"zvX`(QF1$!3_*3h+R! p$*)*F-7$$\"3[**\\(omsa*QF1$!3)oqh;&oJg**F-7$$\"3&GLLed*f:RF1$!3U\"R]H [5[***F-7$$\"3?m;z%[Ed$RF1$!3%\\r%*[$y%p***F-7$$\"37***\\PR`e&RF1$!3%R %Gxh]rm**F-7$$\"3[K$3FI!)f(RF1$!3MMe'3,3V!**F-7$$\"3#emm;@2h*RF1$!3%** )Hw`485)*F-7$$\"37LLL))3E!3%F1$!3Y&e(\\%o(R*3*F-7$$\"3]*****\\c9W;%F1$ !3!QL*)H!el4zF-7$$\"3#HLLe;!pYUF1$!3&p@)e\"eN$RkF-7$$\"3Lmmmmd'*GVF1$! 3v\"Q8M?DX\"[F-7$$\"3)HLLep+^T%F1$!3!G%=[/.xPJF-7$$\"3j*****\\iN7]%F1$ !3=*Roz3)*)z;F-7$$\"39mm;*z$>%e%F1$!3D+Xmrv\"3V'FC7$$\"3aLLLt>:nYF1$!3 _&[O]>PK;)FN7$$\"3VLL$3Lq&4ZF1$!3')oNp`/)y<$FT7$$\"3KLLL)o))>v%F1$\"3S +()[\"\\V%fiFN7$$\"3ALL$e/2Wz%F1$\"3sz#f!GstmEFC7$$\"35LLL.a#o$[F1$\"3 7wlr*zWp1'FC7$$\"3E++]F'f4#\\F1$\"3]`i@Yeq'Q3'F-7$$\"3kLLLomF]5xF-7$$\"3gmmmc%GpL&F1$\"3:j m=s!*>+!*F-7$$\"3?LL$e*Qbw`F1$\"3MIz`M4`B%*F-7$$\"3#)*****\\LzhT&F1$\" 3ub#*4]P'ft*F-7$$\"37LLea?*fV&F1$\"3y6KL+3.[)*F-7$$\"3Umm;uZ!eX&F1$\"3 c\"*pj[ZmH**F-7$$\"3u***\\P\\zw` &F1$\"3E*oeo\"*zk$**F-7$$\"35L$3xo.)ebF1$\"3APrnMq\"=&)*F-7$$\"3;mm;z \"G*zbF1$\"3!*)4^.CLDt*F-7$$\"3=LL3ird[XF-7$$\"\"'F*$ \"3?6]Jj\\5zGF--%*THICKNESSG6#\"\"#-%&COLORG6&%$RGBG$\"\"%!\"\"$F*F*$ \"\"\"F*-%+AXESLABELSG6$Q\"x6\"Q!F\\im-%%VIEWG6$;F(Ffgm%(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 "" }}{PARA 0 "" 0 "" {TEXT 400 62 "Calculat ion of the (non-zero) coefficients of the cosine terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "f := x -> sin(2*x)^2:\n'f(x)'=f(x);\nassume(k_,integ er):\na[2*k-1]=4/Pi*Int('f(x)'*cos((2*k-1)*x),x=0..Pi/2);\nfactor(subs (k_=k,value(subs(k=k_,%))));\naa := unapply(eval(rhs(%),k=(k+1)/2),k): \nmatrix([[k,`|`,seq(2*k-1,k=1..8)],['a'[2*k-1],`|`,seq(aa(2*k-1),k=1. .8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$)-%$sinG6#, $*&\"\"#\"\"\"F'F0F0F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#, &*&\"\"#\"\"\"%\"kGF*F*F*!\"\",$,$-%$IntG6$*&-%\"fG6#%\"xGF*-%$cosG6#* &F'F*F6F*F*/F6;\"\"!,$*&F)F,%#PiGF*F**&\"\"%F*F@F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#,&*&\"\"#\"\"\"%\"kGF*F*F*!\"\",$*.\"#KF*)F ,F+F*%#PiGF,,&*&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\"\"\"\"\" $\"\"&\"\"(\"\"*\"#6\"#8\"#:7,&%\"aG6#,&*&\"\"#F*F(F*F*F*!\"\"F),$*(\" #KF*F1F9%#PiGF9F*,$*(FF9F=F9 F9,$*(F " 0 "" {MPLTEXT 1 0 81 "FS := (x,n) -> sum(32*(-1)^k /(Pi*(2*k-5)*(2*k+3)*(2*k-1))*cos((2*k-1)*x),k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF)-%$s umG6$,$*0\"#K\"\"\")!\"\"%\"kGF3%#PiGF5,&*&\"\"#F3F6F3F3\"\"&F5F5,&*&F :F3F6F3F3\"\"$F3F5,&*&F:F3F6F3F3F3F5F5-%$cosG6#*&F?F39$F3F3F3/F6;F39%F )F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 404 44 "The first few terms of the Fourier series of" }{TEXT -1 1 " " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "FS(x,5); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&#\"#K\"#:\"\"\"*&%#PiG!\"\"-%$ cosG6#%\"xGF(F(F(*&#F&\"#@F(*&F*F+-F-6#,$*&\"\"$F(F/F(F(F(F(F+*&#F&\"# XF(*&F*F+-F-6#,$*&\"\"&F(F/F(F(F(F(F+*&#F&\"$J#F(*&F*F+-F-6#,$*&\"\"(F (F/F(F(F(F(F(*&#F&\"$&eF(*&F*F+-F-6#,$*&\"\"*F(F/F(F(F(F(F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 405 39 "Graphs of some truncated Fourier series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 335 "f := x -> sin(2*x )^2:\nf_psod := unapply(simplify(f(piecewise(x f_psod(x-2*Pi*floor(x/(2*Pi)+1/4)):FS := (x,n) \+ -> sum(32*(-1)^k/(Pi*(2*k-5)*(2*k+3)*(2*k-1))*cos((2*k-1)*x),k=1..n): \nplot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4)],x=-2..6,\n color=[b lack,red,blue,magenta,brown],numpoints=60);" }}{PARA 13 "" 1 "" {GLPLOT2D 617 283 283 {PLOTDATA 2 "6)-%'CURVESG6$7`u7$$!\"#\"\"!$!3!Q2 V!p,]Fd!#=7$$!3C$fN\"))o1H>!#<$!3R54^0Sg8VF-7$$!3['=rixL\"e=F1$!3)\\A \"RAefaHF-7$$!3Ka_\"\\D:kz\"F1$!3Ev!GY8c;!>F-7$$!3(>KfNt'pMj'[aU9D!#?7$$!3;)GPUh6%y:F1$!3*[$\\=^Z_> B!#@7$$!3!Q.A$fV%4c\"F1$\"3oC11-\"[>)QFN7$$!3VznS/rZV:F1$\"3#)*)))3R+R #)HFH7$$!3HD:\\\\)4g_\"F1$\"3c7M,#Gz=+)FH7$$!3!o,h'R`2\"\\\"F1$\"3m\"* f[M]q?D!#>7$$!3I30$)H39c9F1$\"3Wl/ax$Ho;&F]o7$$!3!)GPU&)Qg'Q\"F1$\"3)> @m&[fy'H\"F-7$$!3K\\p,Tp1<8F1$\"3A$H7O7]:O#F-7$$!3!**Q.AH(f_7F1$\"3!y$ ozzV&>`$F-7$$!3WI)*QVw7)=\"F1$\"3'Q6\")4&eW*z%F-7$$!3b'=ri0t87\"F1$\"3 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "h := x -> piecewise(xG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 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 "Find the firs t few terms of the Fourier Series of the function " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 15 " is defined by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([-Pi*(x+Pi), -Pi <= x and x < _Pi/2],[Pi*x, -Pi/2 <= x and x < 0],[x*(Pi-x), 0 <= x and x < Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$,$*&%#PiG\"\"\",&F'F/F.F/F /!\"\"31,$F.F1F'2F'*&%$_PiGF/\"\"#F17$*&F.F/F'F/31,$*&F.F/F8F1F1F'2F' \"\"!7$*&F'F/,&F.F/F'F1F/31F@F'2F'F." }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " } {XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Plot the \+ graphs of some truncated Fourier series. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x )" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " is shown below. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "f := x -> piecewise(x<-Pi/2,-Pi*(Pi+x),x<0,Pi*x,x*(Pi-x)):\n'f(x)'=f(x);\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n'f_(x)'='f(x-2*Pi*floor((x+Pi)/ (2*Pi)))';\nplot(f_(x),x=-Pi..4*Pi,color=COLOR(RGB,.5,0,1),thickness=2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$, $*&%#PiG\"\"\",&F'F/F.F/F/!\"\"2F',$*&\"\"#F1F.F/F17$*&F.F/F'F/2F'\"\" !7$*&F'F/,&F.F/F'F1F/%*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 619 183 183 {PLOTDATA 2 "6'-%'CURVESG6#7gq7$$!3*****4tk#fTJ!#<$!3_&*4J'>bG(>!#D7$$ !3[HZ:*>)RqHF*$!3#[3&)QSK#y`!#=7$$!3Sf%**4v.#*z#F*$!3uj%[5YYc2\"F*7$$! 3c)HJqP[-l#F*$!3-e&>'pNgV:F*7$$!3tPJ1.IH,DF*$!3K_1>y1c6?F*7$$!3#>qzi5x PL#F*$!3E+])))QFy`#F*7$$!3mli\\47Em@F*$!3)*\\$z&*4%4kIF*7$$!3q4/h3\\j( *>F*$!31xVkZ#\\Qf$F*7$$!3w`Xs2'3!H=F*$!3=/%4dR/O7%F*7$$!3uF\"oH='4X;F*$!3QR' 3'Q5l#y%F*7$$!3#fFbMLrsd\"F*$!3*e[)=n.Y9\\F*7$$!3\"G1x57:``\"F*$!3#Q-@ `qML#[F*7$$!3!*\\))p3*eL\\\"F*$!3Ix6uw`_\"p%F*7$$!3xRV7!zjxL\"F*$!3-kI H&*)3F?%F*7$$!3iH)\\:no@=\"F*$!3%)\\\\%QT#*Qr$F*7$$!3xBpoQ*e5-\"F*$!3c I5mg7v2KF*7$$!3C\"=S#e?\\*f)F3$!3<7rZ2,h,FF*7$$!3!>Yti*GHLpF3$!3'3!>\" G?e\"y@F*7$$!3aUnIMP4n_F3$!37*oY\")H1Zl\"F*7$$!3A^Q=%4Qig$F3$!3Uk$y.6L H8\"F*7$$!3!*f41aCQX>F3$!3'[R+hA*f6hF37$$!3-*yk3$G))pB!#>$!3!RS88\\1_W (Fjr7$$\"33-!))y)eSr9F3$\"3;ml9*RagS%F37$$\"3ec1e./;wHF3$\"3'fo4]eITY) F37$$\"336LF>\\\"4[%F3$\"3)>LPS&\\$p?\"F*7$$\"3YDb!Q57\\<'F3$\"3_t0Hb/ he:F*7$$\"3%)RxL)G4*oyF3$\"3[`(>tU$*G&=F*7$$\"3-Tb1)=i)p&*F3$\"3+zOg^# Q14#F*7$$\"3AM$z(3:3F6F*$\"3obdag#=0F#F*7$$\"3m(*)e&Q)**4H\"F*$\"39UJW D]6*Q#F*7$$\"35h%Q$o\"=\\X\"F*$\"3m%3lENtRX#F*7$$\"3*3q')=%\\MH:F*$\"3 /K:'H)GolCF*7$$\"3oS\\V:y;F*$\"3- @'He(e'eX#F*7$$\"3E?9`i_i_F*$\"3Y&Ro t)>kQBF*7$$\"3S!\\tBzFm5#F*$\"3mN#o2u&G!=#F*7$$\"3D3!Q(pV^1CF*$\"3su;( HXw*oL^fG-Z9F*7$$\"3S)4p4v``v#F*$\"3]>h54[A k5F*7$$\"3LnIWJZu4HF*$\"3sxXi*p#=YnF37$$\"3EOq\">rNT1$F*$\"3!=p`6\"eQt BF37$$\"3VM5:(3FNB$F*$!3WY$z3T0#))GF37$$\"3fK]Qi%=HS$F*$!3S_`/6Cz4#)F3 7$$\"3%ohy\"4\">Uc$F*$!3]T04o,sF8F*7$$\"34,A(fv>bs$F*$!3'yaw]4hW$=F*7$ $\"3#HG_$f+#Q*QF*$!3MYBx\"4#>jBF*7$$\"3?lBti.7iSF*$!3QY\"o%)3B>*GF*7$$ \"3tsR!*yDn;UF*$!31!*))Q&>juP$F*7$$\"3E!ev]zC7P%F*$!3=M'4BI.I'QF*7$$\" 3'y#e\\R-$z`%F*$!3U0P\"=!Rs'Q%F*7$$\"3Xvg\"RoNYq%F*$!3mwxJ,XW5\\F*7$$ \"3U+r&*)y&zx[F*$!3'*f)zeQh^T%F*7$$\"3SD\")*R*e&40&F*$!3-R\"))>(G;rQF* 7$$\"3d.F*7$$ \"3u!=krA:i%eF*$!3)=%\\@$*>ys8F*7$$\"3l!)RWeHS9gF*$!3h$)Q-Al/W%)F37$$ \"3gz'G6%=%*yhF*$!3I_EIr]!\\F$F37$$\"3dyL\"Qs![VjF*$\"3U_\"[n+\")y&=F3 7$$\"3CUDC`0o-lF*$\"3l*G][*['QT'F37$$\"3!fqrEQ!)=m'F*$\"3)Hp9pq&HY5F*7 $$\"3)p\"\\U]YkQoF*$\"3G3)3,\">\\O9F*7$$\"31G\"y\"=*3a,(F*$\"3[lUf+p>k Jm7(p^`+#F*7$$\"3Ctu9J+2LtF*$\"3]NDJ-A0'>#F* 7$$\"3mO$)*=bbE](F*$\"3rk`K%3rRM#F*7$$\"33+#\\E2TAn(F*$\"3b5:ko9PMCF*7 $$\"3G0:B;f2\\xF*$\"3Mqrl!*eRcCF*7$$\"3Y5Q\")f2\"f#yF*$\"3+&>'HEJhmCF* 7$$\"3l:hR.cu-zF*$\"3U&eeb\"Q#F*7$$\"3-yp3iNe:$)F*$\"3Qm^BgZKaAF*7$ $\"37g!p!3jtJ')F*$\"3Ixj>n()\\i=F*7$$\"3M07[l;(pz)F*$\"3')*G^2!4L+Cx]&)F3 7$$\"3c!H\"=^DI&G*F*$\"3M?gQA`@(=%F37$$\"3AjCx)3GWX*F*$!3JF&[7Vi[J*Fjr 7$$\"3lPOOEObB'*F*$!3pKMV2O6F*7$ $\"3=Q\"eQhH$\\**F*$!3_E?x>w#zk\"F*7$$\"3I#*)*Q61f65!#;$!3Ac3cGJCr@F*7 $$\"3hqRRh#[#G5F[gl$!3#*z'\\tjeXp#F*7$$\"3H\\dkxzwW5F[gl$!34Q%4&H3a8KF *7$$\"3)z_(*Qp(Gh5F[gl$!3D'>p;-BDt$F*7$$\"3=`:Y(fnk2\"F[gl$!3%yieov8%4 UF*7$$\"3Ayb-,vk\"4\"F[gl$!3w_![?\\/jo%F*7$$\"3%4[f;(p*f4\"F[gl$!3^'y_ >9ZH#[F*7$$\"3$QQ$HUkM+6F[gl$!3UK=B4Y,5\\F*7$$\"3t'GFH\"fp/6F[gl$!3C%4 F$f>PtZF*7$$\"3X*=hNQX!46F[gl$!3]gBU4$Hnj%F*7$$\"31&**G[KWx6\"F[gl$!3e ))Gh4SWjVF*7$$\"3o+o4mKWE6F[gl$!3m;M!)4(e,4%F*7$$\"3![JQ'RT+U6F[gl$!36 6))y\\%*H,OF*7$$\"3uG)zJ,lv:\"F[gl$!3L6Ux*=SC6$F*7$$\"3K-2%*\\h:u6F[gl $!3]rv9cU@\"f#F*7$$\"3sv:q'GZ2>\"F[gl$!3+P4_A$))*p?F*7$$\"39\"3j:7Fm? 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0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 " " {TEXT -1 63 "Find the first few terms of the Fourier Series of the f unction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 15 " is de fined by " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=PIECEWISE([-Pi-x,-Pi<=x and x<_Pi/2],[x, -Pi/2<=x and x<0],[2*x,0<=x and x " 0 "" {MPLTEXT 1 0 220 "f := x -> piecewise(x<-Pi/2,-Pi-x,x<0,x,x f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n'f_(x)'='f (x-2*Pi*floor((x+Pi)/(2*Pi)))';\nplot(f_(x),x=-Pi..5*Pi,color=COLOR(RG B,.5,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#% \"xG-%*PIECEWISEG6&7$,&%#PiG!\"\"F'F.2F',$*&\"\"#F.F-\"\"\"F.7$F'2F'\" \"!7$,$*&F2F3F'F3F32F',$*&F2F.F-F3F37$,&*&F2F3F-F3F3*&F2F3F'F3F.%*othe rwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\" \"\"*(\"\"#F,%#PiGF,-%&floorG6#,$*(F.!\"\",&F'F,F/F,F,F/F5F,F,F5" }} {PARA 13 "" 1 "" {GLPLOT2D 619 183 183 {PLOTDATA 2 "6'-%'CURVESG6#7`p7 $$!3*****4tk#fTJ!#<$!3!*H&[*HJzzi!#E7$$!3i%RP<(fsIFF*$!3(\\)R_=om3T!#= 7$$!3WZQ@uIBtBF*$!3yc%fPz&f$o(F37$$!35VO$>#\\>r>F*$!3-]hlJxRq6F*7$$!3K P1(3OV)oFH\\FP\"F*7$$!3wJw!)*z\"\\m:F*FD7$$!33JAR!)>5l8F*FG7 $$!3;Io(4;7P;\"F*FJ7$$!3:`/)RwQG!zF3FM7$$!3mB6Fv^?OSF3FP7$$!3gt7\"4=mn .#F3FS7$$!3-`B9b'=Ft$!#?FV7$$\"3OLef\"e*pb>F3$\"3pm;>j\"*R6RF37$$\"3A! 4V(\\js[RF3$\"3W!='[*p_u*yF37$$\"3!)\\A\"\\N*z)*fF3$\"3%*\\C)4()f(*>\" F*7$$\"3Q493gB()[!)F3$\"3)=G;?Zu(4;F*7$$\"3S.C\"*yxda)*F3$\"3o![#ybb\" 4(>F*7$$\"3uRVx>$Gg;\"F*$\"3Zz'[&Rm0KBF*7$$\"3q$==>)zIp8F*$\"3Tnj$Q'fh 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-1 34 "_____________ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________ ______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7 $,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 45 " can be extended t o the psuedo-even function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 13 " with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#P iGF%" }{TEXT -1 7 " where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = PIECEWISE([x, -Pi/2 <= x and x < Pi/2],[Pi-x, Pi/2 <= x and x < 3*Pi/2]);" "6#/-%#f_G6#% \"xG-%*PIECEWISEG6$7$F'31,$*&%#PiG\"\"\"\"\"#!\"\"F3F'2F'*&F0F1F2F37$, &F0F1F'F331*&F0F1F2F3F'2F'*(\"\"$F1F0F1F2F3" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " } {XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 25 " is periodic wit h period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Fi nd the the Fourier Series of " }{XPPEDIT 18 0 "f_(x);" "6#-%#f_G6#%\"x G" }{TEXT -1 55 " and plot the graphs of some truncated Fourier series . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Not e that " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 4 " is " }{TEXT 262 3 "odd" }{TEXT -1 27 ", so the Fourier series of " } {XPPEDIT 18 0 "f_(x);" "6#-%#f_G6#%\"xG" }{TEXT -1 6 " is a " }{TEXT 262 11 "sine series" }{TEXT -1 24 " and the coefficient of " } {XPPEDIT 18 0 "sin*k*x" "6#*(%$sinG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 14 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[ k] = 4/Pi;" "6#/&%\"bG6#%\"kG*&\"\"%\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin*k*x,x = 0 .. Pi/2);" "6#-%$IntG6$**-%\"fG 6#%\"xG\"\"\"%$sinGF+%\"kGF+F*F+/F*;\"\"!*&%#PiGF+\"\"#!\"\"" }{TEXT -1 7 ", when " }{TEXT 407 1 "k" }{TEXT -1 9 " is odd, " }{XPPEDIT 18 0 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT -1 6 " when " }{TEXT 408 1 "k" }{TEXT -1 10 " is even. " }}{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 18 "Code for pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 561 "pi := evalf (Pi): pi2 := pi/2:\np1 := plot(exp(sin(x)),x=pi2..5*pi2,0..2.73):\np2 \+ := plot([[[pi2,0],[pi2,3]],[[5*pi2,0],[5*pi2,3]]],color=navy,linestyle =2):\np3 := plot([[3*pi2,0],[3*pi2,3]],color=blue,linestyle=3):\nt1 := plots[textplot]([[5*pi2+.7,-.1,`x`],[-.2,3.2,`y`],[-pi2,-.1,`L`],\n \+ [pi2,-.12,`a`],[5*pi2,-.12,`b`],[3*pi2,-.12,`a+b`],[3*pi2,-.15,`___` ],\n [3*pi2,-.4,`2`]],color=black,font=[HELVETICA,9]):\nt2 := plots [textplot]([6.6,2.4,`y = f(x)`],color=COLOR(RGB,.9,0,0)):\nplots[displ ay]([p1,p2,p3,t1,t2],tickmarks=[0,0],view=[-.2..5*pi2+.7,-.4..3.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 620 "pi := evalf(Pi): pi2 := pi/2:\np1 := plot(exp(sin(x))*cos(x), x=pi2..5*pi2,0..2.73):\np2 := plot([[[pi2,-1.6],[pi2,1.6]],[[5*pi2,-1. 6],[5*pi2,1.6]]],color=navy,linestyle=2):\np3 := plot([[[3*pi2,0]]$3], color=black,\n style=point,symbol=[circle,diamond,cross]):\nt1 \+ := plots[textplot]([[5*pi2+.7,-.1,`x`],[-.2,1.6,`y`],[-pi2,-.1,`L`],\n [pi2-.2,-.12,`a`],[5*pi2+.2,-.12,`b`],[3*pi2,-.16,`a+b`],[3*pi2,-. 19,`___`],\n [3*pi2,-.44,`2`]],color=black,font=[HELVETICA,9]):\nt2 := plots[textplot]([6.6,2.4,`y = f(x)`],color=COLOR(RGB,.9,0,0)):\npl ots[display]([p1,p2,p3,t1,t2],tickmarks=[0,0],view=[-.2..5*pi2+.7,-1.6 ..1.6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 839 "p1 := plot(sin(x),x=0..Pi,thickness=2):\npp := op(1, op(1,plot(sin(x),x=0..Pi/2,adaptive=false,numpoints=20))):\np2 := plot s[polygonplot]([seq([pp[j],pp[j+1],[pp[j+1,1],0],[pp[j,1],0]],j=1..19) ],\n style=patchnogrid,color=COLOR(RGB,1,.85,.85)):\npp := op(1,op(1 ,plot(sin(x),x=Pi/2..Pi,adaptive=false,numpoints=20))):\np3 := plots[p olygonplot]([seq([pp[j],pp[j+1],[pp[j+1,1],0],[pp[j,1],0]],j=1..19)], \n style=patchnogrid,color=COLOR(RGB,.85,1,.85)):\np4 := plot([[Pi/2 ,0],[Pi/2,1]],color=black):\nt1 := plots[textplot]([[3.5,-.05,`x`],[-. 06,1.17,`y`]],font=[HELVETICA,9],color=black):\nt2 := plots[textplot]( [.9,.97,`y = sin x`],color=COLOR(RGB,.9,0,0),font=[HELVETICA,9]):\nplo ts[display]([p1,p2,p3,p4,t1,t2],labelfont=[HELVETICA,9],ytickmarks=3,f ont=[SYMBOL,9],\n xtickmarks=[0=`0`,evalf(Pi/2)=`p/2`,evalf(Pi)=`p`] ,view=[-.1..3.5,-.05..1.17]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 837 "p1 := plot(cos(x),x=0..Pi,t hickness=2):\npp := op(1,op(1,plot(cos(x),x=0..Pi/2,adaptive=false,num points=20))):\np2 := plots[polygonplot]([seq([pp[j],pp[j+1],[pp[j+1,1] ,0],[pp[j,1],0]],j=1..19)],\n style=patchnogrid,color=COLOR(RGB,1,.8 5,.85)):\npp := op(1,op(1,plot(cos(x),x=Pi/2..Pi,adaptive=false,numpoi nts=20))):\np3 := plots[polygonplot]([seq([pp[j],pp[j+1],[pp[j+1,1],0] ,[pp[j,1],0]],j=1..19)],\n style=patchnogrid,color=COLOR(RGB,.85,.85 ,1)):\np4 := plot([[Pi,0],[Pi,-1]],color=black):\nt1 := plots[textplot ]([[3.25,-.05,`x`],[-.06,1.17,`y`]],font=[HELVETICA,9],color=black):\n t2 := plots[textplot]([.9,.9,`y = cos x`],color=COLOR(RGB,.9,0,0),font =[HELVETICA,9]):\nplots[display]([p1,p2,p3,p4,t1,t2],labelfont=[HELVET ICA,9],ytickmarks=3,font=[SYMBOL,9],\n xtickmarks=[0=`0`,evalf(Pi/2) =`p/2`,evalf(Pi)=`p`],view=[-.1..3.25,-1.1..1.17]);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 809 "pi := eva lf(Pi): pi2 := pi/2:\np1 := plot(exp(sin(x)),x=-4..7.15,0..2.73):\np2 \+ := plot([[[-pi,0],[-pi,3]],[[pi,0],[pi,3]],\n [[2*pi,0],[2*pi, 3]]],color=navy,linestyle=2):\np3 := plot([[[-pi2,0],[-pi2,3]],[[pi2,0 ],[pi2,3]],\n [[3*pi2,0],[3*pi2,3]]],color=blue,linestyle=3):\nt 1 := plots[textplot]([[7.25,-.1,`x`],[-.2,3.2,`y`]],color=black,font=[ HELVETICA,9]):\nt2 := plots[textplot]([[-pi,-.15,`-L`],[-pi2,-.1,`L`], [pi2,-.1,`L`],[pi,-.15,`L`],\n [3*pi2,-.1,`3L`],[2*pi,-.15,`2L`],[pi ,-.15,`L`],[-pi2,-.12,`_`],[pi2,-.12,`_`],\n [3*pi2,-.12,`_`],[-pi2- .16,-.165,`-`],\n [-pi2,-.28,`2`],[pi2,-.28,`2`],[3*pi2,-.28,`2`]] ,color=black,font=[HELVETICA,8]):\nt3 := plots[textplot]([4,.9,`y = f( x)`],color=COLOR(RGB,.9,0,0)):\nplots[display]([p1,p2,p3,t1,t2,t3],tic kmarks=[0,0],view=[-4.2..7.25,-.28..3.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 945 "pi := evalf(Pi): \+ pi2 := pi/2:\np1 := plot(exp(sin(x))*cos(x),x=-4..7.15,0..2.73):\np2 : = plot([[[-pi,-1.5],[-pi,1.5]],[[pi,-1.5],[pi,1.5]],\n [[2*pi,-1 .5],[2*pi,1.5]]],color=navy,linestyle=2):\np3 := plot([[[-pi2,-1.5],[- pi2,1.5]],[[pi2,-1.5],[pi2,1.5]],\n [[3*pi2,-1.5],[3*pi2,1.5]]] ,color=blue,linestyle=3):\np4 := plot([[[-pi2,0]],[[pi2,0],[3*pi2,0]]$ 3],color=black,\n style=point,symbol=[circle,diamond,cross]):\n t1 := plots[textplot]([[4.2,-.1,`x`],[-.2,1.6,`y`]],color=black,font=[ HELVETICA,9]):\nt2 := plots[textplot]([[-3,-.1,`-L`],[-1.35,-.08,`L`], [1.4,-.08,`L`],[4.95,-.08,`3L`],\n [6.48,-.08,`2L`],[3,-.1,`L`],[-1 .35,-.1,`_`],[1.4,-.1,`_`],[-1.47,-.165,`-`],\n [4.95,-.1,`_`],[-1 .35,-.24,`2`],[1.4,-.24,`2`],\n [4.95,-.24,`2`]],color=black,f ont=[HELVETICA,8]):\nt3 := plots[textplot]([4,-.7,`y = f(x)`],color=CO LOR(RGB,.9,0,0)):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],tickmarks=[0, 0],view=[-4.2..7.25,-1.6..1.6]);" }}}{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 }