{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Em phasis" -1 271 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Headi ng 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple 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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "Fourier sine and cosine series" } }{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" } }{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 52 "load Fourier series and Fourier transform procedures" } }{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 271 9 "fourier .m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 13 " FourierSeries" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command si milar to the one that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/procd rs/fourier.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 32 "Fourier sine series coefficients" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " is an " }{TEXT 260 3 "odd" }{TEXT -1 13 " function on " }{XPPEDIT 18 0 "[-L,L]" "6#7$ ,$%\"LG!\"\"F%" }{TEXT -1 47 " the Fourier series coefficients are giv en by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1/(2*L )" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x),x=-L..L)" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$%\"LG !\"\"F-" }{TEXT -1 6 " = 0. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[k] = 1/L" "6#/&%\"aG6#%\"kG*&\"\"\"F)%\"LG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos(k*Pi*x/L),x=-L..L)" "6#-%$ IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F *;,$F2F3F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 " ``= 0" "6#/%!G\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " k = 1,2,3,` . . . `;" "6&/%\"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 " because the function " } {XPPEDIT 18 0 "f(x)*cos(k*Pi*x/L);" "6#*&-%\"fG6#%\"xG\"\"\"-%$cosG6#* *%\"kGF(%#PiGF(F'F(%\"LG!\"\"F(" }{TEXT -1 4 " is " }{TEXT 260 3 "odd " }{TEXT -1 5 ", and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k] = 1/L" "6#/&%\"bG6#%\"kG*&\"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin(k*Pi*x/L),x=-L..L)" "6#-%$IntG6$*&-%\"fG6 #%\"xG\"\"\"-%$sinG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3F2" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = 2/L;" "6#/%!G*&\"\"#\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin(k*Pi*x/L),x = 0 .. L)" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\" \"-%$sinG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;\"\"!F2" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 21 "because the function " }{XPPEDIT 18 0 "f(x)*sin(k*Pi*x/L) " "6#*&-%\"fG6#%\"xG\"\"\"-%$sinG6#**%\"kGF(%#Pi GF(F'F(%\"LG!\"\"F(" }{TEXT -1 4 " is " }{TEXT 260 4 "even" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "T he " }{TEXT 260 19 "Fourier sine series" }{TEXT -1 20 " of an odd func tion " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " on " } {XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\"\"F%" }{TEXT -1 4 " is:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = Sum(b[k]*sin(k *Pi*x/L),k = 1 .. infinity);" "6#/-%\"FG6#%\"xG-%$SumG6$*&&%\"bG6#%\"k G\"\"\"-%$sinG6#**F/F0%#PiGF0F'F0%\"LG!\"\"F0/F/;F0%)infinityG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = b[1]*sin(Pi*x/L)+b[2]*sin(2*Pi*x/L )+b[3]*sin(3*Pi*x/L)+` . . . `;" "6#/%!G,**&&%\"bG6#\"\"\"F*-%$sinG6#* (%#PiGF*%\"xGF*%\"LG!\"\"F*F**&&F(6#\"\"#F*-F,6#**F6F*F/F*F0F*F1F2F*F* *&&F(6#\"\"$F*-F,6#**F=F*F/F*F0F*F1F2F*F*%(~.~.~.~GF*" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 8 " wh ere " }{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 = 0 .. L);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#**%\"kGF+%#PiGF+F*F+% \"LG!\"\"F+/F*;\"\"!F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 18 "periodic extensio n" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " is the function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" } {TEXT -1 13 " with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LG F%" }{TEXT -1 20 " defined as follows." }}{PARA 0 "" 0 "" {TEXT -1 20 "For any real number " }{TEXT 267 1 "u" }{TEXT -1 21 ", there is a int eger " }{TEXT 266 1 "k" }{TEXT -1 45 ", which may be positive or negat ive, so that " }{XPPEDIT 18 0 "x=u+2*k*L" "6#/%\"xG,&%\"uG\"\"\"*(\"\" #F'%\"kGF'%\"LGF'F'" }{TEXT -1 14 " implies that " }{XPPEDIT 18 0 "-L \+ <= x;" "6#1,$%\"LG!\"\"%\"xG" }{XPPEDIT 18 0 "``f(L)" "6#0-%#f_G6#% \"LG-%\"fG6#F'" }{TEXT -1 97 ", but this does not affect the value of \+ the integrals used to determine the Fourier coefficients." }}{PARA 0 " " 0 "" {TEXT -1 14 "Note that the " }{TEXT 263 1 "L" }{TEXT -1 36 " wh ich appears in these formulas is " }{TEXT 260 15 "half the period" } {TEXT -1 18 " of the functions " }{XPPEDIT 18 0 "f_(x);" "6#-%#f_G6#% \"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "We often blur the disti nction between the functions " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 12 " defined on " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\" \"F%" }{TEXT -1 28 " and the periodic extension " }{XPPEDIT 18 0 "f_(x )" "6#-%#f_G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Examples of Fourier sine series " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exa mple 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " be the function defined on the interval " }{XPPEDIT 18 0 "-1<= x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" } {TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x)=x" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " is an " }{TEXT 260 3 "odd" } {TEXT -1 38 " function, so its Fourier series is a " }{TEXT 260 11 "si ne series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "The Fourier sine series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 50 " is the same as that of the the periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 11 " given by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_ (x) = x;" "6#/-%#f_G6#%\"xGF'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1 <= \+ x;" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\"" } {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 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "f := x -> x:\n'f(x)'=f(x); \nf_ := x -> f(x-2*floor((x+1)/2)):\n'f_(x)'='f(x-2*floor((x+1)/2))'; \n``=value(rhs(%));\nplot(f_(x),x=-2..4,color=COLOR(RGB,.4,0,1),thickn ess=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*&\"\"#F,-%& floorG6#,&*&F.!\"\"F'F,F,#F,F.F,F,F4" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G,&%\"xG\"\"\"*&\"\"#F'-%&floorG6#,&*&F)!\"\"F&F'F'#F'F)F'F'F/" } }{PARA 13 "" 1 "" {GLPLOT2D 530 207 207 {PLOTDATA 2 "6'-%'CURVESG6#7^p 7$$!\"#\"\"!$F*F*7$$!3!******\\2<#p=!#<$\"30,++]#HyI\"!#=7$$!31++D^NUb iUCF2Fcq7$$!3B++]7YY08F2Ffq7$$\"3%z-+++XDn%!#?Fiq7$$\"3C+ +++y?#>\"F2F]r7$$\"3h****\\(3wY_#F2F`r7$$\"3F)******HOTq$F2Fcr7$$\"3I, +](3\">)*\\F2Ffr7$$\"3_,+]isVIiF2Fir7$$\"3&=++](o:;vF2F\\s7$$\"3#>++v$ )[op)F2F_s7$$\"3p++DJnhL$*F2Fbs7$$\"3W*****\\i%Qq**F2Fes7$$\"3Z7y]bB<, 5F/$!3Gv=#\\Ww#))**F27$$\"3+Dc^[iI05F/$!3+]P%[^Pp%**F27$$\"3`PM_T,W45F /$!3rCcw%e)f0**F27$$\"31]7`MSd85F/$!3W*\\(oa'fU')*F27$$\"36voa?=%=-\"F /$!3))[7`%z\"e\"y*F27$$\"3&**\\ilg4,.\"F/$!3a+]PMR!*)p*F27$$\"3%)\\Pfy ^kY5F/$!3k,D19#[N`*F27$$\"3%***\\i]2=j5F/$!3_++v$\\#>o$*F27$$\"3%**\\( o%*=D'4\"F/$!3]+]7`5[P!*F27$$\"3&****\\(QIKH6F/$!3[++]7'pnq)F27$$\"3#* *****\\4+p=\"F/$!3n+++]!**48)F27$$\"3#****\\7:xWC\"F/$!3'3++v[G_b(F27$ $\"37++]Zn%)o8F/$!3t)****\\_K:J'F27$$\"3y******4FL(\\\"F/$!36-+++HnE]F 27$$\"3#)****\\d6.B;F/$!3y,++D%)opPF27$$\"3(****\\(o3lW.xHF/$\"3'3Dc^$*>.x *F27$$\"3=]PMA$4[)HF/$\"3#=]PMA$4[)*F27$$\"3_7yv')zp))HF/$\"33D\"yv')z p))*F27$$\"3Gv=<^me#*HF/$\"3z_(=<^me#**F27$$\"3/Qfe:`Z'*HF/$\"3[!Qfe:` Z'**F27$$\"3P+++!)RO+IF/$!3C'******>gj***F27$$\"3A++D;:*R1$F/$!3'y*** \\P[3g$*F27$$\"30++]_!>w7$F/$!3[*****\\Z4Qs)F27$$\"3O++v)Q?QD$F/$!3_'* **\\7hzhuF27$$\"3G+++5jypLF/$!3;(*******o8-jF27$$\"3<++]Ujp-NF/$!3I)** **\\dOI(\\F27$$\"3++++gEd@OF/$!3+++++MF%y$F27$$\"39++v3'>$[PF/$!3d)*** \\7R!o^#F27$$\"37++D6EjpQF/$!3o)***\\()Qn.8F27$$\"\"%F*F+-%*THICKNESSG 6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F^bl-%&COLORG6&%$RGBG$Feal!\"\"F+$\"\" \"F*-%%VIEWG6$;F(Fdal%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 43.000000 38.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "The coefficients of the sine terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 48 "T he coefficients of the sine terms are given by:" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "b[k] = 2*Int(f(x)*sin*k*Pi*x,x = 0 . . 1);" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\"-%$IntG6$*,-%\"fG6#%\"xGF*%$sinGF *F'F*%#PiGF*F2F*/F2;\"\"!F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Int(x*sin*k*Pi*x,x = 0 .. 1);" " 6#/%!G*&\"\"#\"\"\"-%$IntG6$*,%\"xGF'%$sinGF'%\"kGF'%#PiGF'F,F'/F,;\" \"!F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " } {XPPEDIT 18 0 "Int(x*sin*k*Pi*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6 $*,%\"xG\"\"\"%$sinGF)%\"kGF)%#PiGF)F(F)F(-F%6$*&%\"uGF)-%!G6#*&%#dvGF )%#dxG!\"\"F)F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x;" "6#/% \"uG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*k*Pi*x/(k*Pi); " "6#/%\"vG,$*,%$cosG\"\"\"%\"kGF(%#PiGF(%\"xGF(*&F)F(F*F(!\"\"F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the integra tion by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int( v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F) %\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b[ k] = 2*``(-x*cos*k*Pi*x/(k*Pi));" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\"-%!G6# ,$*.%\"xGF*%$cosGF*F'F*%#PiGF*F0F**&F'F*F2F*!\"\"F4F*" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWIS EG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-2*Int (-cos*k*Pi*x/(k*Pi),x = 0 .. 1);" "6#,$*&\"\"#\"\"\"-%$IntG6$,$*,%$cos GF&%\"kGF&%#PiGF&%\"xGF&*&F-F&F.F&!\"\"F1/F/;\"\"!F&F&F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = -2*x*cos*k*Pi*x/(k*Pi)+2*``(sin*k*Pi*x/(k^2*Pi^ 2));" "6#/%!G,&*0\"\"#\"\"\"%\"xGF(%$cosGF(%\"kGF(%#PiGF(F)F(*&F+F(F,F (!\"\"F.*&F'F(-F$6#*,%$sinGF(F+F(F,F(F)F(*&F+F'F,F'F.F(F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIE CEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = -2*cos*k*Pi/(k*Pi)+0;" "6#/%!G,&*,\"\"#\"\"\"%$cosGF(%\"kGF(%#PiGF(* &F*F(F+F(!\"\"F-\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -2/(k*Pi)*( -1)^k;" "6#/%!G,$*(\"\"#\"\"\"*&%\"kGF(%#PiGF(!\"\"),$F(F,F*F(F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*(-1)^(k+1)/(k*Pi)" "6#/%!G*(\"\"# \"\"\"),$F'!\"\",&%\"kGF'F'F'F'*&F,F'%#PiGF'F*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " } {XPPEDIT 18 0 "PIECEWISE([2/(k*Pi), `if k is odd`],[-2/(k*Pi), `if k i s even`]);" "6#-%*PIECEWISEG6$7$*&\"\"#\"\"\"*&%\"kGF)%#PiGF)!\"\"%,if ~k~is~oddG7$,$*&F(F)*&F+F)F,F)F-F-%-if~k~is~evenG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [b[k], `|`, 2/Pi, -1/Pi, 2/3/Pi, -1/(2*Pi), 2/5/Pi, -1/(3*Pi), 2/7/Pi, -1/(4 *Pi), 2/9/Pi, -1/(5*Pi)]])" "6#-%'matrixG6#7$7.%\"kG%\"|grG\"\"\"\"\"# \"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"bG6#F(F)*&F+F*%#PiG!\"\", $*&F*F*F9F:F:*(F+F*F,F:F9F:,$*&F*F**&F+F*F9F*F:F:*(F+F*F.F:F9F:,$*&F*F **&F,F*F9F*F:F:*(F+F*F0F:F9F:,$*&F*F**&F-F*F9F*F:F:*(F+F*F2F:F9F:,$*&F *F**&F.F*F9F*F:F:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "interface(showassumed=0): k := 'k': assume(k,integer):\nb[k]=2*Int(x*sin(k*Pi*x),x=0..1);\n``=val ue(rhs(%));\nbb := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k, k=1..10)],[b[k],`|`,seq(bb(k),k=1..10)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$*&\"\"#\"\"\"-%$IntG6$*&%\"xGF+-%$sin G6#*(F'F+%#PiGF+F0F+F+/F0;\"\"!F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\"%#k|irG!\"\"%#PiGF*)F*F)F(F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7$7.%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\" &\"\"'\"\"(\"\")\"\"*\"#57.&%\"bG6#F(F),$*&F+F*%#PiG!\"\"F*,$*&F*F*F:F ;F;,$*(F+F*F,F;F:F;F*,$*&F*F**&F+F*F:F*F;F;,$*(F+F*F.F;F:F;F*,$*&F*F** &F,F*F:F*F;F;,$*(F+F*F0F;F:F;F*,$*&F*F**&F-F*F:F*F;F;,$*(F+F*F2F;F:F;F *,$*&F*F**&F.F*F:F*F;F;Q)pprint266\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " is the sine seri es: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(x) = Sum(` `(-2*(-1)^k/(k*Pi))*sin*k*Pi*x,k = 1 .. infinity);" "6#/-%\"FG6#%\"xG- %$SumG6$*,-%!G6#,$*(\"\"#\"\"\"),$F2!\"\"%\"kGF2*&F6F2%#PiGF2F5F5F2%$s inGF2F6F2F8F2F'F2/F6;F2%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ 2*sin*Pi*x/Pi-sin*2*Pi*x/Pi+2*sin*3*Pi*x/(3*Pi)-sin*4*Pi*x/(2*Pi)+2*si n*5*Pi*x/(5*Pi)-sin*6*Pi*x/(3*Pi)+2*sin*7*Pi*x/(7*Pi)-sin*8*Pi*x/(4*Pi )+` . . . `;" "6#/%!G,4*,\"\"#\"\"\"%$sinGF(%#PiGF(%\"xGF(F*!\"\"F(*,F )F(F'F(F*F(F+F(F*F,F,*.F'F(F)F(\"\"$F(F*F(F+F(*&F/F(F*F(F,F(*,F)F(\"\" %F(F*F(F+F(*&F'F(F*F(F,F,*.F'F(F)F(\"\"&F(F*F(F+F(*&F5F(F*F(F,F(*,F)F( \"\"'F(F*F(F+F(*&F/F(F*F(F,F,*.F'F(F)F(\"\"(F(F*F(F+F(*&F;F(F*F(F,F(*, F)F(\"\")F(F*F(F+F(*&F2F(F*F(F,F,%(~.~.~.~GF(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The foll owing pictures compare the graphs of some truncated Fourier series wit h the graph of the function " }{XPPEDIT 18 0 "f_(x);" "6#-%#f_G6#%\"xG " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 215 "f_ := x -> x-2*floor(x/2+1/2):\nFS := (x,n) - > sum(-2*(-1)^k/(k*Pi)*sin(k*Pi*x),k=1..n);\nplot([f_(x),FS(x,1),FS(x, 2),FS(x,3),FS(x,4),FS(x,5)],x=-2..4,\n color=[black,red,blue,green,m agenta,coral],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF)-%$sumG6$,$*,\"\"#\"\"\") !\"\"%\"kGF3F6F5%#PiGF5-%$sinG6#*(F6F3F7F39$F3F3F5/F6;F39%F)F)F)" }} {PARA 13 "" 1 "" {GLPLOT2D 622 280 280 {PLOTDATA 2 "6*-%'CURVESG6%7^p7 $$!\"#\"\"!$F*F*7$$!3!******\\2<#p=!#<$\"30,++]#HyI\"!#=7$$!31++D^NUb< F/$\"3a****\\([kdW#F27$$!36++]K3XF;F/$\"3*))****\\n\"\\DPF27$$!3%)**** \\F)H')\\\"F/$\"3b,++DiUCF2Fcq7$$!3B++]7YY08F2Ffq7$$\"3%z-+++XDn%!#?Fiq7$$\"3C++ ++y?#>\"F2F]r7$$\"3h****\\(3wY_#F2F`r7$$\"3F)******HOTq$F2Fcr7$$\"3I,+ ](3\">)*\\F2Ffr7$$\"3_,+]isVIiF2Fir7$$\"3&=++](o:;vF2F\\s7$$\"3#>++v$) [op)F2F_s7$$\"3p++DJnhL$*F2Fbs7$$\"3W*****\\i%Qq**F2Fes7$$\"3Z7y]bB<,5 F/$!3Gv=#\\Ww#))**F27$$\"3+Dc^[iI05F/$!3+]P%[^Pp%**F27$$\"3`PM_T,W45F/ $!3rCcw%e)f0**F27$$\"31]7`MSd85F/$!3W*\\(oa'fU')*F27$$\"36voa?=%=-\"F/ $!3))[7`%z\"e\"y*F27$$\"3&**\\ilg4,.\"F/$!3a+]PMR!*)p*F27$$\"3%)\\Pfy^ 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" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f := \+ x -> x:\n'f(x)'=f(x);\nFourierSeries(f(x),x=-1..1,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 221 "f := x -> x:\n'f(x)'=f(x);\nFourierSeries(f (x),x=0..1,type=sine,numterms=10,info=1):\nF := unapply(%,x):\n'F(x)'= F(x);\nf_ := x -> x-2*floor(x/2+1/2):\nplot([f_(x),F(x)],x=-2..4,numpo ints=100,color=[black,red],linestyle=[3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%:k ~th~sine~coefficient~..~G/,$**\"\"#\"\"\",&-%$sinG6#*&%\"kGF(%#PiGF(! \"\"*(F.F(F/F(-%$cosGF,F(F(F(F.!\"#F/F4F0,$**F'F(F.F0F/F0)F0F.F(F0" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,6*(\"\"#\"\"\"%#PiG!\" \"-%$sinG6#*&F,F+F'F+F+F+*&F,F--F/6#,$*(F*F+F,F+F'F+F+F+F-*&#F*\"\"$F+ *&F,F--F/6#,$*(F9F+F,F+F'F+F+F+F+F+*&#F+F*F+*&F,F--F/6#,$*(\"\"%F+F,F+ F'F+F+F+F+F-*&#F*\"\"&F+*&F,F--F/6#,$*(FIF+F,F+F'F+F+F+F+F+*&#F+F9F+*& F,F--F/6#,$*(\"\"'F+F,F+F'F+F+F+F+F-*&#F*\"\"(F+*&F,F--F/6#,$*(FYF+F,F +F'F+F+F+F+F+*&#F+FFF+*&F,F--F/6#,$*(\"\")F+F,F+F'F+F+F+F+F-*&#F*\"\"* F+*&F,F--F/6#,$*(FcoF+F,F+F'F+F+F+F+F+*&#F+FIF+*&F,F--F/6#,$*(\"#5F+F, F+F'F+F+F+F+F-" }}{PARA 13 "" 1 "" {GLPLOT2D 540 191 191 {PLOTDATA 2 " 6&-%'CURVESG6%7as7$$!\"#\"\"!$F*F*7$$!3(pppp4!fO>!#<$\"3w-....*4M'!#>7 $$!3KRRRzuT\")=F/$\"3q1111_#e=\"!#=7$$!3`ggg+,P>=F/$\"3v%RRR**)H1=F87$ $!3]^^^J9\"pv\"F/$\"3+&[[[o&)3V#F87$$!3[ddd<'\\Zp\"F/$\"3;DCCCQ]_IF87$ $!3&)yyyew6P;F/$\"3^6777M#)GOF87$$!3kjjj.JWx:F/$\"3ejjjj*obA%F87$$!3=L LL$fFd^\"F/$\"3AommmSsU[F87$$!3[XXX0+@a9F/$\"3;XXXX***yX&F87$$!3w%[[[g J4R\"F/$\"3G_^^^Ro!4'F87$$!37:::bf>N8F/$\"3w[[[[//[mF87$$!3]aaa93Xs7F/ $\"3+baaa=\\vsF87$$!3SOOOO![%47F/$\"3(fjjjj>b!zF87$$!3]XXX&=L([6F/$\"3 )\\aaa9oE^)F87$$!3KCCC%G)f$4\"F/$\"3mcdddr,k!*F87$$!31444>z\"31\"F/$\" 3U44443#=R*F87$$!3y$RRRbP!G5F/$\"3AigggWi>(*F87$$!3PRRR>H:95F/$\"3M111 13Ze)*F87$$!3t%[[[Go-+\"F/$\"3m_^^^rJ(***F87$$!3MHIIbqK&)**F8Fiq7$$!3K 677i7(z'**F8F\\r7$$!3I$RR*oah]**F8F_r7$$!3Hvvvv'fK$**F8Fbr7$$!3ERRR*3[ &)*)*F8Fer7$$!39-...l$Q')*F8Fhr7$$!3'*GIIILT%z*F8F[s7$$!3\"pvvv:!*\\s* F8F^s7$$!3:UUUUv'>S*F8Fas7$$!3SFFFF\\%*y!*F8Fds7$$!3)=...V&32&)F8Fgs7$ $!3/FFFF\\mzyF8Fjs7$$!3OMOOOA@#G(F8F]t7$$!3aGIIII$)emF8F`t7$$!37>@@@jP '3'F8Fct7$$!3_CCCCW!*oaF8Fft7$$!3cfgggY_F[F8Fit7$$!3=!444p,#pUF8F\\u7$ $!3!yyyye(>mOF8F_u7$$!3)oaaaMNK/$F8Fbu7$$!3(ysssK)yLCF8Feu7$$!3n++++q6 W=F8Fhu7$$!3AFFFFvQ*=\"F8F[v7$$!3%*pddd(R3,'F2F^v7$$\"3ORNOOOc0F!#?Fav 7$$\"3GV[[[[VifF2Fev7$$\"3,\\XXXh^=7F8Fhv7$$\"3?BCCCi-/=F8F[w7$$\"3!fO OO'H0;CF8F^w7$$\"3Fpppp4U9IF8Faw7$$\"3%o\"===o&3k$F8Fdw7$$\"33&RRR**)= WUF8Fgw7$$\"3%[[[[G$=h[F8Fjw7$$\"3E_^^^&oIZ&F8F]x7$$\"3pljjjvKNgF8F`x7 $$\"3!QCCC%)R(zmF8Fcx7$$\"3!ommm')4hD(F8Ffx7$$\"3cFFFF0kqyF8Fix7$$\"3k \"===yC)e%)F8F\\y7$$\"3/4111y#>6*F8F_y7$$\"3[@@@@up%R*F8Fby7$$\"3#Rjjj .nun*F8Fey7$$\"3&evvv0Sz$)*F8Fhy7$$\"3wxyyyIT)***F8F[z7$$\"3FRRk?s%=+ \"F/$!3E21c$zF:)**F87$$\"3x!44M8`Q+\"F/$!3I#44fmo9'**F87$$\"3FUU.\"F/$!3Qyyyy3k!o*F87$$\"3oppp fm=h5F/$!3=....M8)Q*F87$$\"3?FFF2uV!4\"F/$!3+GFFFfi&4*F87$$\"3Q444\\)G W:\"F/$!3I1444:rb%)F87$$\"3G@@@TEk47F/$!3<(yyyetN!zF87$$\"3mXXX0Z_s7F/ $!3UVXXXHvusF87$$\"3]:::vcEL8F/$!3%\\%[[[KMnmF87$$\"3QCCCkp'RR\"F/$!35 cddd.LggF87$$\"3KCCCW\\Wa9F/$!3!ovvvb]bX&F87$$\"3&)ppp\\`a7:F/$!3Y,... lau[F87$$\"3mmmmEMNv:F/$!3KLLLLdYYUF87$$\"3I===yS@N;F/$!3-<===#fyk$F87 $$\"3G:::&f9#)p\"F/$!31Z[[[S&y,$F87$$\"3-%RRRBX_v\"F/$!3yfgggwaZCF87$$ \"3B111mZF==F/$!3rPRRRBD<=F87$$\"3_aaaugky=F/$!3'[XXXDRN@\"F87$$\"3]dd dP/()Q>F/$!32]UUUi&H6'F27$$\"3->===]x,?F/$\"3Q@!>===]x\"Fcv7$$\"3)QLLL V7(f?F/$\"3)pQLLLV7(fF27$$\"3'>:::F[!>@F/$\"3e>:::F[!>\"F87$$\"3QFFF(3 iX=#F/$\"3#QFFF(3iX=F87$$\"3W@@@\"[zQC#F/$\"3T9777[zQCF87$$\"3jFFFFEa/ BF/$\"3GwsssiUXIF87$$\"3YCCC%[NiO#F/$\"3qWUUU[NiOF87$$\"3o&[[[AVHU#F/$ \"3xc[[[AVHUF87$$\"3O+++S=R$[#F/$\"3].+++%=R$[F87$$\"3fOOO;.QVDF/$\"3& fOOO;.QV&F87$$\"3=SRRR$\\vg#F/$\"3!=SRRR$\\vgF87$$\"3QUUUAdedd<%3o%)F87$$\"3!)ppp4-a4HF/$\"3/)pp p4-a4*F87$$\"3WFFF()H.THF/$\"3Russs)H.T*F87$$\"33&[[[wDD(HF/$\"3v][[[w DD(*F87$$\"3y[[)f?*ozHF/$\"3m([[)f?*oz*F87$$\"3Y777ZE&o)HF/$\"3cC@@rk_ o)*F87$$\"33%R*onVV!*HF/$\"3!3%R*onVV!**F87$$\"3qvvD)3;S*HF/$\"3.ddd#) 3;S**F87$$\"3_m;a[p!e*HF/$\"39lmT&[p!e**F87$$\"3Kdd#)3yf(*HF/$\"3EtvD) 3yf(**F87$$\"39[)4\"p')Q**HF/$\"3Q\"[)4\"p')Q***F87$$\"3RRRRH&z6+$F/$! 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" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " is an " }{TEXT 260 3 "odd" } {TEXT -1 38 " function, so its Fourier series is a " }{TEXT 260 11 "si ne series" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "The Fouri er sine series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 50 " is the same as that of the the periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 11 " given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = x-x^3;" "6#/-%#f_G6#%\"xG, &F'\"\"\"*$F'\"\"$!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1 <= x;" "6 #1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f_(x);" "6#-%#f_G6#%\"xG" }{TEXT -1 28 " \+ is periodic with period 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "f := x -> x-x^3:\n'f(x)'=f(x);\nf_ := x -> f(x-2*floor((x+1)/2)):\n'f_(x)'='f(x-2*floor((x+1)/2))';\nplo t(f_(x),x=-2..5,color=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&F'\"\"\"*$)F'\"\"$F)!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*&\"\" #F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,F.F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 623 208 208 {PLOTDATA 2 "6'-%'CURVESG6#7ct7$$!\"#\"\"!$F*F*7 $$!3vmmTg*4P#>!#<$\"3#*R**Hd=g%e(!#>7$$!3GLL$3#*>u%=F/$\"3Mf*R36z-\\\" !#=7$$!3)**\\7`OS5y\"F/$\"3GdL*\\#)>Y3#F87$$!3om;z43m9Ix$F87$$ !3OL3x@tUr9F/$\"3K*H)pfC&*3QF87$$!3WL$3_#4k_9F/$\"3@v\\N\\ApLQF87$$!3^ LekGX&QV\"F/$\"3_/va:8%o%QF87$$!3PLL3K\"o]T\"F/$\"3UH'pF%=+[QF87$$!3% \\P%)R-rjR\"F/$\"363h'3Hfo$QF87$$!3a;a)e\"Rnx8F/$\"3day'[nbI\"QF87$$!3 Meky2o(*e8F/$\"3o;^fC))>wPF87$$!3$**\\(o*pz-M\"F/$\"3WV\"o)pl*es$F87$$ !3L$e*[$[&)GI\"F/$\"3K([Kj;(Q$e$F87$$!3]m;Hn7\\l7F/$\"3AuG#G5!R#Q$F87$ $!37LL3U?#3B\"F/$\"3Gw#ftjb59$F87$$!3'***\\(o\"G:'>\"F/$\"3\\#\\.i7]U% GF87$$!3#omm;f$[h6F/$\"33iJ2=LZ*[#F87$$!3WL$ekO9o7\"F/$\"33Afoh\\Au?F8 7$$!3++](=R;44\"F/$\"3CuTY'3pyd\"F87$$!3cm;H<%=]0\"F/$\"3:))Q_!)HA65F8 7$$!38L$3FW?\">5F/$\"3$Qgk'p,6:PF27$$!3:***\\7oCA$)*F8$!3=%3xb'H`rKF27 $$!3%GL$e9t'4Y*F8$!3BrNIJpkC**F27$$!3Ulm\"z%*4(*3*F8$!3M55bGY`z:F87$$! 3+)**\\7e_%=()F8$!3W.5EsLV\"4#F87$$!3qJLe9_>Z$)F8$!3gp;q$>J7`#F87$$!3' em;athqg(F8$!3t&pOoy`]?$F87$$!3-++Dc#Gp'oF8$!3i;&Q`e[)GOF87$$!3=mTgF\" pi['F8$!3SulcpiQdPF87$$!3MK$e*)**4c5'F8$!3@,M'fzJ&HQF87$$!3'[TNYV!G:fF 8$!3]y#RIr\"\\XQF87$$!3])\\7.(3&\\s&F8$!35gVLYYf[QF87$$!38#e*)fI@Y`&F8 $!3m9V2\"Ga#RQF87$$!3lkmmT(**z&>Q;O$F87$$!3[km\"z>]#[KF8$!3R#= zz6Bb!HF87$$!3!p****\\sZL\\#F8$!3A7\"\\0rT$QBF87$$!38)*****\\bMN1!**HF87$$\"3)eLLeRWg0% F8$\"3wVnd3Ww)Q$F87$$\"3gOLLLCCCZF8$\"3!Rq:_'R')pOF87$$\"3r-v=&>$QF87$$\"3-,+vo()yyiF8$\"3e#eXI/!\\.QF87$$\"3#pmTgA2o' pF8$\"3EjT\")*fp`e$F87$$\"3!GLLLoD[l(F8$\"375k?cnPpJF87$$\"3`+voH$eA.) 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" }} {PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "Int((x-x^3)*sin*k*P i*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*,,&%\"xG\"\"\"*$F)\"\"$!\" \"F*%$sinGF*%\"kGF*%#PiGF*F)F*F)-F%6$*&%\"uGF*-%!G6#*&%#dvGF*%#dxGF-F* F)" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x-x^3;" "6#/%\"uG,&%\"x G\"\"\"*$F&\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*k* Pi*x/(k*Pi);" "6#/%\"vG,$*,%$cosG\"\"\"%\"kGF(%#PiGF(%\"xGF(*&F)F(F*F( !\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "Hence using th e integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#d xG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" } {TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b[k] = 2*``(-(x-x^3)*cos*k*Pi*x/(k*Pi));" "6#/&%\"bG6#%\"kG*&\" \"#\"\"\"-%!G6#,$*.,&%\"xGF**$F1\"\"$!\"\"F*%$cosGF*F'F*%#PiGF*F1F**&F 'F*F6F*F4F4F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, \+ ``],[0, ``]);" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-2*Int((1-3*x^2)*``(-cos*k*Pi*x/(k*Pi)),x = 0 .. 1);" "6#,$*&\"\"#\"\"\"-%$IntG6$*&,&F&F&*&\"\"$F&*$%\"xGF%F&!\"\"F &-%!G6#,$*,%$cosGF&%\"kGF&%#PiGF&F/F&*&F7F&F8F&F0F0F&/F/;\"\"!F&F&F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0+2/(k*Pi);" "6#/%!G,&\"\"!\"\"\"* &\"\"#F'*&%\"kGF'%#PiGF'!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(( 1-3*x^2)*cos*k*Pi*x,x = 0 .. 1);" "6#-%$IntG6$*,,&\"\"\"F(*&\"\"$F(*$% \"xG\"\"#F(!\"\"F(%$cosGF(%\"kGF(%#PiGF(F,F(/F,;\"\"!F(" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int((1-3*x^2) *cos*k*Pi*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*,,&\"\"\"F)*&\"\"$ F)*$%\"xG\"\"#F)!\"\"F)%$cosGF)%\"kGF)%#PiGF)F-F)F--F%6$*&%\"uGF)-%!G6 #*&%#dvGF)%#dxGF/F)F-" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = 1-3* x^2" "6#/%\"uG,&\"\"\"F&*&\"\"$F&*$%\"xG\"\"#F&!\"\"" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "v = sin*k*Pi*x/(k*Pi);" "6#/%\"vG*,%$sinG\"\"\"%\" kGF'%#PiGF'%\"xGF'*&F(F'F)F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the integrat ion by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v *``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)% \"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k]=2/ (k*Pi)" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\"*&F'F*%#PiGF*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((1-3*x^2)*cos*k*Pi*x,x = 0 .. 1);" "6#-%$IntG6 $*,,&\"\"\"F(*&\"\"$F(*$%\"xG\"\"#F(!\"\"F(%$cosGF(%\"kGF(%#PiGF(F,F(/ F,;\"\"!F(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 2/(k*Pi);" "6#/%!G*&\"\"#\"\"\"*&%\"kGF'%#PiGF'!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-3*x^2)*``(sin*k*Pi*x/(k*Pi));" "6#*&,&\"\"\"F%*&\"\"$F%*$%\"xG\"\"#F%!\"\"F%-%!G6#*,%$sinGF%%\"kGF%%# PiGF%F)F%*&F1F%F2F%F+F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F( " }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 2/(k*Pi)" "6#,$*&\"\"#\"\"\"*&%\"k GF&%#PiGF&!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(-6*x*``(sin*k*P i*x/(k*Pi)),x = 0 .. 1);" "6#-%$IntG6$,$*(\"\"'\"\"\"%\"xGF)-%!G6#*,%$ sinGF)%\"kGF)%#PiGF)F*F)*&F0F)F1F)!\"\"F)F3/F*;\"\"!F)" }{TEXT -1 2 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0+12/(k^2*P i^2);" "6#/%!G,&\"\"!\"\"\"*&\"#7F'*&%\"kG\"\"#%#PiGF,!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sin*k*Pi*x,x = 0 .. 1);" "6#-%$IntG6$*, %\"xG\"\"\"%$sinGF(%\"kGF(%#PiGF(F'F(/F';\"\"!F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int(x*sin*k*Pi*x,x) \+ = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*,%\"xG\"\"\"%$sinGF)%\"kGF)%#PiGF )F(F)F(-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxG!\"\"F)F(" }{TEXT -1 8 ", whe re " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "v = -cos*k*Pi*x/(k*Pi);" "6#/%\"vG,$*,%$cosG\"\"\"%\"kG F(%#PiGF(%\"xGF(*&F)F(F*F(!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the integration by parts formula: " } {XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$In tG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6 $*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 9 ", we have" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k] = 12/(k^2*Pi^2);" "6#/&%\"b G6#%\"kG*&\"#7\"\"\"*&F'\"\"#%#PiGF,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sin*k*Pi*x,x = 0 .. 1);" "6#-%$IntG6$*,%\"xG\"\"\"%$sinGF( %\"kGF(%#PiGF(F'F(/F';\"\"!F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ 12/(k^2*Pi^2);" "6#/%!G*&\"#7\"\"\"*&%\"kG\"\"#%#PiGF*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``((-x*cos*k*Pi*x)/(k*Pi));" "6#-%!G6#*&,$*,% \"xG\"\"\"%$cosGF*%\"kGF*%#PiGF*F)F*!\"\"F**&F,F*F-F*F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWI SEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-12/( k^2*Pi^2);" "6#,$*&\"#7\"\"\"*&%\"kG\"\"#%#PiGF)!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(-cos*k*Pi*x/(k*Pi),x = 0 .. 1);" "6#-%$IntG6$,$ *,%$cosG\"\"\"%\"kGF)%#PiGF)%\"xGF)*&F*F)F+F)!\"\"F./F,;\"\"!F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 12/(k^2*Pi^2);" "6#/%!G*&\"#7\"\" \"*&%\"kG\"\"#%#PiGF*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-x*cos* k*Pi*x/(k*Pi))-12/(k^3*Pi^3);" "6#,&-%!G6#,$*.%\"xG\"\"\"%$cosGF*%\"kG F*%#PiGF*F)F**&F,F*F-F*!\"\"F/F**&\"#7F**&F,\"\"$F-F3F/F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(sin*k*Pi*x/(k*Pi));" "6#-%!G6#*,%$sinG\"\"\"% \"kGF(%#PiGF(%\"xGF(*&F)F(F*F(!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "P IECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F( F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -12/(k^3*Pi^3)*cos*k*P i-0;" "6#/%!G,&*,\"#7\"\"\"*&%\"kG\"\"$%#PiGF+!\"\"%$cosGF(F*F(F,F(F- \"\"!F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -12/(k^3*Pi^3)*(-1)^k; " "6#/%!G,$*(\"#7\"\"\"*&%\"kG\"\"$%#PiGF+!\"\"),$F(F-F*F(F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(12/(k^3*Pi^3))*(-1)^(k+1);" "6#/%!G*&-F$ 6#*&\"#7\"\"\"*&%\"kG\"\"$%#PiGF-!\"\"F*),$F*F/,&F,F*F*F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([-12/(k^3*Pi^3), `if k is even`], [12/(k^3*Pi^3), `if k is odd`]);" "6#/%!G-%*PIECEWISEG6$7$,$*&\"#7\"\" \"*&%\"kG\"\"$%#PiGF/!\"\"F1%-if~k~is~evenG7$*&F+F,*&F.F/F0F/F1%,if~k~ is~oddG" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4 , 5, 6, 7, 8], [b[k], `|`, 12/Pi^3, -3/2/Pi^3, 4/9/Pi^3, -3/16/Pi^3, 1 2/125/Pi^3, -1/(18*Pi^3), 12/343/Pi^3, -3/128/Pi^3]])" "6#-%'matrixG6# 7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&%\"bG6#F(F )*&\"#7F**$%#PiGF,!\"\",$*(F,F*F+F:*$F9F,F:F:*(F-F*\"\"*F:*$F9F,F:,$*( F,F*\"#;F:*$F9F,F:F:*(F7F*\"$D\"F:*$F9F,F:,$*&F*F**&\"#=F**$F9F,F*F:F: *(F7F*\"$V$F:*$F9F,F:,$*(F,F*\"$G\"F:*$F9F,F:F:" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "interface(showassumed=0): k := 'k': assume(k,integer):\nb[k]=2*In t((x-x^3)*sin(k*Pi*x),x=0..1);\n``=value(rhs(%));\nbb := unapply(rhs(% ),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..8)],[b[k],`|`,seq(bb(k),k=1 ..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$*&\"\"# \"\"\"-%$IntG6$*&,&%\"xGF+*$)F1\"\"$F+!\"\"F+-%$sinG6#*(F'F+%#PiGF+F1F +F+/F1;\"\"!F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"#7\"\" \")!\"\",&F(F(%#k|irGF(F(F,!\"$%#PiGF-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"( \"\")7,&%\"bG6#F(F),$*&\"#7F*%#PiG!\"$F*,$*(F,F*F+!\"\"F9F:F=,$*(F-F* \"\"*F=F9F:F*,$*(F,F*\"#;F=F9F:F=,$*(F8F*\"$D\"F=F9F:F*,$*&F*F**&\"#=F *)F9F,F*F=F=,$*(F8F*\"$V$F=F9F:F*,$*(F,F*\"$G\"F=F9F:F=Q)pprint276\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 21 " is the sine series: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(x) = Sum(``(12/(Pi^3*k^3))*(-1)^(k+1)*sin*k*Pi* x,k = 1 .. infinity);" "6#/-%\"FG6#%\"xG-%$SumG6$*.-%!G6#*&\"#7\"\"\"* &%#PiG\"\"$%\"kGF4!\"\"F1),$F1F6,&F5F1F1F1F1%$sinGF1F5F1F3F1F'F1/F5;F1 %)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture compares the graphs of s ome truncated Fourier series with the graph of the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "f := x - > x-x^3:\nf_ := x -> f(x-2*floor((x+1)/2)):\nFS := (x,n) -> sum(-12*(- 1)^k/(k^3*Pi^3)*sin(k*Pi*x),k=1..n);\nplot([f_(x),FS(x,1),FS(x,2),FS(x ,3)],x=-2..3,\n color=[black,red,blue,green],linestyle=[3,1$3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG %&arrowGF)-%$sumG6$,$*,\"#7\"\"\")!\"\"%\"kGF3F6!\"$%#PiGF7-%$sinG6#*( F6F3F8F39$F3F3F5/F6;F39%F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 615 243 243 {PLOTDATA 2 "6(-%'CURVESG6%7cs7$$!\"#\"\"!$F*F*7$$!3OLLe9r]X>!#<$ \"3x]k'z,2JV&!#>7$$!3smm;HU,\"*=F/$\"3GiJP*[7p2\"!#=7$$!3&**\\P4E+O%=F /$\"3UL\"o1kSd_\"F87$$!3SL$3FH'='z\"F/$\"35$Q2L!GZ`>F87$$!37+DcEV'Gu\" F/$\"3`!o$eH=M,CF87$$!3gmmTgBa*o\"F/$\"3:S+As]M0GF87$$!3YmmmTp'ej\"F/$ \"3/=%o!fg^eJF87$$!3amm\"H_\">#e\"F/$\"3S]iU+>u[MF87$$!30+D197xG:F/$\" 3)\\v8mQ$*em$F87$$!3ML$3_!4Nv9F/$\"3WIBU1$fB!QF87$$!35+D1R!pHY\"F/$\"3 #pS%p=-]@QF87$$!3imm\"H<(e]9F/$\"37HzLy6qNQF87$$!38L3x1`?Q9F/$\"37Zj1* G[[%QF87$$!3))**\\iSM#eU\"F/$\"3%*4Jfaw#)[QF87$$!3jm\"zWdTMT\"F/$\"3am ;jy`_ZQF87$$!3QLLL3(f5S\"F/$\"31na*[cF3%QF87$$!37+v=Uyn)Q\"F/$\"3\"*fz 4<.iGQF87$$!3km;/wfHw8F/$\"3l%f_*Q(*y5QF87$$!3_L$3_:8]K\"F/$\"3cThm4&z Xn$F87$$!3;+]PM.tt7F/$\"3YUK&QGf=V$F87$$!3[L$3xc$p?7F/$\"31%>>y1#>gIF8 7$$!3em;/,oln6F/$\"3E!R!e_#)*pb#F87$$!3t****\\iMAT6F/$\"3s4HEgHJaAF87$ $!35L$eR7!z96F/$\"3/Q!pf$\\i:>F87$$!3[mmT&yc$)3\"F/$\"3y(\\ht+E)R:F87$ $!3%)**\\(oWB>1\"F/$\"3!f,.@+3e7\"F87$$!31$e9\"4NtM5F/$\"35WCoMr'*)e'F 27$$!3]mTNrNa25F/$\"33\"4Zsb&o\"\\\"F27$$!3H)\\PfLON!)*F8$!3azgCxwB9QF 27$$!3eJLLepjJ&*F8$!3o6k48KW>()F27$$!3])**\\(=(eE0*F8$!3_sT,yy%Rj\"F87 $$!3Ulm;z/ot&)F8$!3=q)zKpR8F#F87$$!33KL3FWYM!)F8$!3)=z*\\Ys+[GF87$$!3u )****\\P[_\\(F8$!33)yBKm6XG$F87$$!3)*)*****\\#=Q&pF8$!3!H>'4H&e7f$F87$ $!3A*****\\7)Q7kF8$!3%*4ae7hpvPF87$$!3)*)***\\ig%>G'F8$!3@'\\'=T8\"H!Q F87$$!3w)*******R]^hF8$!36FSU\"G8P#QF87$$!3_)***\\P>1@gF8$!3-A,j-^BQQF 87$$!3Q*****\\()>1*eF8$!3k)*o8u*4m%QF87$$!3G++]7yj(f>a$f#F87$$!3kJLLLo#)RBF8$!3y<.(pBE<@#F87$$!3gl;a8:#eL[`3@\"F87$$!35omT&Q[*ztF2$ !3%H'zb,XvRtF27$$!3MSLLL3`lCF2$!3)[hBg2KSY#F27$$\"3LYLe9TOEHF2$\"3L$)= Q\"4eQ#HF27$$\"3+L+]i!f#=$)F2$\"3!\\7(pr@qg#)F27$$\"3m-+]7=EX8F8$\"3:m Ks9i\"4K\"F87$$\"3+-+v=xpe=F8$\"3Y3_IDU[%z\"F87$$\"3eO$eRA9WR#F8$\"3- \"yO8?PrD#F87$$\"3*QPu$F87$$\"3iL3-)Q84D&F8$\"3)Gv%* )[n7.QF87$$\"3QM$3_D1l_&F8$\"34b/U;deQQF87$$\"3K&3-))o-Vm&F8$\"3y[U&zS \\p%QF87$$\"39NeRA\"*4-eF8$\"3]$o#G@*f)[QF87$$\"3'\\e*)fb&*)RfF8$\"30p i')o-;WQF87$$\"3!fL$e*)>pxgF8$\"3i9b;jMpKQF87$$\"3Momm\"z+vb'F8$\"3RWO b+>sPPF87$$\"3w++v$f4t.(F8$\"33!yq.lr@b$F87$$\"31p;zWi^bvF8$\"3J\\Bk3u QUKF87$$\"3OPL$e*Gst!)F8$\"3[NpD8l'3\"GF87$$\"3\"*om\"H2\"34')F8$\"3$[ '>k/2NGAF87$$\"3Y+++]#RW9*F8$\"3ycIS5))y(\\\"F87$$\"3s+]i:5J1%*F8$\"3% =qP6aIP3\"F87$$\"3'4+]7y#=o'*F8$\"3U-e3r(*o4jF27$$\"3A,](oaa+$**F8$\"3 K[*RJ=mUQ\"F27$$\"3:++DJE>>5F/$!3w/%)*e4E(GPF27$$\"3=]P4r+`W5F/$!3yaaK Qg'*>$)F27$$\"3A+v$4^n)p5F/$!3;[&erq;VD\"F87$$\"3C]7y]\\?&4\"F/$!3k`5D '*)33k\"F87$$\"3F+]i!RU07\"F/$!3_rKzZ\"\\C*>F87$$\"39+vo/#3o<\"F/$!3Uz _+>@g`EF87$$\"3+++v=S2L7F/$!3)o**H*)Q*QeJF87$$\"3;L$3_NJOG\"F/$!3uQgPH pR([$F87$$\"3Jmmm\"p)=M8F/$!3[/([;FSlq$F87$$\"3!)**\\il!z6O\"F/$!3'o&H yE5B\"y$F87$$\"3GLLeR%p\")Q\"F/$!3Z\"*G[u'**z#QF87$$\"3!**\\ilik;S\"F/ $!3\")Gyqp=GTQF87$$\"3wm;a8)f^T\"F/$!3U'fp?$f-[QF87$$\"3hL3_+]lG9F/$!3 !)GLBE$z$[QF87$$\"3B++](=]@W\"F/$!3iiT'o^*[UQF87$$\"3Y$ekyZ2mY\"F/$!3U \\U[]IQ;QF87$$\"3mm\"H#oZ1\"\\\"F/$!3Y!*>=qL8rPF87$$\"3*)\\Pfe?_::F/$! 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" }}{PARA 0 "" 0 "" {TEXT -1 29 "The tru ncated Fourier series " }{XPPEDIT 18 0 "Sum(b[k]*sin*k*Pi*x,k = 1 .. 2 0)" "6#-%$SumG6$*,&%\"bG6#%\"kG\"\"\"%$sinGF+F*F+%#PiGF+%\"xGF+/F*;F+ \"#?" }{TEXT -1 61 " appears to give a good match with the graph of th e function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "f := x -> x-x^3:\nf_ := x -> f(x-2*floor((x+1)/2)):\nFS := (x,n ) -> sum(12*(-1)^(k+1)/(k^3*Pi^3)*sin(k*Pi*x),k=1..n):\nplot([f_(x),FS (x,20)],x=-2..3,numpoints=80,\n color=[black,COLOR(RGB,.8,0,1)],lines tyle=[3,1],thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 557 195 195 {PLOTDATA 2 "6&-%'CURVESG6&7bs7$$!\"#\"\"!$F*F*7$$!3_a$ouz!yL>!#<$ \"3_$)y&oC$)Gf'!#>7$$!3e@xTBQ;w=F/$\"3[+^\\%*4P>7!#=7$$!3E#eE,^n8\"=F/ $\"3UfjHy_?>=F87$$!3cE,\"[oThu\"F/$\"3%*R^2(e')\\P#F87$$!3E'pbk'eA\"o \"F/$\"3k\"QI#*y7Q'GF87$$!3auz.$fS5i\"F/$\"3UGDi&ep`C$F87$$!3s75[GAse: 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E(>GU$F:7$$\"3PDJqX/%\\!**F1$\"35Eg#f**F1$\"3O9.W@%pb%>F:7$$\"3[\\PMF,%G(**F1$\"3] LC&*yUW(R\"F:7$$\"3uu=nj+U')**F1$\"3d1=$\\k\"o4vF47$$\"\"\"F*$!3p1P)o2 3(QwF--%+AXESLABELSG6$Q\"x6\"Q!Fhat-%'COLOURG6&%$RGBG$F*F*F^bt$\"*++++ \"!\")-%%VIEWG6$;F(F`at%(DEFAULTG" 1 2 0 1 10 0 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 -1 61 "Truncated Fourier series can be obtained \+ using the procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "f := x -> x-x^3:\n'f(x)'=f(x);\nFourierSeries(f(x),x=0..1,type=si ne,numterms=5):\nF := unapply(%,x):\n'F(x)'=F(x);\nplot(F(x),x=-2..3,n umpoints=80,color=COLOR(RGB,.8,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&F'\"\"\"*$)F'\"\"$F)!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,,*(\"#7\"\"\"%#PiG!\"$-%$sinG6#*& F,F+F'F+F+F+*&#\"\"$\"\"#F+*&F,F--F/6#,$*(F5F+F,F+F'F+F+F+F+!\"\"*&#\" \"%\"\"*F+*&F,F--F/6#,$*(F4F+F,F+F'F+F+F+F+F+*&#F4\"#;F+*&F,F--F/6#,$* (F>F+F,F+F'F+F+F+F+F;*&#F*\"$D\"F+*&F,F--F/6#,$*(\"\"&F+F,F+F'F+F+F+F+ F+" }}{PARA 13 "" 1 "" {GLPLOT2D 498 174 174 {PLOTDATA 2 "6'-%'CURVESG 6#7as7$$!\"#\"\"!$\"3#*3#HU)oP\\z!#M7$$!3_a$ouz!yL>!#<$\"3K$RJI\\ccp'! #>7$$!3e@xTBQ;w=F1$\"38eF_UVdG7!#=7$$!3E#eE,^n8\"=F1$\"3#*>8`o=(p\"=F: 7$$!3cE,\"[oThu\"F1$\"3mRQ,Z*GLO#F:7$$!3E'pbk'eA\"o\"F1$\"3K*[K>hmj&GF :7$$!3auz.$fS5i\"F1$\"3%oi8G;e3D$F:7$$!3s75[GAse:F1$\"3S/U%[Ezuc$F:7$$ !3xO\\fdt\\E:F1$\"3ec$o;\">!eo$F:7$$!3#3')3n[sU\\\"F1$\"3[P'RUo>.x$F:7 $$!3'Hj]Ss6#y9F1$\"31-&HA)*G'*z$F:7$$!330CRh4:i9F1$\"3x,%HQEl/#QF:7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 10 "No te that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " is an \+ " }{TEXT 260 3 "odd" }{TEXT -1 38 " function, so its Fourier series is a " }{TEXT 260 11 "sine series" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "The Fourier sine series of " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 50 " is the same as that of the the periodic fu nction " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 11 " given by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = x*sqrt(1-x^2);" "6#/-%#f_G6#%\"xG*&F'\"\" \"-%%sqrtG6#,&F)F)*$F'\"\"#!\"\"F)" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "- 1 <= x;" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\" \"" }{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 2. 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*&F'\"\"\"%#PiGF.F.F(**\"\"#F.F'!\"#F/F(-F*6$F.F-F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = Sum(b[k]*sin(k*Pi*x),k = 1 .. infinit y);" "6#/-%\"fG6#%\"xG-%$SumG6$*&&%\"bG6#%\"kG\"\"\"-%$sinG6#*(F/F0%#P iGF0F'F0F0/F/;F0%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k]= 2*BesselJ(1,k*Pi)/(k^2*Pi)-BesselJ(0,k*Pi)/k" "6#/&%\"bG6#%\"kG,&*(\" \"#\"\"\"-%(BesselJG6$F+*&F'F+%#PiGF+F+*&F'F*F0F+!\"\"F+*&-F-6$\"\"!*& F'F+F0F+F+F'F2F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 106 "The following picture compares the graph s of some truncated Fourier series with the graph of the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 275 "f := x - > x*sqrt(1-x^2):\nf_ := x -> f(x-2*floor((x+1)/2)):\nFS := (x,n)->sum( (2*BesselJ(1,k*Pi)/(k^2*Pi)-BesselJ(0,k*Pi)/k)*sin(k*Pi*x),k=1..n);\np lot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5)],x=-1.5..2.5,\n c olor=[black,red,blue,green,magenta,coral],linestyle=[3,1$5]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%& arrowGF)-%$sumG6$*&,&**\"\"#\"\"\"-%(BesselJG6$F4*&%\"kGF4%#PiGF4F4F9! \"#F:!\"\"F4*&-F66$\"\"!F8F4F9FA*=q7IV!#=7$$!3MLLL$Q6GT\"F*$\"3;zNL$3xC?cJ\"F*$\"3W%Hv)z*))**)\\F-7$$!3#)***\\7Y\"H%H\"F*$ \"33i8\\v4'***\\F-7$$!3EL3-oqi$G\"F*$\"39%=J$*))f#)*\\F-7$$!3qm;zuE'HF \"F*$\"3E79gzK#=*\\F-7$$!3!**\\i:G)Hi7F*$\"3?uc'z)=U!)\\F-7$$!3MLLL))Q j^7F*$\"3AUlltn!Q'\\F-7$$!3F-7$$!3WL$3--&*e,\"F*$\"3e&HM:L\\wu\"F-7$$!3,++vtA%4,\"F*$\"3!>,A 8DZ\"f9F-7$$!3!om\"HF&*)f+\"F*$\"3],ypJHI'3\"F-7$$!3OLL$3yO5+\"F*$\"3W gMQ0utZX!#>7$$!3smm\"zREZ)**F-$!3I:FY#G.k^&Far7$$!3#)****\\()\\3f**F-$ !33zi3'[j(***)Far7$$!3.ML3xNWL**F-$!36H2ZYz:W6F-7$$!38nmmm@!y!**F-$!3W &H))GB*HU8F-7$$!3WML$eM>l&)*F-$!3!e,_Et(oj;F-7$$!3l+++DlB0)*F-$!3U:F-7$$!3;MLL$)3n-(*F-$!3=t$p`q-%[BF-7$$!3onmmT_5+'*F-$!3!e/>U/$o(o #F-7$$!3qMLLeR(\\R*F-$!3>ao%=60$=KF-7$$!3i+++vE%)*=*F-$!3K%3%*[m,Ni$F- 7$$!3'ymm;a[bw)F-$!3=;E<$zv)=UF-7$$!3)RLL$3WDT$)F-$!3,$QX&[c#3g%F-7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " \+ is an " }{TEXT 260 3 "odd" }{TEXT -1 38 " function, so its Fourier ser ies is a " }{TEXT 260 11 "sine series" }{TEXT -1 17 ". Note also that \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " has " }{TEXT 260 6 "period" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\" %#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "The periodic \+ extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 32 " \+ is given by the same formula. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "f := x -> sin(sin(x)):\n'f(x )'=f(x);\nplot(f(x),x=-Pi..5*Pi,color=COLOR(RGB,.4,0,1),thickness=2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinG6#-F)F&" }} {PARA 13 "" 1 "" {GLPLOT2D 639 173 173 {PLOTDATA 2 "6'-%'CURVESG6#7gu7 $$!3*****4tk#fTJ!#<$!3=5KT_Kzzi!#E7$$!3()[oTyf()QIF*$!3K9?K2dcB5!#=7$$ !3I(pB&4$fh$HF*$!3frDQkezD?F37$$!3uX0jSEWLGF*$!3qb#F*$!3O;?a-]Ey_F37$$!3WZQ@uIBtBF *$!3&44&ph7_.kF37$$!3;XP2)*R@s@F*$!3$Gk!=VqJUtF37$$!35VO$>#\\>r>F*$!3w ab:`)*\\hzF37$$!3KS@ST\">+(=F*$!3QO]QQYSm\")F37$$!3KP1(3OV)o5l8F*$!3Ww#o \\o\\*)H)F37$$!3_IXoqqSk7F*$!31iow@A0a\")F37$$!3;Io(4;7P;\"F*$!3u!Ry(z %zb%zF37$$!3UxV(o=!)*p(*F3$!3![HsV!\\NrtF37$$!3:`/)RwQG!zF3$!3Kea*[#3` AlF37$$!3S)yD'p>_pfF3$!31[vL>!\\)H`F37$$!3mB6Fv^?OSF3$!3[%3/QO4t#QF37$ $!3i)>\"4yc[OIF3$!3Q4i6SRoXHF37$$!3gt7\"4=mn.#F3$!3=;op=&[*3?F37$$!3c[ 8t$oYq.\"F3$!3j*ey:&4ML5F37$$!3-`B9b'=Ft$!#?$!3?J0**=8qKPFct7$$\"33\\? 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Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$sinGF+%\"kGF+F*F+/F *;\"\"!%#PiG" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " } {XPPEDIT 18 0 "2/Pi" "6#*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sin(sin*x)*sin*k*x,x = 0 .. Pi);" "6#-%$IntG6$**-%$ sinG6#*&F(\"\"\"%\"xGF+F+F(F+%\"kGF+F,F+/F,;\"\"!%#PiG" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The e xpression for the sine coefficient " }{XPPEDIT 18 0 "b[k]" "6#&%\"bG6# %\"kG" }{TEXT -1 60 " given by Maple 9 cannot be evaluated for certain values of " }{TEXT 264 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "interface(s howassumed=0): k := 'k': assume(k,posint):\nb[k]=2/Pi*Int(sin(sin(x))* sin(k*x),x=0..Pi);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$,$-%$IntG6$*&-%$sinG6#-F/6#%\"xG\"\"\"-F/6#*&F'F4 F3F4F4/F3;\"\"!%#PiG*&\"\"#F4F;!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*0\"\"%\"\"\"%#PiG!\"\"-%$cosG6#,$*(\"\"#F*%#k|irGF(F)F(F( F(-%$sinGF-F(,&F(F*F1F(F*,&F(F(F1F(F*-%*hypergeomG6%7#F(7$,&#\"\"$F0F( *&F0F*F1F(F*,&F " 0 "" {MPLTEXT 1 0 65 "hypergeom([1],[3/2+1/ 2*k,3/2-1/2*k],-1/4);\nsubs(k=5,%);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*hypergeomG6%7#\"\"\"7$,&#\"\"$\"\"#F'*&F,!\"\"%\"kGF 'F',&F*F'*&F,F.F/F'F.#F.\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*hy pergeomG6%7#\"\"\"7$\"\"%!\"\"#F*F)" }}{PARA 8 "" 1 "" {TEXT -1 58 "Er ror, (in hypergeom) numeric exception: division by zero\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We can obtain clo sed formulas for the sine coefficients " }{XPPEDIT 18 0 "b[k]" "6#&%\" bG6#%\"kG" }{TEXT -1 36 " by determining them one at a time. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "for k from 1 to 20 do\n b[k] := simplify(2/Pi*int(sin(sin(x))*s in(k*x),x=0..Pi));\nend do:\nk := 'k':\nseq(b[k],k=1..7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6),$*&\"\"#\"\"\"-%(BesselJG6$F&F&F&F&\"\"!,&*&\" #9F&F'F&F&*&\"\")F&-F(6$F*F&F&!\"\"F*,&*&\"$E'F&F'F&F&*&\"$g$F&F0F&F2F *,&*&\"&MN(F&F'F&F&*&\"&)GUF&F0F&F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Notice \+ that the coefficients " }{XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" } {TEXT -1 6 " with " }{TEXT 265 1 "k" }{TEXT -1 6 " even " }{TEXT 269 3 "are" }{TEXT -1 7 " all 0." }}{PARA 0 "" 0 "" {TEXT -1 112 "However, there is still a problem if the expressions for the coefficients are \+ evaluated with 10 digit precision." }}{PARA 0 "" 0 "" {TEXT -1 11 "Bec ause of " }{TEXT 260 17 "subtraction error" }{TEXT -1 67 ", all signif icance is lost in the evaluation of the coefficient of " }{XPPEDIT 18 0 "sin(7*x)" "6#-%$sinG6#*&\"\"(\"\"\"%\"xGF(" }{TEXT -1 70 ", and oth er coefficients exhibit loss of precision to a lesser extent." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "evalf(73534*BesselJ(1,1));\nevalf(42288*BesselJ(0,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+xz'eB$!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+xz'eB$!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "This problem can be corrected by evaluating the coeffi cients with (much) higher precision." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "seq(evalf(evalf(b[k],120), 15),k=1..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "66$\"0n)*[r65!))!#:$\" \"!F'$\"0oLlzqE\"R!#;F&$\"0pCUga^*\\!#=F&$\"0it[j^Y+$!#?F&$\"0Q#)f.]) \\5!#AF&$\"0F1E\\8gR#!#DF&$\"0Ng*GNB^Q!#GF&$\"0p?W1j]f%!#JF&$\"0_1h8^2 B%!#MF&$\"0JBC)o&p4$!#PF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Alternatively, we can use numerical integration. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "for k from 1 to 20 do\n if irem(k,2)=0 then\n b[k] := \+ 0\n else\n b[k] := evalf[15](evalf[30](2/Pi*Int(sin(sin(x))*sin (k*x),x=0..Pi)));\n end if;\nend do:\nk := 'k':\nseq(b[k],k=1..20); " }}{PARA 12 "" 1 "" {XPPMATH 20 "66$\"0n)*[r65!))!#:\"\"!$\"0oLlzqE\" R!#;F&$\"0pCUga^*\\!#=F&$\"0it[j^Y+$!#?F&$\"0Q#)f.])\\5!#AF&$\"0F1E\\8 gR#!#DF&$\"0Ng*GNB^Q!#GF&$\"0p?W1j]f%!#JF&$\"0X1h8^2B%!#MF&$\"0jHP)o&p 4$!#PF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 22 "The Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "F(x) = Sum(b[k]*sin*k*x,k = 1 .. infinity);" "6#/-%\"FG 6#%\"xG-%$SumG6$**&%\"bG6#%\"kG\"\"\"%$sinGF0F/F0F'F0/F/;F0%)infinityG " }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 24 " where the coeffic ients " }{XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" }{TEXT -1 79 " can be \+ obtained by numerical integration as suggested in the subsection above ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "for k from 1 to 20 do\n if irem(k,2)=0 then\n \+ b[k] := 0\n else\n b[k] := evalf[15](evalf[30](2/Pi*Int(sin(si n(x))*sin(k*x),x=0..Pi)));\n end if;\nend do:\nk := 'k':\nadd(b[k]*s in(k*x),k=1..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6*&$\"0n)*[r65!) )!#:\"\"\"-%$sinG6#%\"xGF(F(*&$\"0oLlzqE\"R!#;F(-F*6#,$*&\"\"$F(F,F(F( F(F(*&$\"0pCUga^*\\!#=F(-F*6#,$*&\"\"&F(F,F(F(F(F(*&$\"0it[j^Y+$!#?F(- F*6#,$*&\"\"(F(F,F(F(F(F(*&$\"0Q#)f.])\\5!#AF(-F*6#,$*&\"\"*F(F,F(F(F( F(*&$\"0F1E\\8gR#!#DF(-F*6#,$*&\"#6F(F,F(F(F(F(*&$\"0Ng*GNB^Q!#GF(-F*6 #,$*&\"#8F(F,F(F(F(F(*&$\"0p?W1j]f%!#JF(-F*6#,$*&\"#:F(F,F(F(F(F(*&$\" 0X1h8^2B%!#MF(-F*6#,$*&\"#F(F,F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture compares the graphs of some trunca ted Fourier series with the graph of the function " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "f := x -> sin(sin(x)):\nF S := (x,n) -> sum(b[k]*sin(k*x),k=1..n);\nplot([f(x),FS(x,1),FS(x,2),F S(x,3)],x=-Pi..3*Pi,\n color=[black,red,blue,green],linestyle =[3,1$3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6 $%)operatorG%&arrowGF)-%$sumG6$*&&%\"bG6#%\"kG\"\"\"-%$sinG6#*&F4F59$F 5F5/F4;F59%F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 533 195 195 {PLOTDATA 2 "6(-%'CURVESG6%7[r7$$!3*****4tk#fTJ!#<$!3=5KT_Kzzi!#E7$$!3yMay)3PY+$ F*$!3%3#[cj2/h8!#=7$$!3-q3EI:onGF*$!3SI(\\uX@@n#F37$$!3'z.'3Js^[FF*$!3 QfXhajLPPF37$$!3*e?6>$HNHEF*$!3>D)*[wAT2ZF37$$!3Y:T[9-M&\\#F*$!3e1sLr% eXm&F37$$!3]Dq0(\\F8O#F*$!3k>%[r-i'okF37$$!3=(*zMckUEAF*$!3Y8VD$QdA7(F 37$$!3))o*QcTD:4#F*$!3!)*y,1mkoi(F37$$!3_X$Gg`ls&>F*$!3!=*ymK#)p$*zF37 $$!3)H=\"F*$!3u')o56ek!*zF37$$!3Q77?Ggo\\5F 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As a consequence, the Fourier series converges rapid ly to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 103 "The following graph shows the error in u sing the truncated Fourier series as far as the term involving " } {XPPEDIT 18 0 "sin*19*x" "6#*(%$sinG\"\"\"\"#>F%%\"xGF%" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "f := x -> sin(sin(x)):\n'f(x)'=f(x);\nFS := (x,n) -> sum(b[k]*sin(k*x),k=1..n):\nevalf[20](plot(f(x)-FS(x,19),x=0..Pi,colo r=blue));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinG6#-F )F&" }}{PARA 13 "" 1 "" {GLPLOT2D 454 146 146 {PLOTDATA 2 "6&-%'CURVES G6#7jo7$$\"\"!F)$!\"!F)7$$\"5=e2#e*p*))QU$!#@$\"%#G#F/7$$\"5P;:k\"*Rzx ZoF/$\"%^XF/7$$\"5qK7ACDa)o#)*F/$\"%0lF/7$$\"5!\\4!o0\"H*f!G\"!#?$\"$V )F?7$$\"5BFGUH&3Jch\"F?$\"%b5F?7$$\"5cfb;`zGm]>F?$\"%g7F?7$$\"57Gn$y>[ :zG#F?$\"%h9F?7$$\"5n'*y]U%3o^i#F?$\"%a;F?7$$\"5u`uD7%y<3'HF?$\"%O=F?7 $$\"5#3,2?Q[nkH$F?$\"%3?F?7$$\"5#[u#>f@xl2OF?$\"%f@F?7$$\"5\")y%yj$fz% )=RF?$\"%*H#F?7$$\"5>=J_5#*oGjXF?$\"%cDF?7$$\"5tGbeAelwH_F?$\"%rFF?7$$ \"5pzqYg:)3T*eF?$\"%LHF?7$$\"5A_!=i`[mud'F?$\"%VIF?7$$\"5S!\\0.\"*Q<%y oF?$\"%uIF?7$$\"5aGHR%GHo$zrF?$\"%%4$F?7$$\"5!=%)>O;,o([tF?$\"%,JF?7$$ \"50bn%G/tn\"=vF?$\"%0JF?7$$\"5IoO2A\\uc(o(F?$\"%2JF?7$$\"5b\"e+8!or'p &yF?$\"%/JF?7$$\"5g]Y%R'[x:(>)F?$\"%\"4$F?7$$\"5h>()eEH$[t`)F?$\"%oIF? 7$$\"5Isn6[*)\\=l))F?$\"%NIF?7$$\"5/D[kp\\;-$>*F?$\"%&*HF?7$$\"5kwpSFR eV)y*F?$\"%/HF?7$$\"5%*pDv.!4W'\\5!#>F\\p7$$\"5V!>ZIcS@'46Fbt$\"%ZEF?7 $$\"5J6Gj,#G*Qz6Fbt$\"%*\\#F?7$$\"5M0>&HVnX6C\"Fbt$\"%qBF?7$$\"5otI(eX A-*38Fbt$\"%VAF?7$$\"5!\\%*H0M[AMP\"Fbt$\"%S@F?7$$\"5tPr?*[gU2W\"Fbt$ \"%f?F?7$$\"5x+f#)=[Ilr9Fbt$\"%L?F?7$$\"5\"QmW%[\"\\jD]\"Fbt$\"%8?F?7$ $\"5]SF_V!e/f`\"Fbt$\"%**>F?7$$\"5?<3gQpcCp:Fbt$\"%&*>F?7$$\"5`2GNY!px Qg\"FbtF^w7$$\"5'yz/T:r4&Q;FbtFiv7$$\"5-O-hp()plo;Fbt$\"%K?F?7$$\"5>uc 6&QE/))p\"Fbt$\"%d?F?7$$\"5\\_6$p@.CRw\"Fbt$\"%M@F?7$$\"5!p/)e\"[7*>J= Fbt$\"%SAF?7$$\"54_([@!yY,(*=Fbt$\"%jBF?7$$\"5f<11#*4Ypg>Fbt$\"%&\\#F? 7$$\"5x)eSjrJ+9.#Fbt$\"%XEF?7$$\"5S&GV%))RD$\\4#Fbt$\"%xFF?7$$\"5&oea? $[nwi@FbtF]t7$$\"5s\"eT4LiMUA#Fbt$\"%(*HF?7$$\"5/K@v)[$\\$yD#Fbt$\"%RI F?7$$\"5P#oilkCN9H#Fbt$\"%rIF?7$$\"5F77r7@00BBFbtFir7$$\"5DFbt$\"%$*HF?7$$\"5?*4`W )f#HIb#Fbt$\"%IHF?7$$\"5RjOQZdW==EFbt$\"%tFF?7$$\"5zQ4Nid]\"[o#Fbt$\"% eDF?7$$\"5*)fyGwAR*3v#Fbt$\"%%H#F?7$$\"5K(Hp^*HPD\"y#Fbt$\"%c@F?7$$\"5 wM209PNh6GFbt$\"%5?F?7$$\"5FSs8G-$4k%GFbt$\"%K=F?7$$\"5yXPAUn]?\")GFbt $\"%U;F?7$$\"5Cxavc9oK7HFbt$\"%l9F?7$$\"5r3sGrh&[M%HFbt$\"%!G\"F?7$$\" 5gDK5IN3jwHFbt$\"%w5F?7$$\"5\\U#>*))3J\")4IFbt$\"$m)F?7$$\"5+6pdnyFdTI Fbt$\"%=mF/7$$\"5^zXBY[CLtIFbt$\"%PXF/7$$\"5m^pg-^DY2JFbt$\"%vAF/7$$\" 5\"QKz*e`EfTJFbt$\"4^z&o#)R$\\(**[!#Q-%+AXESLABELSG6$Q\"x6\"Q!F_cl-%'C OLOURG6&%$RGBGF(F($\"*++++\"!\")-%%VIEWG6$;F($\"5&QKz*e`EfTJFbt%(DEFAU LTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 52 " may also be us ed to find truncated Fourier series. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "f := x -> sin(sin(x)): \n'f(x)'=f(x);\nFourierSeries(f(x),x=0..Pi,type=sine,numterms=9,info=1 ):\nF := unapply(%,x):\n'F(x)'=F(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"fG6#%\"xG-%$sinG6#-F)F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%:k~ th~sine~coefficient~..~G/,$*0\"\"%\"\"\"-%$cosG6#,$*(\"\"#!\"\"%\"kGF( %#PiGF(F(F(-%$sinGF+F(,&F(F/F0F(F/,&F(F(F0F(F/-%*hypergeomG6%7#F(7$,&# \"\"$F.F(*&F.F/F0F(F(,&F " 0 "" {MPLTEXT 1 0 130 "f := x -> sin(sin(x)):\n'f(x)'=f(x);\nFo urierSeries(f(x),x=0..Pi,type=sine,mode=numeric,numterms=9):\nF := una pply(%,x):\n'F(x)'=F(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#% \"xG-%$sinG6#-F)F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,, *&$\"+7<,,))!#5\"\"\"-%$sinGF&F-F-*&$\"+'zqE\"R!#6F--F/6#,$*&\"\"$F-F' F-F-F-F-*&$\"+/Y:&*\\!#8F--F/6#,$*&\"\"&F-F'F-F-F-F-*&$\"+M;l/I!#:F--F /6#,$*&\"\"(F-F'F-F-F-F-*&$\"+M+&)\\5!# " 0 "" {MPLTEXT 1 0 46 "evalf[20](pl ot(f(x)-F(x),x=0..Pi,color=blue));" }}{PARA 13 "" 1 "" {GLPLOT2D 443 119 119 {PLOTDATA 2 "6&-%'CURVESG6#7es7$$\"\"!F)F(7$$\"5=e2#e*p*))QU$! 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" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = 1/L" "6#/&%\"aG6#%\"kG*&\"\"\"F)%\"LG!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos(k*Pi*x/L),x=-L..L)" " 6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\" \"F+/F*;,$F2F3F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 2/L;" "6#/%!G*&\"\"#\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos(k*Pi*x/L),x = 0 .. L);" "6#-%$IntG6$* &-%\"fG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;\"\"! F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k] = 1/L" "6#/&%\"bG6#%\"kG*& \"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin(k*Pi* x/L),x=-L..L)" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#**%\"kGF+%#Pi GF+F*F+%\"LG!\"\"F+/F*;,$F2F3F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " ``= 0" "6#/%!G\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "k=1,2,3,` . . .`" "6&/%\"kG\"\"\"\"\"#\"\"$%'~.~.~.G " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 21 "because the function " }{XPPEDIT 18 0 "f(x)*sin(k*Pi*x/L) " "6#*&-%\"fG6#%\"xG\"\"\"-%$sin G6#**%\"kGF(%#PiGF(F'F(%\"LG!\"\"F(" }{TEXT -1 4 " is " }{TEXT 260 3 " odd" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 260 21 "Fourier cosine series" }{TEXT -1 22 " of the even function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 4 " on " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\"\"F%" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = c ;" "6#/-%\"FG6#%\"xG%\"cG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(a[k]* cos(k*Pi*x/L),k = 1 .. infinity)" "6#-%$SumG6$*&&%\"aG6#%\"kG\"\"\"-%$ cosG6#**F*F+%#PiGF+%\"xGF+%\"LG!\"\"F+/F*;F+%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = c+a[1]*cos( Pi*x/L)+a[2]*cos(2*Pi*x/L)+a[3]*cos(3*Pi*x/L)+` . . . `;" "6#/%!G,,%\" cG\"\"\"*&&%\"aG6#F'F'-%$cosG6#*(%#PiGF'%\"xGF'%\"LG!\"\"F'F'*&&F*6#\" \"#F'-F-6#**F7F'F0F'F1F'F2F3F'F'*&&F*6#\"\"$F'-F-6#**F>F'F0F'F1F'F2F3F 'F'%(~.~.~.~GF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 10 " where " }{XPPEDIT 18 0 "c = 1/L;" " 6#/%\"cG*&\"\"\"F&%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x) ,x = 0 .. L);" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"!%\"LG" }{TEXT -1 1 ", " }}{PARA 256 "" 0 "" {TEXT -1 5 "and " }{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 = 0 .. L);" "6#-%$IntG6$*&-%\"fG6#%\"xG \"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;\"\"!F2" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The Fourier series " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 54 " is \+ also the Fourier series of the periodic extension " }{XPPEDIT 18 0 "f_ (x)" "6#-%#f_G6#%\"xG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Note that the " }{TEXT 268 1 "L" }{TEXT -1 36 " which appears in these formulas is " }{TEXT 260 15 "half the period" }{TEXT -1 18 " of the functions \+ " }{XPPEDIT 18 0 "f_(x);" "6#-%#f_G6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Examples of Fourier cosine series " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 41 " be the function defined on the interval " }{XPPEDIT 18 0 "-Pi <= x;" "6#1,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 "`` < = Pi;" "6#1%!G%#PiG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = x^2;" " 6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " is an " }{TEXT 260 4 "even" }{TEXT -1 38 " function, so its Fourie r series is a " }{TEXT 260 13 "cosine series" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 29 "The Fourier cosine series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 50 " is the same as that of th e the periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" } {TEXT -1 11 " given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f_(x) = x^2;" "6#/-%#f_G6#%\"xG*$F'\"\"#" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "-Pi <= x;" "6#1,$%#PiG!\"\"%\"xG" }{XPPEDIT 18 0 " `` < Pi;" "6#2%!G%#PiG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f_(x);" "6#- %#f_G6#%\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "f := x -> x ^2:\n'f(x)'=f(x);\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n'f_(x)' ='f(x-2*Pi*floor((x+Pi)/(2*Pi)))';\nplot(f_(x),x=-6.28..10,color=COLOR (RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6 #%\"xG*$)F'\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\" xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floorG6#,$*(F.!\"\",&F'F,F/F,F, F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 612 202 202 {PLOTDATA 2 "6'-% 'CURVESG6#7[s7$$!3E++++++!G'!#<$\"3C,A$G==Y,\"!#A7$$!3/n;a#e&G\">'F*$ \"3#*p`_)QfbW)!#?7$$!3(HL$3l6d-hF*$\"3=cl^Ko9iK!#>7$$!3y**\\iZn&Q,'F*$ \"3GU[<*H(y`sF97$$!3emm;IB9DfF*$\"3=TF)QtZ>G\"!#=7$$!3i*\\7'\\CwqdF*$ \"37(=s&\\=xDEFD7$$!3bL$e!pDQ;cF*$\"35***fc\"*eiW%FD7$$!3]mm^D*\\\"p_F *$\"3RjUEbxEG5F*7$$!31nm1iFQ%4&F*$\"3EapE\"\\^KT\"F*7$$!3vmmh)f:'>\\F* $\"31l.6]BKf=F*7$$!3e*\\()H2zcu%F*$\"3mJ&z,UDRO#F*7$$!3IL$etaU#\\UF*$ \"3C#=f$*4;q8%F*7$$!3EM$eL$GC#3%F*$\"3b7/2tx9W[F*7$$!3V+]sofE:RF*$\"3L .sU$*>/2cF*7$$!3kL$ekDyDu$F*$\"3)4auh=%oakF*7$$!3%om\">W0*)pNF*$\"3Opw &3Zo>O(F*7$$!3S++!RfBQ[$F*$\"3@=cx,gUOyF*7$$!3]L$3OkcxR$F*$\"3O'eeDd)p D$)F*7$$!3jmmJ$p*o6LF*$\"3Fql?$='yH))F*7$$!3=+]-VFiDKF*$\"3En&>P$))o[$ *F*7$$!3!>/^V_d8=$F*$\"3)G/FVfN8i*F*7$$!3k$3xcI#4PJF*$\"3'3OQV8[8%)*F* 7$$!3ODJ+(3FG4$F*$\"3%zB+,Rzbc*F*7$$!3km\"H$o=c[IF*$\"3C>1.l%HPH*F*7$$ !3m\\7)4VJ+'HF*$\"3A0pRsgyh()F*7$$!35LLj$*4]rGF*$\"3u=sVcz^X#)F*7$$!3' )**\\#)yca:FF*$\"3]H^yL$)=utF*7$$!3imm,k.ffDF*$\"3)=(pcJG]^lF*7$$!33L$ euh@SQ#F*$\"3)ok3D2fNo&F*7$$!3`*****3(GX3AF*$\"3)*\\k)H3ks([F*7$$!3!** ***>AK;K?F*$\"3X'Gi;O(oHTF*7$$!3E++]tN(e&=F*$\"3wgc\"3smUW$F*7$$!3+++I Je)fo\"F*$\"3?_VMB#[D%GF*7$$!3x****4*3)4;:F*$\"3-rEx:Mb)H#F*7$$!39LekV P#=O\"F*$\"3%=bd(3Rca=F*7$$!3^m;>)R\\v?\"F*$\"3!f0t!\\b6H)F3$\"3'3]r)QhEuoF-7$$\"3a=\\(=$)f:< (F9$\"3-\"=;A/FJ9&F37$$\"3TcmmgJA<:FD$\"3v=LE>h'>I#F97$$\"33F$3d:CGF$F D$\"3U#z=azP62\"FD7$$\"3u(**\\2:D%G]FD$\"3#R^p'\\f]GDFD7$$\"3B++]hC<+n FD$\"3E(H%Q,6B*[%FD7$$\"3s-+Ds(>>P)FD$\"3C`!*HnS!*3qFD7$$\"33nmTl00'= \"F*$\"3r&>DP%fr19F*7$$\"3#p;H$36BY8F*$\"36xJ.(>QB\"=F*7$$\"3wm;C^;T1: F*$\"3'[k'*H1w#pAF*7$$\"3?+DT1!)=z;F*$\"3%oez3Os'>GF*7$$\"3mLLehV'>&=F *$\"3hkZd'*>xHMF*7$$\"3cLepO/VJ?F*$\"3uCU8>'4n7%F*7$$\"3YL$3=^'*3@#F*$ \"35ci#fQj!))[F*7$$\"3%ommxDArO#F*$\"3ZiV@$ynKg&F*7$$\"3B+]s.![L_#F*$ \"3)R%H!z9&GnjF*7$$\"3'o;/$*3w?p#F*$\"3Q;KgqOFZsF*7$$\"3]LL)[@*F*7$$\"3S]i&H&\\pyIF*$\"3A*GeLhi$y%*F*7$$\"3^L3d3JFAJF*$\" 3qwX]k$*e[(*F*7$$\"3izWZZw;LJF*$\"30l2=2&Rn\")*F*7$$\"3HD\"yj=iS9$F*$ \"3k..bnR4a)*F*7$$\"3'4x\"GDn&\\:$F*$\"3w:CggS\"ey*F*7$$\"3i;a=k7&e;$F *$\"3mu.hN:x<(*F*7$$\"3S3F*>MSw=$F*$\"3?HQ\\J')R#e*F*7$$\"3=++!)>%H%4K F*$\"3Gn1?b_(zW*F*7$$\"3y*\\(om[p%H$F*$\"30%e7H6v5$*)F*7$$\"3Q**\\d8.' *zLF*$\"3)GN!\\j`rG%)F*7$$\"3)*)\\i/wD_Y$F*$\"3-sR$p+'*3%zF*7$$\"3c)** \\t?\"\\]NF*$\"3cSMCVqhnuF*7$$\"3K*\\7ch)[:PF*$\"3sjX))*HmIf'F*7$$\"33 +](Q-'[!)QF*$\"3+sB*f%Q'Hx&F*7$$\"3)**\\iW^(ojSF*$\"3K=g(=Xqh#\\F*7$$ \"3!****\\]+*)oC%F*$\"3AFm1IE]YTF*7$$\"33L$eDp.:T%F*$\"3G@7^m?>.NF*7$$ \"3Emm1!Q=hd%F*$\"3PZFk$\\xS\"HF*7$$\"3)GL$G&pz=v%F*$\"3ue\\-xo*[M#F*7 $$\"3Z****\\55kF\\F*$\"3Ub0ES.]P=F*7$$\"3!f;a`)o!p3&F*$\"3Q=^_i?3J9F*7 $$\"3MK$3-wshC&F*$\"352edA]Rv5F*7$$\"3E++!z47Wf&F*$\"3E.I@Q`3WZFD7$$\" 3]K3#o8X#edF*$\"3%RxiX#=ibFFD7$$\"3tk;uv\"y?#fF*$\"3=VKv.O)RI\"FD7$$\" 38)**\\%G:/yF*$\"3f8O'=CU LJ#F*7$$\"3?n;9))fOvzF*$\"39#H@qXvM'GF*7$$\"3!R$3(z3$pK\")F*$\"3$eO27. z1U$F*7$$\"3s****z(=?+H)F*$\"3;\\Z2QiQFSF*7$$\"3]L$e&)zM.Z)F*$\"3Q7$Qw *Gi$y%F*7$$\"3EnmJ4%\\1l)F*$\"3GNx!\\F')[g&F*7$$\"3/+]sY\\#>\"))F*$\"3 wp#3Y;CXR'F*7$$\"3#GLLT[+K(*)F*$\"3d'\\TAl\"=OsF*7$$\"3w\\(oV;x\"f!*F* $\"3XT0=\"zIhq(F*7$$\"3qmTgWQ:X\"*F*$\"3i\">3#4R'3>)F*7$$\"3j$eR[_I6B* F*$\"3')[WK15Q!p)F*7$$\"3d+]20s5<$*F*$\"3>8$HD3#o/#*F*7$$\"3%*\\ivJDDe $*F*$\"3uDcmN%QgX*F*7$$\"3I*\\P%eyR*R*F*$\"37^thk1y5(*F*7$$\"3w5y5!>%o 4%*F*$\"3cN9&\\E&*\\x*F*7$$\"3*R7y<_q*>%*F*$\"3%Q*R_#[@%R)*F*7$$\"3BP% [M&oDI%*F*$\"3lvOP@&4_$)*F*7$$\"3o[(=^=V0W*F*$\"3!ez\"R " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The constant coefficient in the Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "c = 1/Pi;" "6#/%\"cG*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. Pi);" "6#-%$IntG6$-%\"fG6#%\"x G/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(x^2,x = 0 .. Pi);" "6#-%$IntG6$*$%\"xG\"\"#/F'; \"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= ``(1/Pi)*``(Pi^3/3)" "6#/%!G*&-F$6#*&\"\"\"F)%#PiG! \"\"F)-F$6#*&F*\"\"$F/F+F)" }{XPPEDIT 18 0 "`` = Pi^2/3" "6#/%!G*&%#Pi G\"\"#\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 37 "The coefficients of the cosine terms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 51 "The coe fficients of the cosine terms are given by: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = 2/Pi;" "6#/&%\"aG6#%\"kG*&\"\"# \"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*k*x,x = 0 .. Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF+%\"kGF+F*F+/F*;\" \"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2*cos*k*x,x = 0 .. Pi);" "6#-%$IntG6$**%\" xG\"\"#%$cosG\"\"\"%\"kGF*F'F*/F';\"\"!%#PiG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int(x^2*cos*k*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$**%\"xG\"\"#%$cosG\"\"\"%\"kGF+F(F+ F(-F%6$*&%\"uGF+-%!G6#*&%#dvGF+%#dxG!\"\"F+F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x^2" "6#/%\"uG*$%\"xG\"\"#" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "v = sin*k*x/k;" "6#/%\"vG**%$sinG\"\"\"%\"kGF'%\"xGF'F( !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "Hence using the \+ integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u *v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG !\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" } {TEXT -1 10 ", we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = 2/Pi;" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(x^2*sin*k*x/k);" "6#-%!G6#*,%\"xG\"\"#%$si nG\"\"\"%\"kGF*F'F*F+!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE( [Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"! F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-2/Pi;" "6#,$*&\"\"#\"\"\"%#PiG!\" \"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*x*sin*k*x/k,x = 0 .. Pi); " "6#-%$IntG6$*.\"\"#\"\"\"%\"xGF(%$sinGF(%\"kGF(F)F(F+!\"\"/F);\"\"!% #PiG" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 0-4/(k*Pi);" "6#/%!G,&\"\"!\"\"\"*&\"\"%F'*&%\"kGF'%#PiGF'! \"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sin*k*x,x = 0 .. Pi);" " 6#-%$IntG6$**%\"xG\"\"\"%$sinGF(%\"kGF(F'F(/F';\"\"!%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 70 "Then, applying the integration by parts formula to the integral, with " }{XPPEDIT 18 0 "u=x" "6#/%\"uG% \"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*k*x/k;" "6#/%\"vG,$ **%$cosG\"\"\"%\"kGF(%\"xGF(F)!\"\"F+" }{TEXT -1 11 ", we have: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = -4/(k*Pi);" "6 #/&%\"aG6#%\"kG,$*&\"\"%\"\"\"*&F'F+%#PiGF+!\"\"F." }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(-x*cos*k*x/k);" "6#-%!G6#,$*,%\"xG\"\"\"%$cosGF)%\"k GF)F(F)F+!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[ ``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" } {TEXT -1 2 "+ " }{XPPEDIT 18 0 "4/(k*Pi);" "6#*&\"\"%\"\"\"*&%\"kGF%%# PiGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(-cos*k*x/k,x = 0 .. Pi );" "6#-%$IntG6$,$**%$cosG\"\"\"%\"kGF)%\"xGF)F*!\"\"F,/F+;\"\"!%#PiG " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = 4*x*cos*k*x/(Pi*k^2)-4*sin*k*x /(Pi*k^2);" "6#/%!G,&*.\"\"%\"\"\"%\"xGF(%$cosGF(%\"kGF(F)F(*&%#PiGF(* $F+\"\"#F(!\"\"F(*,F'F(%$sinGF(F+F(F)F(*&F-F(*$F+F/F(F0F0" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PI ECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "` ` = 4*Pi*cos*k*Pi/(k^2*Pi)+0;" "6#/%!G,&*.\"\"%\"\"\"%#PiGF(%$cosGF(% \"kGF(F)F(*&F+\"\"#F)F(!\"\"F(\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "` ` = (-1)^k*``(4/(k^2));" "6#/%!G*&),$\"\"\"!\"\"%\"kGF(-F$6#*&\"\"%F(* $F*\"\"#F)F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([4/(k^ 2), `if k is even`],[-4/(k^2), `if k is odd`]);" "6#/%!G-%*PIECEWISEG6 $7$*&\"\"%\"\"\"*$%\"kG\"\"#!\"\"%-if~k~is~evenG7$,$*&F*F+*$F-F.F/F/%, if~k~is~oddG" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2 , 3, 4, 5, 6, 7, 8, 9, 10], [a[k], `|`, -4, 1, -4/9, 1/4, -4/25, 1/9, \+ -4/49, 1/16, -4/81, 1/25]])" "6#-%'matrixG6#7$7.%\"kG%\"|grG\"\"\"\"\" #\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"aG6#F(F),$F-!\"\"F*,$*&F -F*F2F9F9*&F*F*F-F9,$*&F-F*\"#DF9F9*&F*F*F2F9,$*&F-F*\"#\\F9F9*&F*F*\" #;F9,$*&F-F*\"#\")F9F9*&F*F*F?F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "interface(s howassumed=0): k := 'k': assume(k,integer):\na[k]=2/Pi*Int(x^2*cos(k*x ),x=0..Pi);\n``=value(rhs(%));\naa := unapply(rhs(%),k):\nmatrix([[k,` |`,seq(k,k=1..10)],[a[k],`|`,seq(aa(k),k=1..10)]]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6#%#k|irG,$,$-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$co sG6#*&F'F1F/F1F1/F/;\"\"!%#PiG*&F0F1F9!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"%\"\"\"%#k|irG!\"#)!\"\"F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7.%#k|irG%\"|grG\"\"\"\"\"#\"\"$ \"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"aG6#F(F)!\"%F*#F8F2#F*F-#F8\"# D#F*F2#F8\"#\\#F*\"#;#F8\"#\")#F*F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier ser ies of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(x) = Pi^2/3+``;" " 6#/-%\"FG6#%\"xG,&*&%#PiG\"\"#\"\"$!\"\"\"\"\"%!GF." }{XPPEDIT 18 0 "S um(``(4*(-1)^k/(k^2))*cos*k*x,k = 1 .. infinity);" "6#-%$SumG6$**-%!G6 #*(\"\"%\"\"\"),$F,!\"\"%\"kGF,*$F0\"\"#F/F,%$cosGF,F0F,%\"xGF,/F0;F,% )infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi^2/3-4*cos*x+cos*2*x-4 /9*cos*3*x+cos*4*x/4-4/25*cos*5*x+cos*6*x/9-4/49*cos*7*x+` . . . `" "6 #/%!G,4*&%#PiG\"\"#\"\"$!\"\"\"\"\"*(\"\"%F+%$cosGF+%\"xGF+F**(F.F+F(F +F/F+F+*,F-F+\"\"*F*F.F+F)F+F/F+F***F.F+F-F+F/F+F-F*F+*,F-F+\"#DF*F.F+ \"\"&F+F/F+F***F.F+\"\"'F+F/F+F2F*F+*,F-F+\"#\\F*F.F+\"\"(F+F/F+F*%(~. ~.~.~GF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 106 "The following picture compares the graphs of some truncated Fourier series with the graph of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "f := x -> x^2:\nf _ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nFS := (x,n) -> Pi^2/3+sum(4 *(-1)^k/(k^2)*cos(k*x),k=1..n);\nplot([f_(x),FS(x,1),FS(x,2),FS(x,3),F S(x,4),FS(x,5)],x=-6.28..10,\n color=[black,red,blue,green,magenta,c oral],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6 $%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&*&\"\"$!\"\"%#PiG\"\"#\"\"\"-%$ sumG6$,$**\"\"%F3)F0%\"kGF3F;!\"#-%$cosG6#*&F;F39$F3F3F3/F;;F39%F3F)F) F)" }}{PARA 13 "" 1 "" {GLPLOT2D 622 236 236 {PLOTDATA 2 "6*-%'CURVESG 6%7[s7$$!3E++++++!G'!#<$\"3C,A$G==Y,\"!#A7$$!3/n;a#e&G\">'F*$\"3#*p`_) QfbW)!#?7$$!3(HL$3l6d-hF*$\"3=cl^Ko9iK!#>7$$!3y**\\iZn&Q,'F*$\"3GU[<*H 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" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 7 " is an " }{TEXT 260 4 "even" }{TEXT -1 38 " function, so i ts Fourier series is a " }{TEXT 260 13 "cosine series" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 26 "The periodic extension of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 45 " is given by the \+ same formula and has period " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 4 ". 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"" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "The constant coefficient in the Fourier series of " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c = 2/Pi;" "6#/%\"cG*&\"\"#\"\"\"%#PiG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. Pi/2);" "6#-%$IntG6$- %\"fG6#%\"xG/F);\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"# \"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*x,x = 0 .. P i/2);" "6#-%$IntG6$*&%$sinG\"\"\"%\"xGF(/F);\"\"!*&%#PiGF(\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "The coeffici ents of the cosine terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 73 "The coefficients of the cosine terms in the Fourier series are given by: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = 4/Pi;" "6#/&%\"aG6#%\"kG*&\"\"%\"\"\"%#PiG !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*2*k*x,x = 0 .. Pi/ 2);" "6#-%$IntG6$*,-%\"fG6#%\"xG\"\"\"%$cosGF+\"\"#F+%\"kGF+F*F+/F*;\" \"!*&%#PiGF+F-!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = 4/Pi;" "6#/%!G*&\"\"%\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*x*cos*2*k*x,x = 0 .. Pi/2);" "6#-%$In tG6$*.%$sinG\"\"\"%\"xGF(%$cosGF(\"\"#F(%\"kGF(F)F(/F);\"\"!*&%#PiGF(F +!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sin((2*k+1)*x)-sin((2*k-1)*x),x = 0 .. Pi/2);" "6#- %$IntG6$,&-%$sinG6#*&,&*&\"\"#\"\"\"%\"kGF.F.F.F.F.%\"xGF.F.-F(6#*&,&* &F-F.F/F.F.F.!\"\"F.F0F.F6/F0;\"\"!*&%#PiGF.F-F6" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 21 "( using the formula: " }{XPPEDIT 18 0 "co s*alpha*sin*beta = 1/2;" "6#/**%$cosG\"\"\"%&alphaGF&%$sinGF&%%betaGF& *&F&F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(sin(alpha+beta)-sin(alpha-beta)); " "6#-%!G6#,&-%$sinG6#,&%&alphaG\"\"\"%%betaGF,F,-F(6#,&F+F,F-!\"\"F1 " }{TEXT -1 3 " )" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(-cos((2*k+1)*x)/(2*k+1)+cos((2*k-1)*x)/(2*k-1));" "6 #-%!G6#,&*&-%$cosG6#*&,&*&\"\"#\"\"\"%\"kGF/F/F/F/F/%\"xGF/F/,&*&F.F/F 0F/F/F/F/!\"\"F4*&-F)6#*&,&*&F.F/F0F/F/F/F4F/F1F/F/,&*&F.F/F0F/F/F/F4F 4F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi/2, ``],[0, ``]);" " 6#-%*PIECEWISEG6$7$*&%#PiG\"\"\"\"\"#!\"\"%!G7$\"\"!F," }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{XPPEDIT 18 0 "``(0-``(-1/(2*k+1)+1/(2*k-1)));" "6#-%!G6#,&\"\"!\"\"\"-F$6#,&*& F(F(,&*&\"\"#F(%\"kGF(F(F(F(!\"\"F1*&F(F(,&*&F/F(F0F(F(F(F1F1F(F1" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (-4)/(Pi*(4*k^2-1));" "6#/%!G*&,$ \"\"%!\"\"\"\"\"*&%#PiGF),&*&F'F)*$%\"kG\"\"#F)F)F)F(F)F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8], [a[k], ` |`, -4/3/Pi, -4/15/Pi, -4/35/Pi, -4/63/Pi, -4/99/Pi, -4/143/Pi, -4/195 /Pi, -4/255/Pi]])" "6#-%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\" %\"\"&\"\"'\"\"(\"\")7,&%\"aG6#F(F),$*(F-F*F,!\"\"%#PiGF8F8,$*(F-F*\"# :F8F9F8F8,$*(F-F*\"#NF8F9F8F8,$*(F-F*\"#jF8F9F8F8,$*(F-F*\"#**F8F9F8F8 ,$*(F-F*\"$V\"F8F9F8F8,$*(F-F*\"$&>F8F9F8F8,$*(F-F*\"$b#F8F9F8F8" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "interface(showassumed=0): k := 'k': assume(k,in teger):\na[k]=4/Pi*Int(sin(x)*cos(2*k*x),x=0..Pi/2);\n``=simplify(valu e(rhs(%)));\naa := unapply(rhs(%),k):\nmatrix([[k,`|`,seq(k,k=1..8)],[ a[k],`|`,seq(aa(k),k=1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6#%#k|irG,$,$-%$IntG6$*&-%$sinG6#%\"xG\"\"\"-%$cosG6#,$*(\"\"#F2F' F2F1F2F2F2/F1;\"\"!,$*&F8!\"\"%#PiGF2F2*&\"\"%F2F?F>F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"%\"\"\"%#PiG!\"\",&F(F**&F'F()%#k|irG \"\"#F(F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,%#k|i rG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&%\"aG6#F(F),$*(F- F*F,!\"\"%#PiGF8F8,$*(F-F*\"#:F8F9F8F8,$*(F-F*\"#NF8F9F8F8,$*(F-F*\"#j F8F9F8F8,$*(F-F*\"#**F8F9F8F8,$*(F-F*\"$V\"F8F9F8F8,$*(F-F*\"$&>F8F9F8 F8,$*(F-F*\"$b#F8F9F8F8Q)pprint296\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(x) = 2/Pi+``;" "6#/-%\"F G6#%\"xG,&*&\"\"#\"\"\"%#PiG!\"\"F+%!GF+" }{XPPEDIT 18 0 "Sum(``(-4/(P i*(4*k^2-1)))*cos*2*k*x,k = 1 .. infinity);" "6#-%$SumG6$*,-%!G6#,$*& \"\"%\"\"\"*&%#PiGF-,&*&F,F-*$%\"kG\"\"#F-F-F-!\"\"F-F5F5F-%$cosGF-F4F -F3F-%\"xGF-/F3;F-%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2/ Pi-4*cos*2*x/(3*Pi)-4*cos*4*x/(15*Pi)-4*cos*6*x/(35*Pi)-4*cos*8*x/(63* Pi)-4*cos*10*x/(99*Pi)-` . . . `" "6#/%!G,0*&\"\"#\"\"\"%#PiG!\"\"F(*, \"\"%F(%$cosGF(F'F(%\"xGF(*&\"\"$F(F)F(F*F**,F,F(F-F(F,F(F.F(*&\"#:F(F )F(F*F**,F,F(F-F(\"\"'F(F.F(*&\"#NF(F)F(F*F**,F,F(F-F(\"\")F(F.F(*&\"# jF(F)F(F*F**,F,F(F-F(\"#5F(F.F(*&\"#**F(F)F(F*F*%(~.~.~.~GF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture compares the graphs of some truncated Fouri er series with the graph of the function " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "f := x -> abs(sin(x)):\n'f(x)'=f(x );\nFS := (x,n) -> 2/Pi+sum(-4/(Pi*(4*k^2-1))*cos(2*k*x),k=1..n);\nplo t([f(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5)],x=-Pi..2*Pi,\n colo r=[black,red,blue,green,magenta,coral],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$absG6#-%$sinGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowG F),&*&\"\"#\"\"\"%#PiG!\"\"F0-%$sumG6$,$**\"\"%F0F1F2,&*&F8F0)%\"kGF/F 0F0F0F2F2-%$cosG6#,$*(F/F0F)y(**ewN8&!#>7$$!3?wlT yf()QIF*$\"3S7nL![h`-\"!#=7$$!3_k)pRkUSy&\\LfYF97$$!37p-i%y$RcDF* $\"3%*\\5t3F*$\"32' GVHP>%H#*F97$$!3W9H8ACFp=F*$\"3Cw:!HCdyb*F97$$!3kC\\T#e1Ex\"F*$\"3gy'R 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\\?k&F97$Figm$\"1)=T$>YGX^F97$F^hm$\"1Ei-\"\\R0j%F97$Fchm$\"1y?<>_,iSF 97$Fhhm$\"1tBUv$RtZ$F97$F]im$\"1cpB5[$*\\HF97$Fbim$\"1GE$oF\\DT#F97$Fg im$\"1;w-I]/&y\"F97$F\\jm$\"1^s31F@^6F97$Fajm$\"1PCz$ou<\"eF37$Ffjm$\" 1D40jg;.jF][l-%&COLORG6&F^[n$\"\")!\"\"F_[nFc[n-Fa[n6#\"\"#-Fe[nFb[n-% +AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;$!+aEfTJ!\"*$\"+3`=$G'F`io%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 52.000000 48.000000 0 0 "Curve 1" " Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Truncated Fourier series can be obtained using the procedure " } {TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "f := x -> abs(sin( x)):\n'f(x)'=f(x);\nFourierSeries(f(x),x=0..Pi,numterms=6,info=1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$absG6#-%$sinGF&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G,$*&\"\"#\" \"\"%#PiG!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "It is better to use the option \"" }{TEXT 271 8 "type=cos" }{TEXT -1 92 "\" so that the simplified integral formulas for the cosine coefficients given above are used." }}{PARA 0 "" 0 "" {TEXT -1 33 "The second argument is given as \"" }{TEXT 271 11 "x=0 .. Pi/2" }{TEXT -1 39 "\", which corresponds to the interval " } {XPPEDIT 18 0 "[0, Pi/2];" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 46 " over which the integrals are to be evaluated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "FourierS eries(sin(x),x=0..Pi/2,type=cos,numterms=8):\nF := unapply(%,x):\n'F(x )'=F(x);\nplot(F(x),x=-Pi..2*Pi,color=COLOR(RGB,.8,0,1),thickness=2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,4*&\"\"#\"\"\"%#PiG !\"\"F+*&#\"\"%\"\"$F+*&F,F--%$cosG6#,$*&F*F+F'F+F+F+F+F-*&#F0\"#:F+*& F,F--F46#,$*&F0F+F'F+F+F+F+F-*&#F0\"#NF+*&F,F--F46#,$*&\"\"'F+F'F+F+F+ F+F-*&#F0\"#jF+*&F,F--F46#,$*&\"\")F+F'F+F+F+F+F-*&#F0\"#**F+*&F,F--F4 6#,$*&\"#5F+F'F+F+F+F+F-*&#F0\"$V\"F+*&F,F--F46#,$*&\"#7F+F'F+F+F+F+F- *&#F0\"$&>F+*&F,F--F46#,$*&\"#9F+F'F+F+F+F+F-*&#F0\"$b#F+*&F,F--F46#,$ *&\"#;F+F'F+F+F+F+F-" }}{PARA 13 "" 1 "" {GLPLOT2D 557 205 205 {PLOTDATA 2 "6'-%'CURVESG6#7as7$$!3*****4tk#fTJ!#<$\"3?mvR!>A[u$!#>7$$ !3'yGjGJM-4$F*$\"3oGCYJ\\AP^F-7$$!3?wlTyf()QIF*$\"3S.F*$\"3\"yav>,i%3#*F=7$$!3W9H8ACFp= F*$\"3IZN'zD9!\\&*F=7$$!3kC\\T#e1Ex\"F*$\"3k5F3k:H=)*F=7$$!3%fzbvg?Es \"F*$\"3(\\jvu%Q\"R!**F=7$$!3+nmpKYjs;F*$\"3qvx?\"fe>&**F=7$$!34Qv$yl[ Ei\"F*$\"3QaYw>sss**F=7$$!3Q4%yHoiEd\"F*$\"3A**4W%z8\"y**F=7$$!3o&fuP, PG_\"F*$\"3%R/S\"p:dt**F=7$$!3?#yqXM6IZ\"F*$\"3GWBG\"p&[a**F=7$$!3\\op Ovc=B9F*$\"3'*=T'3^9$4**F=7$$!3yaJ;1+Ot8F*$\"3fdb0v1SF)*F=7$$!3![U/fNc 3F\"F*$\"3J,z(zTpTa*F=7$$!3/&pXcq_$o6F*$\"3Yv>4FNe\"=*F=7$$!3k0RqMu1y5 F*$\"3!R(*y\"yT4C))F=7$$!3sk6iP;#y()*F=$\"3)HJN)p$H,P)F=7$$!3G\\#**oKB 9'))F=$\"3#f5%GbocFxF=7$$!3$QLxh,D]%yF=$\"3U!Qw$H#[g/(F=7$$!3)Q0QjF`W# oF=$\"3>>*33;AtL'F=7$$!3#Rx)\\O:)Q!eF=$\"3QT\"))Hf#)R\\&F=7$$!3$**))=y br.#[F=$\"35!fjS#)Q4f%F=7$$!3%f+R\"z:'o$QF=$\"3%*z$o%Q^VWPF=7$$!3'H&yb 'HSP%HF=$\"3%=K6.$ejpHF=7$$!3++n(R,>10#F=$\"3>!GQi#=iG?F=7$$!3jVA@>_h> :F=$\"3V!*zvAP![T\"F=7$$!3qsyZWU6'))*F-$\"3%45x5fH*>')F-7$$!3)3fb1F&4J sF-$\"3](HZ'*zG%[kF-7$$!334L$oHwgd%F-$\"3k4YoXS4b[F-7$$!3=o@#*4oc[KF-$ \"3e:'o:Pl!4VF-7$$!3GF5,Bt0@>F-$\"3lmcIxuBVRF-7$$!3tj))*4Oya$f!#?$\"3/ +(>Ik9Qw$F-7$$\"3IXD63lhRtF^y$\"3E$fq&>\"fQx$F-7$$\"3(>[!H6.EK_F-$\"3E 'H)ya+%))=&F-7$$\"3T4(p<(*e0t*F-$\"3MW,DY(R!z%)F-7$$\"3p$*[Ajd)GU\"F=$ \"3.1#fo.OiI\"F=7$$\"3U;GFHcrs=F=$\"3ITZk6ixA=F=7$$\"3O%y?6qufR#F=$\"3 N;!HjoI]T#F=7$$\"3J_(oHxL#>HF=$\"3'4!H<>+ZYHF=7$$\"3C?n\"[%G\\UMF=$\"3 kRlCx)3VT$F=7$$\"3=)okm\">vlRF=$\"3+1'GVWv:&QF=7$$\"3Wf3^*z(4#*[F=$\"3 =+ZH9(>gl%F=7$$\"3pIqN#oV%=eF=$\"3p)3G'3822bF=7$$\"3s3OwL>zMoF=$\"3>8]qF=7$$\"3!y:Mf1W*=))F=$\"3cT&eB ^,\"*p(F=7$$\"3')G\")pYzu'y*F=$\"35m3@U'GtJ)F=7$$\"3Q>\")pw\\lz5F*$\"3 vX0M?],J))F=7$$\"3)eUE(e^j!=\"F*$\"3$eG%y2aqE#*F=7$$\"3IE'H%)[mLF\"F*$ \"3g[CBCvY_&*F=7$$\"3sEG8=y4m8F*$\"3y,f!f33@\")*F=7$$\"3WG!f9X4hT\"F*$ \"3:bd!4b*3+**F=7$$\"3=I_y%3@hY\"F*$\"3'pkn;.$3]**F=7$$\"3\">V6\"=F8;: F*$\"3?uU#\\)>1s**F=7$$\"3kLwV^V9m:F*$\"3?b[wn43y**F=7$$\"3_t+&HQ#4=;F *$\"3W_n-Doqt**F=7$$\"3Q8DY9//q;F*$\"3A`H(GJ'f`**F=7$$\"3E`\\(fW))>s\" F*$\"3#Q$H)[PTZ!**F=7$$\"39$R([xk$Rx\"F*$\"31#=DB'4Y:)*F=7$$\"3+qF*$\"3woBUj74\\#*F=7$$\"3OD\"Qjy *\\_?F*$\"3%ocs\">e`r))F=7$$\"3'Q5?'Q%z,:#F*$\"3WnNgmS$3R)F=7$$\"3-T#) Qx?4^AF*$\"311G>t)=nv(F=7$$\"3=yj:;Z+_BF*$\"3sZ*>R(=+!3(F=7$$\"3u[uwX! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 7 " is an " }{TEXT 260 4 "even" }{TEXT -1 38 " function, so i ts Fourier series is a " }{TEXT 260 13 "cosine series" }{TEXT -1 4 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 29 "The Fourier cosine series of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 50 " is the same as t hat of the the periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#% \"xG" }{TEXT -1 11 " given by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = PIECEWISE([1+ 2*x/Pi, -Pi <= x and x < 0],[1-2*x/Pi, 0 <= x and x <= Pi]);" "6#/-%#f _G6#%\"xG-%*PIECEWISEG6$7$,&\"\"\"F-*(\"\"#F-F'F-%#PiG!\"\"F-31,$F0F1F '2F'\"\"!7$,&F-F-*(F/F-F'F-F0F1F131F6F'1F'F0" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "f_(x);" "6#-%#f_G6#%\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "f := x -> piecewise(x < 0,1+2*x/Pi,1-2*x/Pi):\n'f(x) '=f(x);\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\n'f_(x)'='f(x-2*Pi *floor((x+Pi)/(2*Pi)))';\nplot(f_(x),x=-2*Pi..5*Pi,color=COLOR(RGB,.4, 0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-% *PIECEWISEG6$7$,&\"\"\"F-*(\"\"#F-F'F-%#PiG!\"\"F-2F'\"\"!7$,&F-F-*(F/ F-F'F-F0F1F1%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#% \"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floorG6#,$*(F.!\"\",&F'F,F/F, F,F/F5F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 737 147 147 {PLOTDATA 2 "6' -%'CURVESG6#7]r7$$!3)****>YH&=$G'!#<$\"25>V+#********F*7$$!3VbD?nI^VgF *$\"3_U*H98*>u%)!#=7$$!3y6^yR3%Q!eF*$\"3y,n\"GM)R[pF27$$!3SI'HiJ.`f&F* $\"3\"R!o0Ns!3i&F27$$!3+\\Tn#zlnQ&F*$\"3;2pHFh@$H%F27$$!3otsQ,#y5?+GF27$$!3C*R!o\"G@x\"\\F*$\"3Nie5Hg=28F27$$!3;O,WgWk\"o%F*$ !3?9_')GpDd>!#>7$$!3'R()*>RwcXWF*$!3u+*y[TP')p\"F27$$!3h<=))\\Gh5UF*$! 3oTA!)fgS%>$F27$$!3QgPcg!ec(RF*$!3G*eDZqu,p%F27$$!3%oQfX*[#yv$F*$!3[EP G5P%p2'F27$$!3K8]bG<**RNF*$!3#f'=%er7PY(F27$$!3V$)oa-\"QWJ$F*$!3k[;I'p J'**))F27$$!3a`(QlZ%)))3$F*$!3'3dQKK\\Wm*F27$$!35\\L'Qf;c&GF*$!3'GtV&f )>%z\")F27$$!35Xz=6([Bi#F*$!3G**)[eR!R%p'F27$$!3;x1d@*G)*Q#F*$!38pyZSM 79_F27$$!3y3M&>84t:#F*$!3xOo5&[cQt$F27$$!3<;p,[R8=>F*$!3'GfqN(*>7@#F27 $$!3aB/3k(e*y;F*$!3[*[V.iMe)oFQ7$$!3]%R$)=8$Ho9F*$\"3rF5:hgbDlFQ7$$!3Y ljo*\\FwD\"F*$\"3ebXEuYp$*>F27$$!3#3z^Q*yY?5F*$\"3hxk2<'*\\.NF27$$!3#> ;s,)G3LyF2$\"3j*R)))fXI8]F27$$!3!))e_0#)[F2$\"3AR*eB!))yv()F27$$!31A] =S9ebxFQ$\"3u;r*3Nki]*F27$$!3?!)=/p1,()[FQ$\"3Uh;.QK)))o*F27$$!3NQ()*y *)R%=?FQ$\"311i;D@]r)*F27$$!3>u;FB^aTe!#?$\"3SyMto:\"G'**F27$$\"32NSWK (38])Fbu$\"3G\\#*p()*ye%**F27$$\"3Wuf\")eiT%G#FQ$\"3%p(>8W&pX&)*F27$$ \"3OXvQW;q=PFQ$\"3i/Zc+,Ej(*F27$$\"31HQn'=VeX*FQ$\"3S;cHEB-)R*F27$$\"3 G6g*GZ)H>:F2$\"33Fl-_XyK!*F27$$\"3X'HZA8[Kg$F2$\"3)G\"[E%*451xF27$$\"3 k\"e)f\"z(>(o&F2$\"3o)4.lV<%zjF27$$\"3AVP\\+b@l\")F2$\"3qDMumB'=![F27$ $\"3[!*)Q4KBV1\"F*$\"3t_P)pH2VA$F27$$\"3W5T4DRCu7F*$\"3wR\"eYA6z)=F27$ $\"3SI$\\#HX;%[\"F*$\"3#zEDL_^^^&FQ7$$\"3Zg66'4_$Gt*z6..5F27$ $\"3c!*H(HmRD(>F*$!3=5*yA^xvb#F27$$\"3[0mjNqo)=#F*$!3ZA&3TU9O$RF27$$\" 3W?-I3W$[S#F*$!3wM\"QfL^'4`F27$$\"3:$)yiLB)>k#F*$!3oWNL=>Q>oF27$$\"3'e ab*e-8zGF*$!3P_*G2]7\"H$)F27$$\"3W&36NNItm(*F27$$\"3dk@A<;'y@$F*$!39%[OtpcW^*F27$$\"39/x(* p?xILF*$!3[h\\-Ikk&z)F27$$\"3G$))4Z\\#RmNF*$!3YNMW=-k&H(F27$$\"3Vi?W>H ,-QF*$!3A2>'o+Mcz&F27$$\"31rB3AgQ=SF*$!3f+&oL0g\"=WF27$$\"3ozEsC\"fZB% F*$!3>'4v)*4'oSIF27$$\"3;\"46puY\"oWF*$!3=>^w]V*[b\"F27$$\"3i-&*4pV`,Z F*$!3*oV9b;g-\"pFbu7$$\"3a/q:;&eR%\\F*$\"3]vj]^u@u9F27$$\"3Y1X@jEQ'=&F *$\"3R&*ym/v`*)HclnF*$\"3>'>%y)H)3HpF27$$\"3E_j&\\UCdh=(z=sF*$\"3-(R3ey4P/%F27$$\"3oJ\"et%pnTuF*$ \"3!R)yRdJ\"[i#F27$$\"3e&p7A#p9*o(F*$\"3A[1!oOq$\\5F27$$\"3[fs1(*ohOzF *$!3kje'zBC2E&FQ7$$\"3]he9w'z*e\")F*$!3`6qzTynT>F27$$\"3kiWAbCM\"Q)F*$ !3/IuzfKGdLF27$$\"3EOdF%=i(=')F*$!3Yw$R%)>W(o[F27$$\"3))4qK8>=c))F*$!3 )GK\"3P^?!Q'F27$$\"3/BvNv%>82*F*$!3/Rrv%R8)\\xF27$$\"3=O!)QPqX'G*F*$!3 WdHV_;U>\"*F27$$\"3U\\?`#ed_M*F*$!31r'z!\\vv$\\*F27$$\"3kignF\"eSS*F*$ !3q%QEdW$4o)*F27$$\"3)e2?GneGY*F*$!3o,pid1dd(*F27$$\"3(34kz@f;_*F*$!3% p@D3.ER'*F*$!3POnonHcM')F27$$\"3cX,a)Rhov*F*$!354LR u6*e)yF27$$\"3Cjws?$o\"y**F*$!3SM%H,b2qZ'F27$$\"34=:HCv%*>5!#;$!3[db'e #R7o]F27$$\"3eC#HVZ!3V5F`gl$!3qCg9(eMaf$F27$$\"31JpOCM@m5F`gl$!3$>\\E% [_uA@F27$$\"33$4Z86I))3\"F`gl$!3rY`CnBAHoFQ7$$\"33bsK)zY96\"F`gl$\"31[ Ux\\x+pvFQ7$$\"3,E+ftV7N6F`gl$\"3=uvPK?EkAF27$$\"3%pz_)[>!)e6F`gl$\"3m Cxx*GB;x$F27$$\"31=u(zE1;=\"F`gl$\"3aA\"*yM8RB_F27$$\"3>R?5(e5W?\"F`gl $\"3a@0!)z$f^n'F27$$\"3Ybx5(HJxA\"F`gl$\"3#)o%H$*)=!)f\")F27$$\"3urM62 ?0^7F`gl$\"34;%e))RWWk*F27$$\"3/of')4'zTF\"F`gl$\"3Yz8<\"o1K)))F27$$\" 3Mk%=E@2tH\"F`gl$\"3,v6?hx&3T(F27$$\"3J0\"3w2f&=8F`gl$\"3U%*>*f,Az0'F2 7$$\"3HYxfU4\")R8F`gl$\"3t7Gyqi)\\q%F27$$\"3u3w9yz;k8F`gl$\"3[hD0G#pV: $F27$$\"3=rup8]_)Q\"F`gl$\"3B5BK&=_Pg\"F27$$\"3w'fbmB5.T\"F`gl$\"3owTk s\"='o@FQ7$$\"3OAPhfa4K9F`gl$!3!QZ$z]&G+<\"F27$$\"3m^*y50B`X\"F`gl$!3g >w\"*\\Hu[EF27$$\"3'4=WDk]&y9F`gl$!3UlD0pa\"F`gl$ !3;&Q$3Mp/z%)F27$$\"3'****\\OK'zq:F`gl$!2Gm*y*z*******F*-%*THICKNESSG6 #\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fc_m-%&COLORG6&%$RGBG$\"\"%!\"\"$\"\"!F ]`m$\"\"\"F]`m-%%VIEWG6$;$!+3`=$G'!\"*$\"+Fjzq:!\")%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 46.000000 43.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The constant coefficient in the Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c = 1/Pi;" "6#/%\"cG*&\"\"\"F&%#PiG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. Pi);" "6#-%$IntG6$-% \"fG6#%\"xG/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "1/Pi;" "6#*&\"\"\"F$%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(1-2*x/Pi),x = 0 .. Pi);" "6#-%$IntG6$-%!G 6#,&\"\"\"F**(\"\"#F*%\"xGF*%#PiG!\"\"F//F-;\"\"!F." }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "x - x^2/Pi " "6#, &%\"xG\"\"\"*&F$\"\"#%#PiG!\"\"F)" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PI ECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F (7$\"\"!F(" }{TEXT -1 6 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 36 "The coefficients of the cosine terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 49 "The Fouri er coefficients of the cosine terms are " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = 2/Pi;" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\" %#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*cos*k*x,x = 0 .. \+ Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF+%\"kGF+F*F+/F*;\"\"!%#P iG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int((1-2*x/Pi)*cos*k*x,x = 0 .. Pi);" "6#-%$IntG6$**,& \"\"\"F(*(\"\"#F(%\"xGF(%#PiG!\"\"F-F(%$cosGF(%\"kGF(F+F(/F+;\"\"!F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "I nt((1-2*x/Pi)*cos(k*x),x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*&,&\"\" \"F)*(\"\"#F)%\"xGF)%#PiG!\"\"F.F)-%$cosG6#*&%\"kGF)F,F)F)F,-F%6$*&%\" uGF)-%!G6#*&%#dvGF)%#dxGF.F)F," }{TEXT -1 8 ", with " }{XPPEDIT 18 0 "u=1-2*x/Pi" "6#/%\"uG,&\"\"\"F&*(\"\"#F&%\"xGF&%#PiG!\"\"F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = sin*k*x/k;" "6#/%\"vG**%$sinG\"\"\"% \"kGF'%\"xGF'F(!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 44 " Hence, by the integration by parts formula: " }{XPPEDIT 18 0 "Int(u*`` (dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6# *&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/ F0F)F1F0" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[k] = 2/Pi" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((1-2*x/Pi)*cos*k*x,x = 0 .. Pi);" "6#-%$IntG6$**,&\"\"\"F(*(\"\"#F(%\"xGF(%#PiG!\"\"F-F(%$cosGF(%\"kGF(F +F(/F+;\"\"!F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " } {XPPEDIT 18 0 "2/Pi" "6#*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 2 " " } {XPPEDIT 18 0 "[(1-2*x/Pi)*``(sin*k*x/k)];" "6#7#*&,&\"\"\"F&*(\"\"#F& %\"xGF&%#PiG!\"\"F+F&-%!G6#**%$sinGF&%\"kGF&F)F&F1F+F&" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEW ISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-2/Pi " "6#,$*&\"\"#\"\"\"%#PiG!\"\"F(" }{XPPEDIT 18 0 "Int(``(-2/Pi)*``(sin *k*x/k),x = 0 .. Pi);" "6#-%$IntG6$*&-%!G6#,$*&\"\"#\"\"\"%#PiG!\"\"F/ F--F(6#**%$sinGF-%\"kGF-%\"xGF-F4F/F-/F5;\"\"!F." }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 0-``(4*cos*k*x/(k^2*Pi^2));" "6#/%!G,&\"\"!\"\"\"- F$6#*,\"\"%F'%$cosGF'%\"kGF'%\"xGF'*&F-\"\"#%#PiGF0!\"\"F2" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PI ECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = -``(4/(k^2*Pi^2))*(cos*k*Pi-1);" "6#/%!G,$*&-F$6#*&\"\"%\"\"\"*&%\" kG\"\"#%#PiGF.!\"\"F+,&*(%$cosGF+F-F+F/F+F+F+F0F+F0" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 4*(1-(-1)^k)/(k^2*Pi^2);" "6#/%!G*(\"\"%\"\"\",&F' F'),$F'!\"\"%\"kGF+F'*&F,\"\"#%#PiGF.F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "PIECEWISE([8/(k^2*Pi^2), `if k is odd`],[0, `if k is even`]);" "6#- %*PIECEWISEG6$7$*&\"\")\"\"\"*&%\"kG\"\"#%#PiGF,!\"\"%,if~k~is~oddG7$ \"\"!%-if~k~is~evenG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, \+ 1, 2, 3, 4, 5, 6, 7, 8], [a[k], `|`, 8/Pi^2, 0, 8/9/Pi^2, 0, 8/25/Pi^2 , 0, 8/49/Pi^2, 0]])" "6#-%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$ \"\"%\"\"&\"\"'\"\"(\"\")7,&%\"aG6#F(F)*&F1F**$%#PiGF+!\"\"\"\"!*(F1F* \"\"*F9*$F8F+F9F:*(F1F*\"#DF9*$F8F+F9F:*(F1F*\"#\\F9*$F8F+F9F:" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "interface(showassumed=0): k := 'k': assume(k,in teger):\na[k]=2/Pi*Int((1-2*x/Pi)*cos(k*x),x=0..Pi);\n``=value(rhs(%)) ;\naa := unapply(rhs(%),k):\nmatrix([[k,`|`,seq(k,k=1..8)],[a[k],`|`,s eq(aa(k),k=1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%#k|i rG,$,$-%$IntG6$*&,&\"\"\"F/*(\"\"#F/%\"xGF/%#PiG!\"\"F4F/-%$cosG6#*&F' F/F2F/F//F2;\"\"!F3*&F1F/F3F4F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G,$**\"\"%\"\"\"%#PiG!\"#,&F(!\"\")F,%#k|irGF(F(F.F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,%#k|irG%\"|grG\"\"\"\"\"#\"\"$\"\" %\"\"&\"\"'\"\"(\"\")7,&%\"aG6#F(F),$*&F1F*%#PiG!\"#F*\"\"!,$*(F1F*\" \"*!\"\"F8F9F*F:,$*(F1F*\"#DF>F8F9F*F:,$*(F1F*\"#\\F>F8F9F*F:Q)pprint3 06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 22 "The Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "F(x) = Sum(``(8*Pi^2/((2*k-1)^2))*cos((2*k-1)*x),k = 1 \+ .. infinity);" "6#/-%\"FG6#%\"xG-%$SumG6$*&-%!G6#*(\"\")\"\"\"*$%#PiG \"\"#F1*$,&*&F4F1%\"kGF1F1F1!\"\"F4F9F1-%$cosG6#*&,&*&F4F1F8F1F1F1F9F1 F'F1F1/F8;F1%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture compares th e graphs of some truncated Fourier series with the graph of the functi on " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "f := x -> piecewise(x < 0,1+2*x/Pi,1-2*x/Pi):\nf_ := x -> f(x-2*Pi*floo r((x+Pi)/(2*Pi))):\nFS := (x,n) -> sum(8/((2*k-1)^2*Pi^2)*cos((2*k-1)* x),k=1..n);\nplot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5)],x=-2 *Pi..3*Pi,\n color=[black,red,blue,green,magenta,coral],linestyle=[3 ,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%) operatorG%&arrowGF)-%$sumG6$,$**\"\")\"\"\",&*&\"\"#F3%\"kGF3F3F3!\"\" !\"#%#PiGF9-%$cosG6#*&F4F39$F3F3F3/F7;F39%F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 506 232 232 {PLOTDATA 2 "6*-%'CURVESG6%7cr7$$!3)****>YH&=$G' !#<$\"25>V+#********F*7$$!3!QHkk%3*>6'F*$\"3y'R))Q@U,\"*)!#=7$$!3j(e3$ 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F27$Fe_m$!3Gzw%zjV2u'F27$Fj_m$!3!yZb(f32UyF27$F_`m$!3#G-#3yje]!*F27$Fd `m$!3-LC+oy/'f*F2-Fi`m6&F[amFc[o$\")AR!)\\Fe[oFf[oFg[o-%+AXESLABELSG6$ Q\"x6\"Q!Ffht-%%VIEWG6$;$!+3`=$G'!\"*$\"+izxC%*F^it%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "Taking a truncated series with a \+ fairly large number of terms appears to give a good match with the gra ph of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "f := x -> piecewise(x < 0,1+2*x/Pi,1-2*x/Pi):\nf_ := x -> f(x-2*Pi*floor((x+Pi)/(2*Pi))):\nFS := (x,n) -> sum(8/((2*k-1)^2 *Pi^2)*cos((2*k-1)*x),k=1..n); \nplot([f_(x),FS(x,10)],x=-2*Pi..3*Pi,n umpoints=80,\n color=[black,COLOR(RGB,.8,0,1)],linestyle=[3,1],th ickness=[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"n 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!f#F27$F_jl$\"3Q%QKo8P9P\"F27$Fdjl$\"3*)Qq\\5.U-#*Fhjl7$Fjjl$!3?R\\&y! p_*>\"F27$F_[m$!3k%f*GCpxnBF27$Fd[m$!3yYFh1Y*3l$F27$Fi[m$!3-MEa??2!*[F 27$F^\\m$!31%yv?1V+A'F27$Fc\\m$!3@d\\)=q\"pSuF27$Fh\\m$!3)Q'>n>08o()F2 7$F]]m$!3?$)H#\\\"f_(z*F2-%&COLORG6&Fd]m$\"\")!\"\"$Fe]mFe]m$Fi]mFe]m- Fg]m6#\"\"#-F[^mFh]m-%+AXESLABELSG6$Q\"x6\"Q!F]dn-%%VIEWG6$;$!+3`=$G'! \"*$\"+izxC%*Fedn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Truncated Fourier series can be obtained using the procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "f := x -> piecewise(x<0,1+2*x/Pi,1-2*x/Pi):\n'f(x)'= f(x);\nFourierSeries(f(x),x=0..Pi,type=cos,numterms=20):\nF := unapply (%,x):\n'F(x)'=F(x);\nplot(F(x),x=-2*Pi..3*Pi,color=COLOR(RGB,.8,0,1), thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIEC EWISEG6$7$,&\"\"\"F-*(\"\"#F-F'F-%#PiG!\"\"F-2F'\"\"!7$,&F-F-*(F/F-F'F -F0F1F1%*otherwiseG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG, 6*(\"\")\"\"\"%#PiG!\"#-%$cosGF&F+F+*&#F*\"\"*F+*&F,F--F/6#,$*&\"\"$F+ F'F+F+F+F+F+*&#F*\"#DF+*&F,F--F/6#,$*&\"\"&F+F'F+F+F+F+F+*&#F*\"#\\F+* &F,F--F/6#,$*&\"\"(F+F'F+F+F+F+F+*&#F*\"#\")F+*&F,F--F/6#,$*&F2F+F'F+F +F+F+F+*&#F*\"$@\"F+*&F,F--F/6#,$*&\"#6F+F'F+F+F+F+F+*&#F*\"$p\"F+*&F, F--F/6#,$*&\"#8F+F'F+F+F+F+F+*&#F*\"$D#F+*&F,F--F/6#,$*&\"#:F+F'F+F+F+ F+F+*&#F*\"$*GF+*&F,F--F/6#,$*&\"#F+F'F+F+F+F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 465 157 157 {PLOTDATA 2 "6'-%'CURVESG6#7er7$$!3)****>YH&=$G'!#<$\"3'4*H#\\\"f_(z*! #=7$$!3!QHkk%3*>6'F*$\"3G;')o#34x\"*)F-7$$!3j(e3$)R'zSfF*$\"3O=fryz+Hy F-7$$!3+[+MC5%=z&F*$\"3quXd\">Ut'oF-7$$!3Z2:P]c)Gk&F*$\"3pz**p1VcDfF-7 $$!3WWwe`(p`Z&F*$\"3a\\5z$4\"[_[F-7$$!3I#y.o&Q&yI&F*$\"3;?ty&)oe(z$F-7 $$!3#p\\54FF-7$$!3k57.b7gq\\F*$\"3;DK!p&=1`;F-7$$! 3$=$z^0kx-[F*$\"3mN?s_!)Hcc!#>7$$!3.`Y+c:&\\j%F*$!3iN6F#*p&*G[Fen7$$!3 wY(HuVc$zWF*$!3I3-tG'eQ\\\"F-7$$!3]S[&)=8wBVF*$!3Q-\"4l!)QKY#F-7$$!31C :*fe^E;%F*$!3#Rvk(G3Z6NF-7$$!3l2#GJ&=a,SF*$!3u+WS8sQ7XF-7$$!3$=6Jp$># \\$QF*$!3@:pz!)z^+cF-7$$!3c:St??IoOF*$!3logqUe-JmF-7$$!3A.8sck@-NF*$!3 oUxmguR@xF-7$$!3J\"f3F*38OLF*$!3#G%*R(pS5Y()F-7$$!3m\\'fT=62D$F*$!3OKO D5/'HP*F-7$$!3+32hv9HlJF*$!37uj&zz/\\x*F-7$$!3$HZt%[l$R9$F*$!3#pm+9F.t z*F-7$$!3SPiL@;eAJF*$!3!\\')*=Zm$Hy*F-7$$!3)=+*>%pE75$F*$!3%Q#*p1llEt* F-7$$!3!owhqwr)zIF*$!3%Gz%)[#*e%\\'*F-7$$!3v&H(y7>;PIF*$!3i%fM)R&oXS*F -7$$!39DG^e?X%*HF*$!3%H%3h>p0)4*F-7$$!3AwJ%pgwR%GF*$!31,5;(F-7$$!3)[()>pV,T_#F*$!3%G@]Y]gI1'F-7$$!3+Ai Y=&Q 6^9?F*$!3TJ))yz/T>GF-7$$!3!\\\")R(3Gf]=F*$!3%))38*GGk(y\"F-7$$!3YM)f*y Wn'o\"F*$!3\"R[M^FBNI(Fen7$$!3kvH'=$4#y`\"F*$\"3v_v%RIit.#Fen7$$!3!o6m ZQn*)Q\"F*$\"3kC83[odi6F-7$$!3)4jW)>h'>@\"F*$\"3#4y=[@$yvAF-7$$!3;XJ# \\&['\\.\"F*$\"3UPAe_WeAMF-7$$!3?V]Sic@]))F-$\"3U!p]YV5ON%F-7$$!3![jyb x#y]tF-$\"3G@!**Hv^OL&F-7$$!3:V(=%pee1cF-$\"3db]YMXF7kF-7$$!3]^)eK'*)Q iQF-$\"3iN!=1N9]c(F-7$$!3,K_^e\"z%=BF-$\"3Qq\\\\r5U)[)F-7$$!37DhrPNpXx Fen$\"33?2c@Azp&*F-7$$!3Q_dGZ8IGcFen$\"3Y&oqH]ONn*F-7$$!3jz`&o:44^$Fen $\"3M2w)4UW#[(*F-7$$!3)o+Dk'p^$R\"Fen$\"3gf>iXEn*y*F-7$$\"3feO0SAvQs!# ?$\"3=TTjuMS&z*F-7$$\"3M6h'[gf'e\\Fen$\"3*>+V;bO0q*F-7$$\"3%o&os&)RW$> *Fen$\"3'y*=zpA+&[*F-7$$\"3yM[uu7Im1i,$)F-7$$\"3sHp&)=YEEUF-$\"3zq6#fQH7L(F-7$$\"3dvoz'3r#R eF-$\"3m&)Gv\")*esE'F-7$$\"3'Q)>g?TFAvF-$\"3=t6`7yF@_F-7$$\"3;#42W:x_? *F-$\"3vAGOd'oN8%F-7$$\"3?5FhJ*z]2\"F*$\"3]s3?u_rhJF-7$$\"3>6ZyZ@jH7F* $\"3;*paN%=7m@F-7$$\"3y$Q0AfPjR\"F*$\"3,1-'e@#G96F-7$$\"3QcgiOI/j:F*$ \"3'4G.%H-!)zZFgz7$$\"3xAvmTJ?OF* $!3)Hq%GAp1]@F-7$$\"3/^H=F'*4g?F*$!30^3=Dqc>JF-7$$\"3!H\"pl2g$3@#F*$!3 3j,)zB\\32%F-7$$\"3Axo7GajtBF*$!3\\Vwq$Gbs6&F-7$$\"3aTof[[VODF*$!3-D#[ !G3jPhF-7$$\"3bpq()zDi/FF*$!3=@^_$)3&eB(F-7$$\"3c(Hd6J5G(GF*$!3Kp!))4( 3Uc#)F-7$$\"3Ac)*\\_(z]&HF*$!3\"=#p6bfS/))F-7$$\"3'[TUQ>\\t.$F*$!3m:f$ HP;eS*F-7$$\"3S%p8X\"R[yIF*$!3s\"))\\:\\;Ik*F-7$$\"3^t\\=N'='>JF*$!3wq -Mm21y(*F-7$$\"3G81_&*f=SJF*$!3gx6_BdW(z*F-7$$\"3i_i&eN`2;$F*$!3I96&H' pq#y*F-7$$\"3Q#*=>;2K\"=$F*$!347z^'4kYt*F-7$$\"3;Kv_w!))=?$F*$!3sf;Sr: &fl*F-7$$\"3#35df!z3hLF*$!3a&>F/N&3r&)F-7$$\"3]pmQNxG?NF*$!3j#3Cc*Rx6w F-7$$\"3)3LSJ+_qp$F*$!3cZD*Q$4?XkF-7$$\"3G#*R*3F;Q(QF*$!3ANpViCA_`F-7$ $\"3))p!zt#okKSF*$!3/3_y=f(eJ%F-7$$\"3YZT'QQx9>%F*$!3*oB8M=^iK$F-7$$\" 3aUah/H1hVF*$!3$=tsQi#4HAF-7$$\"3iPnOD%[1`%F*$!3ANBfY#)*>;\"F-7$$\"3!) Q<`7\"=Vo%F*$!3-&[\\TuXKt\"Fen7$$\"3(*Rnp*z()z$[F*$\"3\\I7(GZ1s$zFen7$ $\"3nY9Dd$*)f+&F*$\"3%zNt!GdG1(F-7 $$\"3%Hbe(R@;#)fF*$\"3%*4&HpdXt1)F-7$$\"3.NH!R!*4P9'F*$\"3+-&pl\"4Ab\" *F-7$$\"3+P3I)G\"*f='F*$\"3'*y)**4qz=X*F-7$$\"33Q()psEFGiF*$\"3'HB^iI/ $z'*F-7$$\"3i)o(*[O8%\\iF*$\"33&p>'f/)=v*F-7$$\"3;Rm4dSbqiF*$\"3Q`4hy5 2\"z*F-7$$\"3q*e&H\\Zp\"H'F*$\"3;?$=vd$f%z*F-7$$\"3CSX\\Ta$GJ'F*$\"3;] zm[3Ci(*F-7$$\"3QU.H5#)R(R'F*$\"3V0T5fR&yL*F-7$$\"3WXh3z4'>['F*$\"3O:4 k*p#o:()F-7$$\"3g4Q$G(*[[k'F*$\"3e,&RmBw^r(F-7$$\"3wt9empt2oF*$\"3O7tC RPAXmF-7$$\"3E#oAmY8V(pF*$\"3Wj+-&)>H9cF-7$$\"3v!*Qmm**)39(F*$\"3d#eH# [DwEXF-7$$\"3d)4#=Hr31tF*$\"3'3J;wQV(*\\$F-7$$\"3S1.q\"H%GruF*$\"3G$=/ 1uScU#F-7$$\"3'R%4MFL3BwF*$\"3s6&)ypWC![\"F-7$$\"3a\"e\")HO#)[x(F*$\"3 ,L9,D#[Z$\\Fen7$$\"3/P\"Q$)=h)[zF*$!3.Cgq%)=&[%fFen7$$\"3n\"p%p8+%G7)F *$!3Q!ePVhL'>U\\`Mn$p#F-7$$\"3Jmd_%[dSV)F* $!3gtU7scE-PF-7$$\"37B\\8_)o**f)F*$!3ym'R]Lj.u%F-7$$\"3%*zSu>-)ew)F*$! 3+o*y'[B\"H\"eF-7$$\"35R&f$o&yY#*)F*$!3JDtDgFA3oF-7$$\"3E)*\\(p\"pZ$3* F*$!3b:'>G%)\\`$yF-7$$\"3!*)\\_%Hu7a#*F*$!3?pS>G\"p<#*)F-7$$\"3^***H>% zxC%*F*$!3?$)H#\\\"f_(z*F--%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q !Fjam-%&COLORG6&%$RGBG$\"\")!\"\"$\"\"!Fdbm$\"\"\"Fdbm-%%VIEWG6$;$!+3` =$G'!\"*$\"+izxC%*F]cm%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 47 "Find the Fourier sine series of the function: \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)=PIECEWISE([ -1,-1<=x and x<0],[0,x=0],[1,0 " 0 "" {MPLTEXT 1 0 174 "f := x -> p iecewise(x=0,0,1):\n'f(x)'=f(x);\nFourierSeries(f(x),x=0..1,type=sine, numterms=9,info=1):\nF := unapply(%,x);\nplot(F(x),x=-2..3,numpoints=7 5,color=COLOR(RGB,.8,0,1));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 " " {TEXT -1 47 "Find the Fourier cosine series of the function " } {XPPEDIT 18 0 "f(x)=abs(x)" "6#/-%\"fG6#%\"xG-%$absG6#F'" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-2,2]" "6#7$,$\"\"#!\"\"F%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1" "6#/%\"cG\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[k]=4*(( -1)^k-1)/k^2/Pi^2" "6#/&%\"aG6#%\"kG**\"\"%\"\"\",&),$F*!\"\"F'F*F*F.F **$F'\"\"#F.*$%#PiGF0F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "Fourier cosine series: " }{XPPEDIT 18 0 "1-8/Pi^2" "6#,&\"\"\"F$* &\"\")F$*$%#PiG\"\"#!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(Pi*x/ 2)-8/(9*Pi^2)" "6#,&-%$cosG6#*(%#PiG\"\"\"%\"xGF)\"\"#!\"\"F)*&\"\")F) *&\"\"*F)*$F(F+F)F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(3/2*Pi*x)-8 /(25*Pi^2)" "6#,&-%$cosG6#**\"\"$\"\"\"\"\"#!\"\"%#PiGF)%\"xGF)F)*&\" \")F)*&\"#DF)*$F,F*F)F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(5/2*Pi* x)-8/(49*Pi^2)" "6#,&-%$cosG6#**\"\"&\"\"\"\"\"#!\"\"%#PiGF)%\"xGF)F)* &\"\")F)*&\"#\\F)*$F,F*F)F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(7/2 *Pi*x)-8/(81*Pi^2)" "6#,&-%$cosG6#**\"\"(\"\"\"\"\"#!\"\"%#PiGF)%\"xGF )F)*&\"\")F)*&\"#\")F)*$F,F*F)F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "co s(9/2*Pi*x)-` . . . `" "6#,&-%$cosG6#**\"\"*\"\"\"\"\"#!\"\"%#PiGF)%\" xGF)F)%(~.~.~.~GF+" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "f := x -> abs(x);\nFourie rSeries(f(x),x=-2..2,numterms=9,info=1):\nF := unapply(%,x);\nplot(F(x ),x=-2..7,numpoints=75,color=COLOR(RGB,.8,0,1));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 49 "(a) Show that the Fourier series of t he function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 11 " g iven by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x ^3;" "6#/-%\"fG6#%\"xG*$F'\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-1 \+ <= x;" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " is periodic with period 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(``(-2*(-1)^k*(k^2*Pi^2-6)/(k^3*Pi^3))*sin*k*Pi*x,k \+ = 1 .. infinity);" "6#-%$SumG6$*,-%!G6#,$**\"\"#\"\"\"),$F-!\"\"%\"kGF -,&*&F1F,%#PiGF,F-\"\"'F0F-*&F1\"\"$F4F7F0F0F-%$sinGF-F1F-F4F-%\"xGF-/ F1;F-%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 74 "(b) Compare the graphs of some truncated \+ Fourier series with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________ _________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_______________________________ ___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 49 "(a) Show that the Fourier series of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 11 " given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x*cos*x;" "6#/-%\"fG6#%\"xG*(F'\"\"\"%$cosGF)F' F)" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-Pi/2 <= x;" "6#1,$*&%#PiG\"\"\" \"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` < Pi/2;" "6#2%!G*&%#PiG\"\"\"\" \"#!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "Pi" "6#%#Pi G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(``(16*(-1)^(1+k)*k/(Pi*(1+2*k )^2*(-1+2*k)^2))*sin*2*k*x,k = 1 .. infinity);" "6#-%$SumG6$*,-%!G6#** \"#;\"\"\"),$F,!\"\",&F,F,%\"kGF,F,F1F,*(%#PiGF,*$,&F,F,*&\"\"#F,F1F,F ,F7F,,&F,F/*&F7F,F1F,F,F7F/F,%$sinGF,F7F,F1F,%\"xGF,/F1;F,%)infinityG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Compare the graphs of some truncated Fourier series w ith the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ _" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 49 "(a) Show that the Fou rier series of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 11 " given by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x*abs(x);" "6#/-%\"f G6#%\"xG*&F'\"\"\"-%$absG6#F'F)" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-1 \+ <= x;" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 27 " is periodic with period 2 " }}{PARA 0 "" 0 "" {TEXT -1 3 "is \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``((4*(-1)^k- 2*k^2*Pi^2*(-1)^k-4)/(k^3*Pi^3))*sin*k*Pi*x,k = 1 .. infinity);" "6#-% $SumG6$*,-%!G6#*&,(*&\"\"%\"\"\"),$F.!\"\"%\"kGF.F.**\"\"#F.*$F2F4F.%# PiGF4),$F.F1F2F.F1F-F1F.*&F2\"\"$F6F:F1F.%$sinGF.F2F.F6F.%\"xGF./F2;F. %)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Compare the graphs of some truncated Fourie r series with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 49 "(a) Show that the Fou rier series of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 11 " given by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x^2-x^4;" "6#/-%\"fG 6#%\"xG,&*$F'\"\"#\"\"\"*$F'\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2% !G\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 27 " is periodic with period 2 " }}{PARA 0 "" 0 "" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2/15+Sum (``(4*(-1)^(1+k)*(k^2*Pi^2-12)/(k^4*Pi^4))*cos*k*Pi*x,k = 1 .. infinit y);" "6#,&*&\"\"#\"\"\"\"#:!\"\"F&-%$SumG6$*,-%!G6#**\"\"%F&),$F&F(,&F &F&%\"kGF&F&,&*&F5F%%#PiGF%F&\"#7F(F&*&F5F1F8F1F(F&%$cosGF&F5F&F8F&%\" xGF&/F5;F&%)infinityGF&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Compare the graphs of some trunc ated Fourier series with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 1 "." }}{PARA 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 34 "____________________ ______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 49 "(a) Show that the Fourier series of t he function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 11 " g iven by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x*sin*x;" "6#/-%\"fG6#%\"xG*(F'\"\"\"% $sinGF)F'F)" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-Pi <= x;" "6#1,$%#PiG! \"\"%\"xG" }{XPPEDIT 18 0 "`` < Pi;" "6#2%!G%#PiG" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-cos(x)/2+``;" "6#,(\"\"\"F$*&-%$cosG6#%\"xGF$\" \"#!\"\"F+%!GF$" }{XPPEDIT 18 0 "Sum(``(2*(-1)^(1+k)/(k^2-1))*cos*k*x, k = 2 .. infinity);" "6#-%$SumG6$**-%!G6#*(\"\"#\"\"\"),$F,!\"\",&F,F, %\"kGF,F,,&*$F1F+F,F,F/F/F,%$cosGF,F1F,%\"xGF,/F1;F+%)infinityG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Compare the graphs of some truncated Fourier series w ith the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ _" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }