{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " 260 266 "" 1 18 0 0 0 0 0 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 "" 260 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{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 "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 277 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Ti mes" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "Differentiation of Fourier series " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canad a" }}{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 277 9 "four ier.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 similar to the one that follows, where the file path gives its locati on." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/pr ocdrs/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 79 "The coefficients in the Fourier series of the derivativ e of a periodic function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "phi(x )" "6#-%$phiG6#%\"xG" }{TEXT -1 36 " is a periodic function with perio d " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 31 " which has the Fourier series: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c+``;" "6#,&%\"cG\"\"\"%!GF%" }{XPPEDIT 18 0 "Sum(a[k]* cos(k*Pi*x/L)+b[k]*sin(k*Pi*x/L),k = 1 .. infinity);" "6#-%$SumG6$,&*& &%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#PiGF,%\"xGF,%\"LG!\"\"F,F,*&&%\"bG 6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F,%)infinityG" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c=1/(2*L)" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%\"LGF&! \"\"" }{XPPEDIT 18 0 "Int(phi(x),x = -L .. L);" "6#-%$IntG6$-%$phiG6#% \"xG/F);,$%\"LG!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[k]=1/L" "6 #/&%\"aG6#%\"kG*&\"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(phi(x)*cos(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\" \"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3F2" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "b[k]=1/L" "6#/&%\"bG6#%\"kG*&\"\"\"F)%\"LG !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*sin(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$sinG6#**%\"kGF+%#PiGF+F* F+%\"LG!\"\"F+/F*;,$F2F3F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Suppose that the " }{TEXT 261 10 "derivative" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi*`'`(x)" "6#*&%$phi G\"\"\"-%\"'G6#%\"xGF%" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(x)" "6# -%$phiG6#%\"xG" }{TEXT -1 27 " also has a Fourier series." }}{PARA 0 " " 0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "cos(k*Pi*x/L) " "6#-%$cosG6#**%\"kG\"\"\"%#PiGF(%\"xGF(%\"LG!\"\"" }{TEXT -1 27 " in the Fourier series for " }{XPPEDIT 18 0 "phi*`'`(x)" "6#*&%$phiG\"\" \"-%\"'G6#%\"xGF%" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[k] = 1/L;" "6#/&%\"AG6#%\"kG*&\"\"\"F)%\"LG!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi*`'`(x)*cos(k*Pi*x/L),x = -L \+ .. L);" "6#-%$IntG6$*(%$phiG\"\"\"-%\"'G6#%\"xGF(-%$cosG6#**%\"kGF(%#P iGF(F,F(%\"LG!\"\"F(/F,;,$F3F4F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 40 " 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$*&F4F)-F+6#*&%#d uGF)F/F0F)F1F0" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(phi*`'`(x)*cos(k*Pi*x/L),x = -L .. L)" "6#-%$IntG6$ *(%$phiG\"\"\"-%\"'G6#%\"xGF(-%$cosG6#**%\"kGF(%#PiGF(F,F(%\"LG!\"\"F( /F,;,$F3F4F3" }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = co s(k*Pi*x/L), v = phi(x)],[du/dx = ``(-k*Pi/L)*sin(k*Pi*x/L), dv/dx = p hi*`'`(x)]);" "6#-%*PIECEWISEG6$7$/%\"uG-%$cosG6#**%\"kG\"\"\"%#PiGF.% \"xGF.%\"LG!\"\"/%\"vG-%$phiG6#F07$/*&%#duGF.%#dxGF2*&-%!G6#,$*(F-F.F/ F.F1F2F2F.-%$sinG6#**F-F.F/F.F0F.F1F2F./*&%#dvGF.F " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Rate of decay of Fourier coeffic ients" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"x G" }{TEXT -1 36 " is a periodic function with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 39 " which has the Fourier se ries expansion" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi (x) = c;" "6#/-%$phiG6#%\"xG%\"cG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "S um(a[k]*cos(k*Pi*x/L)+b[k]*sin(k*Pi*x/L),k = 1 .. infinity);" "6#-%$Su mG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#PiGF,%\"xGF,%\"LG!\"\"F,F, *&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F,%)infinityG" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c=1/(2*L)" "6#/%\"cG*&\"\"\"F&*&\"\" #F&%\"LGF&!\"\"" }{XPPEDIT 18 0 "Int(phi(x),x = -L .. L);" "6#-%$IntG6 $-%$phiG6#%\"xG/F);,$%\"LG!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a [k]=1/L" "6#/&%\"aG6#%\"kG*&\"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(phi(x)*cos(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&- %$phiG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3 F2" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "b[k]=1/L" "6#/&%\"bG6#%\"kG* &\"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*sin(k* Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$sinG6#**% \"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "From Parseval 's formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(2*L)" "6#*&\"\"\"F$*&\"\"#F$%\"LGF$! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)^2,x = -L .. L) = c^2+ 1/2;" "6#/-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F+;,$%\"LG!\"\"F0,&*$%\"cGF, \"\"\"*&F5F5F,F1F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2, k = 1 .. infinity);" "6#-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6 #F+F,F-/F+;F-%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 23 "it follows that, since " }{XPPEDIT 18 0 "Int(phi(x)^2,x = -L .. L) ;" "6#-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F*;,$%\"LG!\"\"F/" }{TEXT -1 23 " is finite, the series " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k = 1 .. in finity)" "6#-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-/F+;F -%)infinityG" }{TEXT -1 11 " converges." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "Limit([a[k]^2+b[k]^2],k = infinity) = 0;" "6#/ -%&LimitG6$7#,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F-F.F//F-%)infinity G\"\"!" }{TEXT -1 3 " so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(a[k],k=infinity)=0" "6#/-%&LimitG6$&%\"aG6#%\"kG/ F*%)infinityG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Limit(b[k],k=i nfinity)=0" "6#/-%&LimitG6$&%\"bG6#%\"kG/F*%)infinityG\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 44 "that is, the Fourier coeffici ents tend to 0." }}{PARA 0 "" 0 "" {TEXT -1 58 "(This is a special cas e of the Riemann - Lebesgue lemma.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Now suppose that " }{XPPEDIT 18 0 "phi( x)" "6#-%$phiG6#%\"xG" }{TEXT -1 41 " is differentiable everywhere, an d that " }{XPPEDIT 18 0 "phi*`'`(x)" "6#*&%$phiG\"\"\"-%\"'G6#%\"xGF% " }{TEXT -1 38 " also has a Fourier series expansion. " }}{PARA 0 "" 0 "" {TEXT -1 33 "Then the Fourier coefficients of " }{XPPEDIT 18 0 "p hi*`'`(x)" "6#*&%$phiG\"\"\"-%\"'G6#%\"xGF%" }{TEXT -1 1 " " }{TEXT 261 17 "also tend to zero" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "The Fourier coefficients of " }{XPPEDIT 18 0 "phi*`'`(x)" "6#*&%$p hiG\"\"\"-%\"'G6#%\"xGF%" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "A[k] = ` `(k*Pi/L)*b[k];" "6#/&%\"AG6#%\"kG*&-%!G6#*(F'\"\"\"%#PiGF-%\"LG!\"\"F -&%\"bG6#F'F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B[k] = ``(-k*Pi/L)* a[k];" "6#/&%\"BG6#%\"kG*&-%!G6#,$*(F'\"\"\"%#PiGF.%\"LG!\"\"F1F.&%\"a G6#F'F." }{TEXT -1 29 ", (see the previous section) " }}{PARA 0 "" 0 " " {TEXT -1 7 "we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(k*a[k],k=infinity)=0" "6#/-%&LimitG6$*&%\"kG\"\"\"&%\"aG6# F(F)/F(%)infinityG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Limit(k*b [k],k=infinity)=0" "6#/-%&LimitG6$*&%\"kG\"\"\"&%\"bG6#F(F)/F(%)infini tyG\"\"!" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 264 20 "____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Not only do the Fourier coeffic ients " }{XPPEDIT 18 0 "a[k]" "6#&%\"aG6#%\"kG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 28 " tend to 0, but they do so \"" }{TEXT 261 12 "faster than " }{XPPEDIT 18 0 "1/k" "6#*&\"\"\"F $%\"kG!\"\"" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 34 "This ar gument can be repeated. If " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG " }{TEXT -1 42 " is twice differentiable everywhere, and " }{XPPEDIT 18 0 "phi*`''`(x)" "6#*&%$phiG\"\"\"-%#''G6#%\"xGF%" }{TEXT -1 41 " al so has a Fourier series expansion then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(k^2*a[k],k = infinity) = 0;" "6#/-%&LimitG 6$*&%\"kG\"\"#&%\"aG6#F(\"\"\"/F(%)infinityG\"\"!" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "Limit(k^2*b[k],k = infinity) = 0;" "6#/-%&LimitG6$*& %\"kG\"\"#&%\"bG6#F(\"\"\"/F(%)infinityG\"\"!" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 268 21 "_____________________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "the Fourier coefficie nts " }{XPPEDIT 18 0 "a[k]" "6#&%\"aG6#%\"kG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 12 " tend to 0 \"" }{TEXT 261 12 "faster than " }{XPPEDIT 18 0 "1/(k^2);" "6#*&\"\"\"F$*$%\"kG\" \"#!\"\"" }{TEXT -1 13 "\", and so on." }}{PARA 0 "" 0 "" {TEXT -1 170 "Roughly speaking, the \"smoother\" the function, the faster its F ourier coefficients tend to 0, which also gives a measure of the rate \+ of convergence of the Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 82 "The Fourier series of two families of infinitely d ifferentiable periodic functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Two trigonometric se ries" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 41 "We consider the two trigonometric series " }{XPPEDIT 18 0 "Sum(cos(2*k*Pi*x)/(n^k),k = 0 .. infinity)" "6#-%$SumG6$*&-%$cos G6#**\"\"#\"\"\"%\"kGF,%#PiGF,%\"xGF,F,)%\"nGF-!\"\"/F-;\"\"!%)infinit yG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Sum(sin(2*k*Pi*x)/(n^k),k = 0 .. infinity)" "6#-%$SumG6$*&-%$sinG6#**\"\"#\"\"\"%\"kGF,%#PiGF,%\"xG F,F,)%\"nGF-!\"\"/F-;\"\"!%)infinityG" }{TEXT -1 8 ", where " }{TEXT 270 1 "n" }{TEXT -1 47 " is a positive integer which is greater than 1 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When \+ " }{XPPEDIT 18 0 "n>=2" "6#1\"\"#%\"nG" }{TEXT -1 29 ", infinite geome tric series " }{XPPEDIT 18 0 "Sum(exp(2*k*Pi*i*x)/(n^k),k = 0 .. infi nity)" "6#-%$SumG6$*&-%$expG6#*,\"\"#\"\"\"%\"kGF,%#PiGF,%\"iGF,%\"xGF ,F,)%\"nGF-!\"\"/F-;\"\"!%)infinityG" }{TEXT -1 30 " of complex terms has the sum" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "1/( 1-exp(2*Pi*i*x)/n)=n/(n-exp(2*Pi*i*x))" "6#/*&\"\"\"F%,&F%F%*&-%$expG6 #**\"\"#F%%#PiGF%%\"iGF%%\"xGF%F%%\"nG!\"\"F1F1*&F0F%,&F0F%-F)6#**F,F% F-F%F.F%F/F%F1F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Using the Euler formula " }{XPPEDIT 18 0 "exp(2*k*Pi*i*x)=cos(2*k*Pi*x)+i*s in(2*k*Pi*x)" "6#/-%$expG6#*,\"\"#\"\"\"%\"kGF)%#PiGF)%\"iGF)%\"xGF),& -%$cosG6#**F(F)F*F)F+F)F-F)F)*&F,F)-%$sinG6#**F(F)F*F)F+F)F-F)F)F)" } {TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((cos(2*k*Pi*x)+i*sin(2*k*Pi*x))/(n^k),k = 0 .. infinity);" " 6#-%$SumG6$*&,&-%$cosG6#**\"\"#\"\"\"%\"kGF-%#PiGF-%\"xGF-F-*&%\"iGF-- %$sinG6#**F,F-F.F-F/F-F0F-F-F-F-)%\"nGF.!\"\"/F.;\"\"!%)infinityG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``= n/(n-cos(2*Pi*x)-i*sin(2*Pi*x))" "6 #/%!G*&%\"nG\"\"\",(F&F'-%$cosG6#*(\"\"#F'%#PiGF'%\"xGF'!\"\"*&%\"iGF' -%$sinG6#*(F-F'F.F'F/F'F'F0F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ n*(n-cos(2*Pi*x)+i*sin(2*Pi*x))/((n-cos(2*Pi*x))^2+sin(2*Pi*x)^2);" "6 #/%!G*(%\"nG\"\"\",(F&F'-%$cosG6#*(\"\"#F'%#PiGF'%\"xGF'!\"\"*&%\"iGF' -%$sinG6#*(F-F'F.F'F/F'F'F'F',&*$,&F&F'-F*6#*(F-F'F.F'F/F'F0F-F'*$-F46 #*(F-F'F.F'F/F'F-F'F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = n*(n-cos(2*P i*x))/(n^2-2*n*cos(2*Pi*x)+1)+i*``(n*sin(2*Pi*x)/(n^2-2*n*cos(2*Pi*x)+ 1));" "6#/%!G,&*(%\"nG\"\"\",&F'F(-%$cosG6#*(\"\"#F(%#PiGF(%\"xGF(!\" \"F(,(*$F'F.F(*(F.F(F'F(-F+6#*(F.F(F/F(F0F(F(F1F(F(F1F(*&%\"iGF(-F$6#* (F'F(-%$sinG6#*(F.F(F/F(F0F(F(,(*$F'F.F(*(F.F(F'F(-F+6#*(F.F(F/F(F0F(F (F1F(F(F1F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "By equating real parts we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((cos(2*k*Pi*x))/(n^k),k = 0 .. infinity)=n*(n-cos(2*Pi*x))/(n^2-2*n*cos(2*Pi*x)+1)" "6#/-%$Su mG6$*&-%$cosG6#**\"\"#\"\"\"%\"kGF-%#PiGF-%\"xGF-F-)%\"nGF.!\"\"/F.;\" \"!%)infinityG*(F2F-,&F2F--F)6#*(F,F-F/F-F0F-F3F-,(*$F2F,F-*(F,F-F2F-- F)6#*(F,F-F/F-F0F-F-F3F-F-F3" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 24 "________________________" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "By eq uating imaginary parts, we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(sin(2*k*Pi*x)/(n^k),k = 0 .. infinity) = n*sin(2*Pi *x)/(n^2-2*n*cos(2*Pi*x)+1);" "6#/-%$SumG6$*&-%$sinG6#**\"\"#\"\"\"%\" kGF-%#PiGF-%\"xGF-F-)%\"nGF.!\"\"/F.;\"\"!%)infinityG*(F2F--F)6#*(F,F- F/F-F0F-F-,(*$F2F,F-*(F,F-F2F--%$cosG6#*(F,F-F/F-F0F-F-F3F-F-F3" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 24 "_____ ___________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "asume(n_,integer): assume(n_>=2):\nSum(cos(2*k*Pi*x)/(n^k),k=0. .infinity);\n``=combine(subs(n_=n,value(subs(n=n_,%))));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%$cosG6#,$**\"\"#\"\"\"%\"kGF-%#PiGF -%\"xGF-F-F-)%\"nGF.!\"\"/F.;\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&*$)%\"nG\"\"#\"\"\"!\"\"*&F)F+-%$cosG6#,$*(F*F+ %#PiGF+%\"xGF+F+F+F+F+,(F'F,*(F*F+F)F+F.F+F+F+F,F," }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "asume(n_, integer): assume(n_>=2):\nSum(sin(2*k*Pi*x)/(n^k),k=0..infinity);\n``= combine(subs(n_=n,value(subs(n=n_,%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%$sinG6#,$**\"\"#\"\"\"%\"kGF-%#PiGF-%\"xGF-F-F-)%\" nGF.!\"\"/F.;\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, $*(%\"nG\"\"\"-%$sinG6#,$*(\"\"#F(%#PiGF(%\"xGF(F(F(,(*$)F'F.F(!\"\"*( F.F(F'F(-%$cosGF+F(F(F(F4F4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the series " }{XPPEDIT 18 0 "Sum(cos(2*k*Pi*x)/n^k,k=0..i nfinity)" "6#-%$SumG6$*&-%$cosG6#**\"\"#\"\"\"%\"kGF,%#PiGF,%\"xGF,F,) %\"nGF-!\"\"/F-;\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "A finite sum " }{XPPEDIT 18 0 "Sum(cos(2*k*Pi*x)/n^k,k=0. .m)" "6#-%$SumG6$*&-%$cosG6#**\"\"#\"\"\"%\"kGF,%#PiGF,%\"xGF,F,)%\"nG F-!\"\"/F-;\"\"!%\"mG" }{TEXT -1 31 " leaves a remainder or error " }{XPPEDIT 18 0 "Sum(cos(2*k*Pi*i*x)/n^k,k=m+1..infinity)" "6#-%$SumG6$ *&-%$cosG6#*,\"\"#\"\"\"%\"kGF,%#PiGF,%\"iGF,%\"xGF,F,)%\"nGF-!\"\"/F- ;,&%\"mGF,F,F,%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 108 "The terms of this last series are, term by term, less than or \+ equal in magnitude to the terms of the series " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n^k),k = m+1 .. infinity) = 1/(n ^(m+1)*(1-1/n));" "6#/-%$SumG6$*&\"\"\"F()%\"nG%\"kG!\"\"/F+;,&%\"mGF( F(F(%)infinityG*&F(F(*&)F*,&F0F(F(F(F(,&F(F(*&F(F(F*F,F,F(F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(n^m*(n-1));" "6#*&\"\"\"F$*&)%\"nG%\"mGF $,&F'F$F$!\"\"F$F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Sum(1/(n^k),k=m+1..infinity) ;\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*& \"\"\"F')%\"nG%\"kG!\"\"/F*;,&%\"mGF'F'F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"nG,$%\"mG!\"\"\"\"\",&F%F)F)F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "This \"error bound\" is independent of \+ " }{TEXT 271 1 "x" }{TEXT -1 44 ", that is, it is uniform, and tends t o 0 as " }{XPPEDIT 18 0 "m->infinity" "6#f*6#%\"mG7\"6$%)operatorG%&ar rowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "Thus the trigonometric series " }{XPPEDIT 18 0 "Sum(cos(2*k*Pi*x)/ (n^k),k = 0 .. infinity)" "6#-%$SumG6$*&-%$cosG6#**\"\"#\"\"\"%\"kGF,% #PiGF,%\"xGF,F,)%\"nGF-!\"\"/F-;\"\"!%)infinityG" }{TEXT -1 11 " conve rges " }{TEXT 261 9 "uniformly" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "Similarly, the trigonometric series " }{XPPEDIT 18 0 "Sum (sin(2*k*Pi*x)/(n^k),k = 0 .. infinity);" "6#-%$SumG6$*&-%$sinG6#**\" \"#\"\"\"%\"kGF,%#PiGF,%\"xGF,F,)%\"nGF-!\"\"/F-;\"\"!%)infinityG" } {TEXT -1 16 " also converges " }{TEXT 261 9 "uniformly" }{TEXT -1 1 ". " }}{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 26 "For each positive integer " } {TEXT 269 1 "n" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "n>= 2" "6#1\"\" #%\"nG" }{TEXT -1 16 ", the function " }{XPPEDIT 18 0 "xi[n](x) = n*( n-cos(2*Pi*x))/(n^2-2*n*cos(2*Pi*x)+1);" "6#/-&%#xiG6#%\"nG6#%\"xG*(F( \"\"\",&F(F,-%$cosG6#*(\"\"#F,%#PiGF,F*F,!\"\"F,,(*$F(F2F,*(F2F,F(F,-F /6#*(F2F,F3F,F*F,F,F4F,F,F4" }{TEXT -1 65 " is an infinitely different iable periodic function with period 1." }}{PARA 0 "" 0 "" {TEXT -1 47 "From the discussion in the previous subsection " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "Sum(cos(2*k*Pi*x)/(n^k),k = 0 .. inf inity) = n*(n-cos(2*Pi*x))/(n^2-2*n*cos(2*Pi*x)+1);" "6#/-%$SumG6$*&-% $cosG6#**\"\"#\"\"\"%\"kGF-%#PiGF-%\"xGF-F-)%\"nGF.!\"\"/F.;\"\"!%)inf inityG*(F2F-,&F2F--F)6#*(F,F-F/F-F0F-F3F-,(*$F2F,F-*(F,F-F2F--F)6#*(F, F-F/F-F0F-F-F3F-F-F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "xi := (n,x) -> n*(n-cos(2 *Pi*x))/(n^2-2*n*cos(2*Pi*x)+1);\nplot([xi(2,x),xi(3,x),xi(4,x),xi(5,x )],x=-1..3,\n color=[red,blue,green,magenta]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiGf*6$%\"nG%\"xG6\"6$%)operatorG% &arrowGF)*(9$\"\"\",&F.F/-%$cosG6#,$*(\"\"#F/%#PiGF/9%F/F/!\"\"F/,(*$) F.F6F/F/*(F6F/F.F/F1F/F9F/F/F9F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 660 261 261 {PLOTDATA 2 "6(-%'CURVESG6$7iv7$$!\"\"\"\"!$\"\"#F*7$$!3%p mm\"HU,\"*)*!#=$\"3AIf'>[og)>!#<7$$!3!RLL$e%G?y*F0$\"3%oCoOxXe%>F37$$! 3u****\\(oUIn*F0$\"3yUTtZVi$)=F37$$!3ommm;p0k&*F0$\"3EC#ys5[`!=F37$$!3 kLL$e9r]X*F0$\"3W$)y)>$eN<)yl`i\"F37 $$!3Umm;/'*4P#*F0$\"31u1$>00R`\"F37$$!3PLLLLQ6G\"*F0$\"3+Gi8C#fhW\"F37 $$!3s+](o*eGL!*F0$\"3sHb\\B`Ru8F37$$!3'pm;/'zXQ*)F0$\"3^f,Gf!\\wI\"F37 $$!3?L$eR-IO%))F0$\"3+S&e.m1iC\"F37$$!3a++](3-)[()F0$\"3khy-\"F37 $$!3-LLe9i9f&)F0$\"3QZSL-q#G4\"F37$$!3immmT.\\p$)F0$\"3&[vm3xxL,\"F37$ 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\\^n$\"3h*GEQRz.9\"F-7$Fa^nFf^p-Fd^n6&Ff^nF^^pF^^pF[^pF_^p-%+AXESLABEL SG6$Q\"x6\"Q!F`_r-%%VIEWG6$;F(Fa^n%(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" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "xi := (n,x) -> n*(n-cos(2*Pi*x))/(n^2-2*n*cos(2*Pi*x)+1):\nFourie rSeries(xi(2,x),x=0..1,mode=termbyterm);\nFourierSeries(xi(8,x),x=0..1 ,mode=termbyterm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"\"\"F$*&#F$ \"\"#F$-%$cosG6#,$*(F'F$%#PiGF$%\"xGF$F$F$F$*&#F$\"\"%F$-F)6#,$*(F1F$F -F$F.F$F$F$F$*&#F$\"\")F$-F)6#,$*(\"\"'F$F-F$F.F$F$F$F$*&#F$\"#;F$-F)6 #,$*(F8F$F-F$F.F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"\"\"F$* &#F$\"\")F$-%$cosG6#,$*(\"\"#F$%#PiGF$%\"xGF$F$F$F$*&#F$\"#kF$-F)6#,$* (\"\"%F$F.F$F/F$F$F$F$*&#F$\"$7&F$-F)6#,$*(\"\"'F$F.F$F/F$F$F$F$*&#F$ \"%'4%F$-F)6#,$*(F'F$F.F$F/F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "For each positive integer " } {TEXT 275 1 "n" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "n>= 2" "6#1\"\" #%\"nG" }{TEXT -1 16 ", the function " }{XPPEDIT 18 0 "psi[n](x) = n* sin(2*Pi*x)/(n^2-2*n*cos(2*Pi*x)+1);" "6#/-&%$psiG6#%\"nG6#%\"xG*(F(\" \"\"-%$sinG6#*(\"\"#F,%#PiGF,F*F,F,,(*$F(F1F,*(F1F,F(F,-%$cosG6#*(F1F, F2F,F*F,F,!\"\"F,F,F:" }{TEXT -1 65 " is an infinitely differentiable \+ periodic function with period 1." }}{PARA 0 "" 0 "" {TEXT -1 51 "From \+ the discussion in the first subsection we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(sin(2*k*Pi*x)/(n^k),k = 1 .. infini ty) = n*sin(2*Pi*x)/(n^2-2*n*cos(2*Pi*x)+1);" "6#/-%$SumG6$*&-%$sinG6# **\"\"#\"\"\"%\"kGF-%#PiGF-%\"xGF-F-)%\"nGF.!\"\"/F.;F-%)infinityG*(F2 F--F)6#*(F,F-F/F-F0F-F-,(*$F2F,F-*(F,F-F2F--%$cosG6#*(F,F-F/F-F0F-F-F3 F-F-F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "psi := (n,x) -> n*sin(2*Pi*x)/(n^2 -2*n*cos(2*Pi*x)+1);\nplot([psi(2,x),psi(3,x),psi(4,x),psi(5,x)],x=-1. .3,\n color=[red,blue,green,magenta]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)*(9$ \"\"\"-%$sinG6#,$*(\"\"#F/%#PiGF/9%F/F/F/,(*$)F.F5F/F/*(F5F/F.F/-%$cos GF2F/!\"\"F/F/F>F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 527 281 281 {PLOTDATA 2 "6(-%'CURVESG6$7_w7$$!\"\"\"\"!$\"3')4%*e'>(e)*[!#L7$$!3/M Le9r]X**!#=$\"3Sqpu)*pUIo!#>7$$!3%pmm\"HU,\"*)*F1$\"35nymIaxb8F17$$!3( )***\\PM@l$)*F1$\"3Z?wM$R!Q3?F17$$!3!RLL$e%G?y*F1$\"3xq<]'ev>j#F17$$!3 u****\\(oUIn*F1$\"3W*)H.o'*HjPF17$$!3ommm;p0k&*F1$\"3c_*QhwMzq%F17$$!3 [*****\\P&3Y$*F1$\"3s$*=(4M3I*fF17$$!3PLLLLQ6G\"*F1$\"3e\"\\P*e/]qlF17 $$!3/nT5l)*p!3*F1$\"3wo(e.[.Gi'F17$$!3s+](o*eGL!*F1$\"3m(>_w**3Sl'F17$ $!3SMekG>(e)*)F1$\"3AXWC+aHmmF17$$!3'pm;/'zXQ*)F1$\"3c1R)[@<_W\"z`%H\\F17$$!3LLLL$))Qj^(F1$\"3/_UA(>5H.%F17$$!3#R LLL`Np3(F1$\"30;R=x_^2KF17$$!3ULLL$=Kvl'F1$\"3Mbe**>b9fCF17$$!3'***** \\7(p,B'F1$\"3O*QFG]$F17$$!3'4++]d(Q&\\#F1$!3tSp&>`y#4SF17$$!3cLL$3d[.1#F1$!3c&3l \\3yB#\\F17$$!3:mmmm&4`i\"F1$!3qQQL_3UfeF17$$!3#HLL3F=PV\"F1$!3>iGDoyX IiF17$$!3p*****\\(p7U7F1$!3+X?25c,AlF17$$!33LL3F8LY6F1$!36[rrl*Guh'F17 $$!3Ymm;zc`]5F1$!3#>pKC-!=kmF17$$!3:L$3_&yj-5F1$!3c*H5j2U\\m'F17$$!3W) ***\\7.SZ&*F4$!3Kzx#Ru(*zk'F17$$!3Olm\"H2A%o!*F4$!3uH+oYd>6mF17$$!3GKL LLQW*e)F4$!3m_Wu`3E_lF17$$!39,++D'zDV'F4$!3+U_*4\"p8[fF17$$!3+qmm;arvU F4$!3t$F4$!3gxs0.)=Pp$F17$$!3%)QLL37&)=@F4$!3] &oPHx8Sc#F17$$!311+Dc^jz:F4$!3UfWgE;^V>F17$$!3Ftm;/\">//\"F4$!3%zB+1HI aH\"F17$$!3%[SL3_I?,&Fbs$!3y(=I>!zz%G'F47$$\"3HI#*******H,Q!#@$\"3_%GL 2U&zwZFbs7$$\"3?E****\\7V%z&Fbs$\"3'\\nk^kU1E(F47$$\"3%H*****\\K(37\"F 4$\"3u;0I(y[NR\"F17$$\"3E$****\\P.Bm\"F4$\"3QAlr>XjS?F17$$\"3d$****** \\LP?#F4$\"3]a\"z\"e*o'eEF17$$\"3A%*****\\Pf'G$F4$\"3i!Q5W#>czPF17$$\" 3%[*******RXpVF4$\"3S\"y'GFno:ZF17$$\"37'******\\u^`'F4$\"3)4B)QvoO\"* fF17$$\"3Q(*******\\*3q)F4$\"3I*QfMZt!olF17$$\"3\\(*****\\KmA#*F4$\"3Q 8/8G6NDmF17$$\"3e(******\\JWu*F4$\"3#G1X)\\lCdmF17$$\"3x*****\\(*>m-\" F1$\"3%>#f3gYkmmF17$$\"3y*******z'zy5F1$\"3g$H(*H-Xil'F17$$\"3!)****** \\/:$=\"F1$\"3%H)[^@U\"ee'F17$$\"3!)*******4/vG\"F1$\"3E+y?`&QLY'F17$$ \"3')*******R6i\\\"F1$\"3[H!**3o1c6'F17$$\"3!********p=\\q\"F1$\"3mk@ \\7&pGp&F17$$\"3?LL$e9rR3#F1$\"3'R%4%y9@7([F17$$\"3_mmm\"fBIY#F1$\"35[ ^y$eYY2%F17$$\"3l****\\7Ov8HF1$\"3IB5?RcD1KF17$$\"3yKLLLO[kLF1$\"3uIN< j]fACF17$$\"3SKLL$36ju$F1$\"3&)G&=%Q@@7=F17$$\"3.KLLL&Q\"GTF1$\"3I(z@! pd(yB\"F17$$\"3+*****\\s]k,&F1$!3iA9Ne\"opH#Fbs7$$\"3Alm;HTg4aF1$!35Zt /o9DSdF47$$\"3WJLLLvv-eF1$!3Y!3J0,do8\"F17$$\"3:mm;H\"4TB'F1$!3#HD\"Hy t)>y\"F17$$\"3'3++]sgam'F1$!3oPCHx#[BZ#F17$$\"3c+++]%4i2(F1$!3]1m'F17$$\"3oL3-jgk$)*)F1$!3]$oeVCUkm'F17$$\"3]L $e9cTG.*F1$!3A9l]Y'3Ul'F17$$\"3KLe*)fq.#3*F1$!30X)G9,G;i'F17$$\"39LLLe DBJ\"*F1$!3sKvf&4mic'F17$$\"3Umm\"H_)[V$*F1$!3.w(Hn6#o.gF17$$\"3s**** \\([Wdb*F1$!3q)HTv8*prZF17$$\"3!em\"zpC(=m*F1$!3c%47,1z\"pQF17$$\"3+LL 3_/+o(*F1$!3oEli&f`oy#F17$$\"3A+]PM%GT()*F1$!3oR:a-\")eg:F17$$\"3Immm; 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cjl$\"3e(\\un)y\\$>$F47$Fhjl$\"3+DAan26*p%F47$F][m$\"3RXDO$3d\"GhF47$F b[m$\"3i#zWlY:FX)F47$Fg[m$\"3oT'4:d![M5F17$F\\\\m$\"3eS(4.$[0$=\"F17$F a\\m$\"3]6#[ek(Qm7F17$Ff\\m$\"3;ZGQ?V\\w7F17$F[]m$\"33yyokl7F17$F_^m$\"3s&e@7c8GD\"F17$Fd^ m$\"3i&45$Go,q6F17$Fi^m$\"3aL2!>![1[5F17$F^_m$\"3\\p\\72<:8!*F47$Fc_m$ \"3gJ/*[2brO(F47$Fh_m$\"3?m$y/Q%*\\`&F47$F]`m$\"3o\\#Q2M)\\^OF47$Fb`m$ \"3qNG9POo@=F47$Fg`m$!3OV^<)e4?2\"Fhr7$F\\am$!3WTq^-Og%)=F47$Faam$!31D uhpa^bPF47$Ffam$!3'okfhJQSf&F47$F[bm$!39Z)3uOIAQ(F47$F`bm$!3.9(y]2\")> $*)F47$Febm$!3=q?H(*[\"G.\"F17$Fjbm$!3c@jF$otS;\"F17$F_cm$!3K^N1b@g_7F 17$F\\go$!3)H/q(4)*Hl7F17$Fdcm$!3-)e6v=G^F\"F17$Fdgo$!3/Tuc!zr>G\"F17$ Ficm$!3)R.`GUBdG\"F17$F^dm$!3CaZNF_G'G\"F17$Fcdm$!3a(=]0opNG\"F17$Fhdm $!3MhA%[`+vF\"F17$F]em$!3#=r0/q8!o7F17$Fbem$!3u^6MrMD)=\"F17$Fgem$!3y \"*4$HayZ/\"F17$F\\fm$!3<\"[JD>2d^)F47$Fafm$!3mDhF47$Fffm$!3!>Df K[(>!o%F47$F[gm$!3)[)RE4G5kJF47$F`gm$!3uQO**f " 0 "" {MPLTEXT 1 0 150 "psi := (n,x) -> n *sin(2*Pi*x)/(n^2-2*n*cos(2*Pi*x)+1):\nFourierSeries(psi(2,x),x=0..1,m ode=termbyterm);\nFourierSeries(psi(8,x),x=0..1,mode=termbyterm);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"#F&-%$sinG6#,$*(F'F&%#P iGF&%\"xGF&F&F&F&*&#F&\"\"%F&-F)6#,$*(F1F&F-F&F.F&F&F&F&*&#F&\"\")F&-F )6#,$*(\"\"'F&F-F&F.F&F&F&F&*&#F&\"#;F&-F)6#,$*(F8F&F-F&F.F&F&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\")F&-%$sinG6#,$*(\"\"#F& %#PiGF&%\"xGF&F&F&F&*&#F&\"#kF&-F)6#,$*(\"\"%F&F.F&F/F&F&F&F&*&#F&\"$7 &F&-F)6#,$*(\"\"'F&F.F&F/F&F&F&F&*&#F&\"%'4%F&-F)6#,$*(F'F&F.F&F/F&F&F &F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 56 "In each of these two examples, the Fourie r coefficients " }{XPPEDIT 18 0 "1/n^k" "6#*&\"\"\"F$)%\"nG%\"kG!\"\" " }{TEXT -1 15 " tend to 0 as " }{XPPEDIT 18 0 "k -> infinity" "6#f*6 #%\"kG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 19 " mor e rapidly than " }{XPPEDIT 18 0 "1/k^m" "6#*&\"\"\"F$)%\"kG%\"mG!\"\" " }{TEXT -1 20 " for any choices of " }{TEXT 272 1 "n" }{TEXT -1 5 " a nd " }{TEXT 273 1 "m" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "m>=2" "6#1 \"\"#%\"mG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n >= 2" "6#1\"\"#%\"nG " }{TEXT -1 115 ". The corresponding Fourier series converges rapidly, and the convergence is uniform on the whole of the real line." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 75 "The Fourier series of a twice continuously differentiable periodic function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Conside r the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 10 " given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([3*x^2-4*x^3, 0 <= x and x <= 1/2],[4*x^3-9*x^2+6*x-1, 1/2 \+ <= x and x <= 1]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&*&\"\"$\"\"\"* $F'\"\"#F/F/*&\"\"%F/*$F'F.F/!\"\"31\"\"!F'1F'*&F/F/F1F57$,**&F3F/*$F' F.F/F/*&\"\"*F/*$F'F1F/F5*&\"\"'F/F'F/F/F/F531*&F/F/F1F5F'1F'F/" } {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 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "The first derivative of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " is given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = PIECEWISE([6*x*(1-2*x), 0 <= x and x <= 1/2],[6*(2*x-1)*(x-1), 1/2 <= x and x <= 1]);" "6#/-% $f~'G6#%\"xG-%*PIECEWISEG6$7$*(\"\"'\"\"\"F'F.,&F.F.*&\"\"#F.F'F.!\"\" F.31\"\"!F'1F'*&F.F.F1F27$*(F-F.,&*&F1F.F'F.F.F.F2F.,&F'F.F.F2F.31*&F. F.F1F2F'1F'F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f ~'G6#%\"xG" }{TEXT -1 28 " is periodic with period 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The second derivativ e of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " is given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f ''`(x) = P IECEWISE([6-24*x, 0 <= x and x <= 1/2],[24*x-18, 1/2 <= x and x <= 1]) ;" "6#/-%%f~''G6#%\"xG-%*PIECEWISEG6$7$,&\"\"'\"\"\"*&\"#CF.F'F.!\"\"3 1\"\"!F'1F'*&F.F.\"\"#F17$,&*&F0F.F'F.F.\"#=F131*&F.F.F7F1F'1F'F." } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "`f ''`(x)" "6#-%%f~''G6#%\"xG" } {TEXT -1 28 " is periodic with period 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f ''`(x)" "6#-% %f~''G6#%\"xG" }{TEXT -1 64 " is continuous, but not differentiable, a t integer multiples of " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 241 "f := x -> piecewise(x<=1/2,3*x^2-4*x^3,4*x^3- 9*x^2+6*x-1):\n'f(x)'=f(x);\nf_ := x -> f(x-floor(x)):\n'f_(x)'='f(x-f loor(x))';\nFourierSeries(f(x),x=0..1,info=true):\nF := unapply(%,x); \nplot([f_(x),F(x)],x=-1..3,color=[red,green],thickness=[1,3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&*&\" \"$\"\"\")F'\"\"#F/F/*&\"\"%F/)F'F.F/!\"\"1F'#F/F17$,**&F3F/F4F/F/*&\" \"*F/F0F/F5*&\"\"'F/F'F/F/F/F5%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"-%&floorGF&!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G#\"\"\"\"\" )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%G,$*&\"#7F*F+!\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*(\"\"%\"\"\"%#PiGF*%\"xGF* F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$ cosG6#,$*(\"\"'\"\"\"%#PiGF*%\"xGF*F*%%~-->G,$*(\"\"%F*\"#F!\"\"F+!\"% F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*(\" \")\"\"\"%#PiGF*%\"xGF*F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%:k~th~sine~coefficient~..~G/,$**\"\"$\"\"\",**&%\"kGF(%#PiGF(!\"\" *(\"\"#F(-%$sinG6#F*F(-%$cosGF2F(F-*&F/F(F0F(F(*(F+F(F,F()F3F/F(F(F(F+ !\"%F,F8F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-% $sinG6#,$*(\"\"#\"\"\"%#PiGF*%\"xGF*F*%%~-->G\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$*(\"\"%\"\"\"%#PiGF*%\"xG F*F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG- %$sinG6#,$*(\"\"'\"\"\"%#PiGF*%\"xGF*F*%%~-->G\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$*(\"\")\"\"\"%#PiGF*%\"xG F*F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\" 6$%)operatorG%&arrowGF(,(#\"\"\"\"\")F.*(\"#7F.%#PiG!\"%-%$cosG6#,$*( \"\"#F.F2F.9$F.F.F.!\"\"*&#\"\"%\"#FF.*&F2F3-F56#,$*(\"\"'F.F2F.F:F.F. 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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/8-12/Pi^4" "6#,&*&\"\"\"F%\"\")!\"\"F%*&\"#7F%*$%#PiG \"\"%F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(cos(2*(2*k-1)*Pi*x)/(2* k-1)^4,k=1..infinity)" "6#-%$SumG6$*&-%$cosG6#**\"\"#\"\"\",&*&F+F,%\" kGF,F,F,!\"\"F,%#PiGF,%\"xGF,F,*$,&*&F+F,F/F,F,F,F0\"\"%F0/F/;F,%)infi nityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "This series conv erges uniformly to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "This follows from the fact tha t the terms of the series " }{XPPEDIT 18 0 "Sum(cos(2*(2*k-1)*Pi*x)/(( 2*k-1)^4),k = 1 .. infinity)" "6#-%$SumG6$*&-%$cosG6#**\"\"#\"\"\",&*& F+F,%\"kGF,F,F,!\"\"F,%#PiGF,%\"xGF,F,*$,&*&F+F,F/F,F,F,F0\"\"%F0/F/;F ,%)infinityG" }{TEXT -1 84 " are term by term less than or equal in ma gnitude to those of the convergent series " }{XPPEDIT 18 0 "Sum(1/((2* k-1)^4),k = 1 .. infinity)" "6#-%$SumG6$*&\"\"\"F'*$,&*&\"\"#F'%\"kGF' F'F'!\"\"\"\"%F-/F,;F'%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Sum(1/((2*k- 1)^4),k=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $SumG6$*&\"\"\"F'*$),&%\"kG\"\"#!\"\"F'\"\"%F'F-/F+;F'%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#PiG\"\"%\"\"\"#F(\"#'*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Note that, when " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 24 ", the Fourier series is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/8-12/Pi^4" "6#,&*& \"\"\"F%\"\")!\"\"F%*&\"#7F%*$%#PiG\"\"%F'F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(1/((2*k-1)^4),k = 1 .. infinity);" "6#-%$SumG6$*&\" \"\"F'*$,&*&\"\"#F'%\"kGF'F'F'!\"\"\"\"%F-/F,;F'%)infinityG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 21 "which converges to 0." }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "FS := (x,n) -> 1/8-12/Pi^4*S um(cos(2*(2*k-1)*Pi*x)/(2*k-1)^4,k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&#\"\"\" \"\")F/*(\"#7F/%#PiG!\"%-%$SumG6$*&-%$cosG6#,$**\"\"#F/,&*&F>F/%\"kGF/ F/F/!\"\"F/F3F/9$F/F/F/F?F4/FA;F/9%F/FBF)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "xx := sqrt(5 )/4:\nevalf(f(xx));\nevalf(FS(xx,20));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*3KPS#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+gEt.C!#5" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 74 "The Fourier series of a once continuously differentiable periodic function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Conside r the function " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 10 " given by " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g( x) = PIECEWISE([x-2*x^2, 0 <= x and x <= 1/2],[2*x^2-3*x+1, 1/2 <= x a nd x <= 1]);" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$,&F'\"\"\"*&\"\"#F-*$F 'F/F-!\"\"31\"\"!F'1F'*&F-F-F/F17$,(*&F/F-*$F'F/F-F-*&\"\"$F-F'F-F1F-F -31*&F-F-F/F1F'1F'F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x);" "6#-% \"gG6#%\"xG" }{TEXT -1 28 " is periodic with period 1. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "1/6" "6#*&\"\"\"F$\"\"'!\"\"" }{TEXT -1 38 " \+ times the derivative of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 21 " in the last section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The derivative of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`g '`(x) = PIECEWISE([1-4*x, 0 <= x and x <= 1/2],[4*x-3, 1/2 <= x and x <= 1]);" "6#/-%$g~'G6#%\"xG-%*PIECEW ISEG6$7$,&\"\"\"F-*&\"\"%F-F'F-!\"\"31\"\"!F'1F'*&F-F-\"\"#F07$,&*&F/F -F'F-F-\"\"$F031*&F-F-F6F0F'1F'F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "` g '`(x);" "6#-%$g~'G6#%\"xG" }{TEXT -1 28 " is periodic with period 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The \+ derivative " }{XPPEDIT 18 0 "`g '`(x);" "6#-%$g~'G6#%\"xG" }{TEXT -1 64 " is continuous, but not differentiable at integer multiples of " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "g := x -> piecewise(x<=1/2,x-2*x^2,2*x^2-3*x+1):\n'g(x)'=g(x);\ng_ := x -> g(x-floor(x)):\n'g_(x)'='g(x-floor(x))';\nFourierSeries(g(x),x=0.. 1,info=true):\nG := unapply(%,x);\nplot([g_(x),G(x)],x=-1..3,color=[re d,green],thickness=[1,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6 #%\"xG-%*PIECEWISEG6$7$,&F'\"\"\"*&\"\"#F-)F'F/F-!\"\"1F'#F-F/7$,(*&F/ F-F0F-F-*&\"\"$F-F'F-F1F-F-%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#g_G6#%\"xG-%\"gG6#,&F'\"\"\"-%&floorGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%G\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*(\"\"%\"\"\"%#PiGF* %\"xGF*F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient ~ofG-%$cosG6#,$*(\"\"'\"\"\"%#PiGF*%\"xGF*F*%%~-->G\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*(\"\")\"\"\"%#PiGF* %\"xGF*F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%:k~th~sine~c oefficient~..~G/**-%$cosG6#*&%\"kG\"\"\"%#PiGF+F+,(*(F*F+F,F+-%$sinGF( F+F+*&\"\"#F+F&F+F+F2!\"\"F+F*!\"$F,F4,$**F2F+,&)F3F*F+F+F3F+F*F4F,F4F 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$*(\" \"#\"\"\"%#PiGF*%\"xGF*F*%%~-->G,$*&\"\"%F*F+!\"$F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$*(\"\"%\"\"\"%#PiGF*%\"xG F*F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG- %$sinG6#,$*(\"\"'\"\"\"%#PiGF*%\"xGF*F*%%~-->G,$*(\"\"%F*\"#F!\"\"F+! \"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$* (\"\")\"\"\"%#PiGF*%\"xGF*F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 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4fv.al<(F27$Fdjn$!3?naYbUA(e&F27$Fijn$!3V_%)z'=Yy#QF27$F^[o$!3Ia%=1#[* f%>F27$Fc[o$!3aBAjdjC`5!#L-Ff[o6&Fh[oF+Fi[oF+-F]\\o6#Fd[o-%+AXESLABELS G6$Q\"x6\"Q!Fhdq-%%VIEWG6$;F(Fc[o%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of " }{XPPEDIT 18 0 "g (x)" "6#-%\"gG6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4/(Pi^3);" "6#*&\"\"%\"\"\"*$%#PiG\"\"$!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(sin(2*(2*k-1)*Pi*x)/((2*k-1)^3), k = 1 .. infinity);" "6#-%$SumG6$*&-%$sinG6#**\"\"#\"\"\",&*&F+F,%\"kG F,F,F,!\"\"F,%#PiGF,%\"xGF,F,*$,&*&F+F,F/F,F,F,F0\"\"$F0/F/;F,%)infini tyG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "This series conver ges uniformly to " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "This follows from the fact that t he terms of the series " }{XPPEDIT 18 0 "Sum(sin(2*(2*k-1)*Pi*x)/((2*k -1)^3),k = 1 .. infinity);" "6#-%$SumG6$*&-%$sinG6#**\"\"#\"\"\",&*&F+ F,%\"kGF,F,F,!\"\"F,%#PiGF,%\"xGF,F,*$,&*&F+F,F/F,F,F,F0\"\"$F0/F/;F,% )infinityG" }{TEXT -1 84 " are term by term less than or equal in magn itude to those of the convergent series " }{XPPEDIT 18 0 "Sum(1/((2*k- 1)^3),k = 1 .. infinity);" "6#-%$SumG6$*&\"\"\"F'*$,&*&\"\"#F'%\"kGF'F 'F'!\"\"\"\"$F-/F,;F'%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "FS := (x,n) -> 4/Pi^3*Sum(sin(2*(2*k-1)*Pi*x)/(2*k -1)^3,k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG 6\"6$%)operatorG%&arrowGF),$*(\"\"%\"\"\"%#PiG!\"$-%$SumG6$*&-%$sinG6# ,$**\"\"#F0,&*&F " 0 "" {MPLTEXT 1 0 48 "xx := sqrt(2)/2:\nevalf(g(xx));\nevalf(FS(xx,20));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$!*V.K@\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+]a?87!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "xx := 1/4:\nevalf(g(xx));\nevalf(FS (xx,20));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++]7!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+e**)*\\7!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x = 1/4" "6#/%\"xG*&\"\"\"F&\"\"%! \"\"" }{TEXT -1 29 ", the Fourier series becomes " }{XPPEDIT 18 0 "4/P i^3" "6#*&\"\"%\"\"\"*$%#PiG\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^(k+1)/((2*k-1)^3),k=1..infinity)" "6#-%$SumG6$*&),$\"\"\"! \"\",&%\"kGF)F)F)F)*$,&*&\"\"#F)F,F)F)F)F*\"\"$F*/F,;F)%)infinityG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "Since this series conve rges to " }{XPPEDIT 18 0 "g(1/4) = 1/8;" "6#/-%\"gG6#*&\"\"\"F(\"\"%! \"\"*&F(F(\"\")F*" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "Sum((-1)^ (k+1)/((2*k-1)^3),k=1..infinity)=Pi^3/32" "6#/-%$SumG6$*&),$\"\"\"!\" \",&%\"kGF*F*F*F**$,&*&\"\"#F*F-F*F*F*F+\"\"$F+/F-;F*%)infinityG*&%#Pi GF2\"#KF+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "Maple does n ot give the value " }{XPPEDIT 18 0 "Pi^3/32" "6#*&%#PiG\"\"$\"#K!\"\" " }{TEXT -1 28 " for the sum of this series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Sum((-1)^(k+1)/((2 *k-1)^3),k=1..infinity);\nvalue(%);\nevalf(%);\nevalf(Pi^3/32);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\",&%\"kG\"\"\"F+F+F+,& *&\"\"#F+F*F+F+F+F(!\"$/F*;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*hypergeomG6%7&#\"\"\"\"\"#F'F'F(7%#\"\"$F)F+F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+j9Y*o*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+m9Y*o*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }