{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 "" -1 264 "" 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{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 "" -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 "" 258 273 "" 1 18 0 0 0 0 0 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 "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "Grey Emphasis" -1 280 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " 260 282 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 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 "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 56 "Dirichlet's theorem on the conver gence of Fourier 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 29 "Absolutely integral functio ns" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 11 "A function " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" } {TEXT -1 15 " is said to be " }{TEXT 261 19 "absolutely integral" } {TEXT -1 18 " over an interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"b G" }{TEXT -1 15 " provided that " }{XPPEDIT 18 0 "Int(abs(phi(x)),x = \+ a .. b);" "6#-%$IntG6$-%$absG6#-%$phiG6#%\"xG/F,;%\"aG%\"bG" }{TEXT -1 26 " is a finite real number." }}{PARA 0 "" 0 "" {TEXT -1 18 "In p articular, if " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 55 " is continuous (and therefore bounded) on the interval " } {XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 7 ", then " } {XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 27 " is absolutely integrable. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi (x)" "6#-%$phiG6#%\"xG" }{TEXT -1 34 " be a periodic function of perio d " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 18 " and s uppose that " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 261 19 "absolutely integral" }{TEXT -1 25 " over a period , that is, " }{XPPEDIT 18 0 "Int(abs(phi(x)),x = -L .. L);" "6#-%$IntG 6$-%$absG6#-%$phiG6#%\"xG/F,;,$%\"LG!\"\"F0" }{TEXT -1 26 " is a fini te real number." }}{PARA 0 "" 0 "" {TEXT -1 33 "Then the Fourier coeff icients of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 1 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1/(2*L)" "6#/ %\"cG*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Int(phi(x),x = -L .. L);" "6#-%$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\"\"\"-%$cosG6#**%\"kG F+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$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*;,$F2F3F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "al l exist as finite real numbers." }}{PARA 0 "" 0 "" {TEXT -1 12 "For ex ample " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(a[k]) = 1/(2*L)" "6#/-%$absG6#&%\"aG6#%\"kG*&\"\"\"F,*&\"\"#F,%\"LGF,!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(Int(phi(x)*cos(k*Pi*x/L),x = -L .. \+ L));" "6#-%$absG6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$cosG6#**%\"kGF.%#P iGF.F-F.%\"LG!\"\"F./F-;,$F5F6F5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``<= \+ 1/(2*L)" "6#1%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(abs(phi(x)*cos(k*Pi*x/L)),x = -L .. L);" "6#-%$IntG 6$-%$absG6#*&-%$phiG6#%\"xG\"\"\"-%$cosG6#**%\"kGF.%#PiGF.F-F.%\"LG!\" \"F./F-;,$F5F6F5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "``<= 1/(2*L)" "6#1%!G*&\"\"\"F&*&\"\" #F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(phi(x)),x = - L .. L) < infinity;" "6#2-%$IntG6$-%$absG6#-%$phiG6#%\"xG/F-;,$%\"LG! \"\"F1%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "Thus, if " }{XPPEDIT 18 0 "phi(x)" "6#-%$p hiG6#%\"xG" }{TEXT -1 70 " is absolutely integral, we can at least con struct the Fourier series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Phi(x) = c+``;" "6#/-%$Phi G6#%\"xG,&%\"cG\"\"\"%!GF*" }{XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/L)+b[k ]*sin(k*Pi*x/L),k = 1 .. infinity);" "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\" \"-%$cosG6#**F+F,%#PiGF,%\"xGF,%\"LG!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#** F+F,F1F,F2F,F3F4F,F,/F+;F,%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "but " }{XPPEDIT 18 0 "Phi(x);" "6#-%$PhiG6#%\"xG" } {TEXT -1 32 " may not be exactly the same as " }{XPPEDIT 18 0 "phi(x) " "6#-%$phiG6#%\"xG" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "Phi(x) ;" "6#-%$PhiG6#%\"xG" }{TEXT -1 44 " may not be the Fourier series exp ansion of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The Dirichlet kerne l" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 34 " be a periodic function of period " }{XPPEDIT 18 0 "2*L" "6#*& \"\"#\"\"\"%\"LGF%" }{TEXT -1 20 ", and suppose that " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 48 " is absolutely integral ov er a period, that is, " }{XPPEDIT 18 0 "Int(abs(phi(x)),x = -L .. L); " "6#-%$IntG6$-%$absG6#-%$phiG6#%\"xG/F,;,$%\"LG!\"\"F0" }{TEXT -1 26 " is a finite real number." }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "n=1,2,` . . `" "6%/%\"nG\"\"\"\"\"#%&~.~.~G" }{TEXT -1 6 ", let " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "phi[n]( x) = c+``;" "6#/-&%$phiG6#%\"nG6#%\"xG,&%\"cG\"\"\"%!GF-" }{XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/L)+b[k]*sin(k*Pi*x/L),k = 1 .. n);" "6#-%$Su mG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#PiGF,%\"xGF,%\"LG!\"\"F,F, *&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F,%\"nG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "be \+ the " }{TEXT 261 24 "truncated Fourier series" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "c,a[k]" "6$%\"cG&%\"aG6#%\"kG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" }{TEXT -1 29 " are Fourier coef ficients of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 10 ", that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1/ (2*L)" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(phi(x),x = -L .. L);" "6#-%$IntG6$-%$phiG6#%\"xG/F) ;,$%\"LG!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[k] = 1/L" "6#/&% \"aG6#%\"kG*&\"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(p hi(x)*cos(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"- %$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3F2" }{TEXT -1 7 " a nd " }{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*;,$F2F3F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "There is a shortcut which can be u sed to evaluate " }{XPPEDIT 18 0 "phi[n](x[0]);" "6#-&%$phiG6#%\"nG6#& %\"xG6#\"\"!" }{TEXT -1 19 " for a fixed value " }{XPPEDIT 18 0 "x=x[0 ]" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 4 " of " }{TEXT 272 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n](x[0]) = Sum(c[k]*exp(k*Pi*i*x[0] /L),k = -n .. n);" "6#/-&%$phiG6#%\"nG6#&%\"xG6#\"\"!-%$SumG6$*&&%\"cG 6#%\"kG\"\"\"-%$expG6#*,F5F6%#PiGF6%\"iGF6&F+6#F-F6%\"LG!\"\"F6/F5;,$F (F@F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " } {XPPEDIT 18 0 "c[0]=c,c[k]=(a[k]-i*b[k])/2" "6$/&%\"cG6#\"\"!F%/&F%6#% \"kG*&,&&%\"aG6#F+\"\"\"*&%\"iGF1&%\"bG6#F+F1!\"\"F1\"\"#F7" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[-k]=(a[k]+i*b[k])/2" "6#/&%\"cG6#,$% \"kG!\"\"*&,&&%\"aG6#F(\"\"\"*&%\"iGF/&%\"bG6#F(F/F/F/\"\"#F)" }{TEXT -1 7 ", for " }{XPPEDIT 18 0 "k = 1,2,` . . . `,n;" "6&/%\"kG\"\"\"\" \"#%(~.~.~.~G%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "c[k] =1/(2*L)" "6# /&%\"cG6#%\"kG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(phi(x)*exp(-k*Pi*i*x/L),x = -L .. L);" "6#-%$IntG6$ *&-%$phiG6#%\"xG\"\"\"-%$expG6#,$*,%\"kGF+%#PiGF+%\"iGF+F*F+%\"LG!\"\" F5F+/F*;,$F4F5F4" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n](x[0]) = Sum(``(1/(2*L))*``(Int(phi(x)* exp(-i*k*Pi*x/L),x = -L .. L))*exp(i*k*Pi*x[0]/L),k = -n .. n);" "6#/- &%$phiG6#%\"nG6#&%\"xG6#\"\"!-%$SumG6$*(-%!G6#*&\"\"\"F6*&\"\"#F6%\"LG F6!\"\"F6-F36#-%$IntG6$*&-F&6#F+F6-%$expG6#,$*,%\"iGF6%\"kGF6%#PiGF6F+ F6F9F:F:F6/F+;,$F9F:F9F6-FD6#*,FHF6FIF6FJF6&F+6#F-F6F9F:F6/FI;,$F(F:F( " }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = 1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(phi(x)*Sum(exp(k*Pi*i*(x[0]-x)/L),k = -n .. n),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$SumG6$-%$expG6#*,% \"kGF+%#PiGF+%\"iGF+,&&F*6#\"\"!F+F*!\"\"F+%\"LGF:/F3;,$%\"nGF:F?F+/F* ;,$F;F:F;" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*Delta[n](Pi*(x[0]-x)/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-&%&DeltaG6#%\"nG6#*(%#P iGF+,&&F*6#\"\"!F+F*!\"\"F+%\"LGF8F+/F*;,$F9F8F9" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 18 "__________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Delta[n](x) = Sum(exp(k*i*x),k = - n .. n);" "6#/-&%&DeltaG6#%\"nG6#%\"xG-%$SumG6$-%$expG6#*(%\"kG\"\"\"% \"iGF3F*F3/F2;,$F(!\"\"F(" }{TEXT -1 16 ", which is the " }{TEXT 261 16 "Dirichlet kernel" }{TEXT -1 10 " of order " }{TEXT 266 1 "n" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "If we reverse the order \+ of the terms in the sum " }{XPPEDIT 18 0 "Sum(exp(k*i*x),k = -n .. n) " "6#-%$SumG6$-%$expG6#*(%\"kG\"\"\"%\"iGF+%\"xGF+/F*;,$%\"nG!\"\"F1" }{TEXT -1 62 ", we see that it is a finite geometric series with first term " }{XPPEDIT 18 0 "a = exp(n*i*x);" "6#/%\"aG-%$expG6#*(%\"nG\"\" \"%\"iGF*%\"xGF*" }{TEXT -1 18 " and common ratio " }{XPPEDIT 18 0 "r \+ = exp(-i*x);" "6#/%\"rG-%$expG6#,$*&%\"iG\"\"\"%\"xGF+!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Hence, if " }{XPPEDIT 18 0 "x< >0" "6#0%\"xG\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Delta[n](x) = a*(1-r^(2*n+1))/(1-r);" "6#/-&%&DeltaG 6#%\"nG6#%\"xG*(%\"aG\"\"\",&F-F-)%\"rG,&*&\"\"#F-F(F-F-F-F-!\"\"F-,&F -F-F0F4F4" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(n*i*x)*(1-exp(-(2* n+1)*i*x))/(1-exp(-i*x));" "6#/%!G*(-%$expG6#*(%\"nG\"\"\"%\"iGF+%\"xG F+F+,&F+F+-F'6#,$*(,&*&\"\"#F+F*F+F+F+F+F+F,F+F-F+!\"\"F6F+,&F+F+-F'6# ,$*&F,F+F-F+F6F6F6" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (exp(n*i*x)-exp (-(n+1)*i*x))/(1-exp(-i*x));" "6#/%!G*&,&-%$expG6#*(%\"nG\"\"\"%\"iGF, %\"xGF,F,-F(6#,$*(,&F+F,F,F,F,F-F,F.F,!\"\"F4F,,&F,F,-F(6#,$*&F-F,F.F, F4F4F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (exp((2*n+1)*i*x/2)-exp(-(2 *n+1)*i*x/2))/(exp(i*x/2)-exp(-i*x/2));" "6#/%!G*&,&-%$expG6#**,&*&\" \"#\"\"\"%\"nGF.F.F.F.F.%\"iGF.%\"xGF.F-!\"\"F.-F(6#,$**,&*&F-F.F/F.F. F.F.F.F0F.F1F.F-F2F2F2F.,&-F(6#*(F0F.F1F.F-F2F.-F(6#,$*(F0F.F1F.F-F2F2 F2F2" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sin((2*n+1)*x/2)/sin(x/2);" " 6#/%!G*&-%$sinG6#*(,&*&\"\"#\"\"\"%\"nGF-F-F-F-F-%\"xGF-F,!\"\"F--F'6# *&F/F-F,F0F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Also " } {XPPEDIT 18 0 "Delta[n](0) = Sum(1,k = -n .. n);" "6#/-&%&DeltaG6#%\"n G6#\"\"!-%$SumG6$\"\"\"/%\"kG;,$F(!\"\"F(" }{TEXT -1 4 " = " } {XPPEDIT 18 0 "2*n+1" "6#,&*&\"\"#\"\"\"%\"nGF&F&F&F&" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 12 "Note that " }{XPPEDIT 18 0 "Limit(s in((2*n+1)*x/2)/sin(x/2),x=0)=2*n+1" "6#/-%&LimitG6$*&-%$sinG6#*(,&*& \"\"#\"\"\"%\"nGF/F/F/F/F/%\"xGF/F.!\"\"F/-F)6#*&F1F/F.F2F2/F1\"\"!,&* &F.F/F0F/F/F/F/" }{TEXT -1 6 ", so " }{XPPEDIT 18 0 "Delta[n](x);" "6 #-&%&DeltaG6#%\"nG6#%\"xG" }{TEXT -1 21 " is continuous at 0. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The Fejer kernel of order " }{TEXT 283 1 "n" }{TEXT -1 30 " can be defined by t he formula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta[n ](x) = PIECEWISE([sin((2*n+1)*x/2)/sin(x/2), x <> 0],[2*n+1, x = 0]); " "6#/-&%&DeltaG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$*&-%$sinG6#*(,&*&\"\"# \"\"\"F(F7F7F7F7F7F*F7F6!\"\"F7-F16#*&F*F7F6F8F80F*\"\"!7$,&*&F6F7F(F7 F7F7F7/F*F=" }{TEXT -1 3 " ." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 282 20 "____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Delta[n](x) = Sum(exp(k*i*x),k = -n .. n);" "6#/-&%&DeltaG6#%\"nG6#%\"xG-%$SumG6$-%$expG6#*(%\"kG\"\"\"% \"iGF3F*F3/F2;,$F(!\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "Sum(cos*k*x-i*sin*k*x,k = 1 .. n);" "6#-%$S umG6$,&*(%$cosG\"\"\"%\"kGF)%\"xGF)F)**%\"iGF)%$sinGF)F*F)F+F)!\"\"/F* ;F)%\"nG" }{TEXT -1 7 " + 1 + " }{XPPEDIT 18 0 "Sum(cos*k*x+i*sin*k*x, k = 1 .. n);" "6#-%$SumG6$,&*(%$cosG\"\"\"%\"kGF)%\"xGF)F)**%\"iGF)%$s inGF)F*F)F+F)F)/F*;F)%\"nG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 10 " = 1 + 2 " }{XPPEDIT 18 0 "Sum(cos*k*x,k = 1 .. n);" "6#-%$Sum G6$*(%$cosG\"\"\"%\"kGF(%\"xGF(/F);F(%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "1+ 2*Sum(cos(k*x),k=1..n)=sin((2*n+1)*x/2)/sin(x/2);\ntesteq(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&\"\"\"F%*&\"\"#F%-%$SumG6$-%$cosG 6#*&%\"kGF%%\"xGF%/F/;F%%\"nGF%F%*&-%$sinG6#,$*(F'!\"\",&*&F'F%F3F%F%F %F%F%F0F%F%F%-F66#,$*&F'F:F0F%F%F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "From this we see tha t the Dirichlet kernel " }{XPPEDIT 18 0 "Delta[n](x);" "6#-&%&DeltaG6# %\"nG6#%\"xG" }{TEXT -1 7 " is an " }{TEXT 261 13 "even periodic" } {TEXT -1 22 " function with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"# \"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 60 "The following picture shows the graphs of Dirichlet kernels " }{XPPEDIT 18 0 "Delta[n](x);" "6#-&%&DeltaG6#%\"n G6#%\"xG" }{TEXT -1 19 " of various orders " }{TEXT 267 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "Delta := (n,x) -> sin((2*n+1)*x/2)/sin(x/2);\nplot([ Delta(1,x),Delta(2,x),Delta(4,x),Delta(8,x)],x=-Pi..3*Pi,\n color=[r ed,blue,green,magenta]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaGf *6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)*&-%$sinG6#,$*&#\"\"\"\"\"#F4*& ,&*&F5F49$F4F4F4F4F49%F4F4F4F4-F/6#,$*&F3F4F:F4F4!\"\"F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 642 323 323 {PLOTDATA 2 "6(-%'CURVESG6$7]s7$$!3** ***4tk#fTJ!#<$!\"\"\"\"!7$$!3-f)ywv`t5$F*$!3CalED\"y#))**!#=7$$!3g$HNHEF*$!3r\">7 !o(yHV(F37$$!3Y:T[9-M&\\#F*$!3w.dR>N#p'fF37$$!3]Dq0(\\F8O#F*$!3M7I\"yn SX@%F37$$!3=(*zMckUEAF*$!3%o*)4f!p<$>#F37$$!3))o*QcTD:4#F*$\"3GX5K')Rb u\\!#?7$$!3_X$Gg`ls&>F*$\"3#o&zFl@fhCF37$$!3\" 4]F37$$!3)ozd:cH&)p\"F*$\"3w&fA1`wAX(F37$$!3#=(ypmM0u:F*$\"3wEL7&)o3g]7F*7$$!3/ml6%*yF;8F*$\"3?)H=\"F*$\"3[XsD\"*>Lc F*7$$!3;F/\"*pd)HF*7$$!3c$ p\"fuhX/5F3$\"3g%>\\z:>**)HF*7$$!3Y!)Q%oPnll'!#>$\"3#oospkqb*HF*7$$!3G D3x2IdoKF^u$\"3wZ9;Q<$*)*HF*7$$\"37*HAIh8U>\"Feo$\"3Y;S&Qd)****HF*7$$ \"3W'\\m2;F8_$F^u$\"3y^ai`,w)*HF*7$$\"3'HwI-'HBBpF^u$\"3#HIG()z3_*HF*7 $$\"3&H]pf(Q^K5F3$\"3o?ht='[$*)HF*7$$\"3gHf\"fX/FP\"F3$\"3kH&f\\R'=\") HF*7$$\"3!Hy3eh&3`?F3$\"339]\")\\i*z&HF*7$$\"3@O;qvnYLFF3$\"3z!)fOtcuD HF*7$$\"3`\"[TRT8[/%F3$\"3v[.^pKhQGF*7$$\"3'oK\"=_+;c`F3$\"3A6#oE!)3*> FF*7$$\"3[(>c*G%))pa'F3$\"3cp%R&*oike#F*7$$\"36o5t0o\"yt(F3$\"37$H\\E' faICF*7$$\"3[_H5Dp#Q:*F3$\"3![$=/YW(*=AF*7$$\"3o$[ZWq$)p0\"F*$\"3'zh!* =F-I)>F*7$$\"3E'f$RN$Qp<\"F*$\"3A)3)=Ox]nF*$\"3M=q:L9nQCF37$$\"3_e$f/C ;S4#F*$\"3yB*H%4()zil!#@7$$\"3PaX*yvcIA#F*$!3vHVJ8qoR@F37$$\"3A](H`F(4 _BF*$!3e*=LXrtS3%F37$$\"3))eT.=vt'[#F*$!3%3g1fIBF'eF37$$\"3an&Q2wx8i#F *$!33%)*GInZUN(F37$$\"3DohnL&>]u#F*$!375?Ji9!yW)F37$$\"3%*oPh18moGF*$! 3C>q5([(pf#*F37$$\"32Q?Q%[V`$HF*$!33bxl<'=hd*F37$$\"3>2.:ic--IF*$!3Fj& f[`Eb!)*F37$$\"3`TW.^nONIF*$!3[y2\\UmE())*F37$$\"3Kw&=*RyqoIF*$!397mQ^ :!p%**F37$$\"356F!)G*[?5$F*$!3)p\"4%e(\\O%)**F37$$\"3WXoo<+RNJF*$!3C4? 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NjtCJ6F*7$$\"3YwMm'=QiH)F*$!3e%zDdvOo;\"F*7$F][p$!3!Gq9=a\"zw6F*7$$\"3 n'yM<&R.J$)F*$!3!*f&[o[w6;\"F*7$$\"3mU/FM=V[$)F*$!3K(3UhK)f?6F*7$$\"3) o41orHeO)F*$!3%GQ[0T(=c5F*7$F\\`m$!3[,;_mio&p*F37$Fa`m$\"3C*Gn&[)*[plF ^u7$Ff`m$\"3)>2)zTVOV(*F37$F_hr$\"3)zirXSf31\"F*7$Fihr$\"3#>a#R5*QK1\" F*7$F^ir$\"3C]CO)H5P#)*F37$F[am$\"3[w9()yDab#)F37$Ffir$\"3X`dj)fD/T$F3 7$F`am$!3)))QO_#HJ!Q#F37$$\"3_Zui.\"z6'*)F*$!3kqi#\\K4f;(F37$Feam$!3Ct J^BC$H')*F37$$\"3IAnz7\"ek0*F*$!3g[&G6GSr,\"F*7$$\"3]9)>Dy<#)3*F*$!3gJ msgA#fu*F37$$\"3o1HC_u(*>\"*F*$!37;uLER/@')F37$Fjam$!3-![11=\"y!)oF37$ Fdbm$\"3ejC[,N#))*RF37$Fhcm$\"2C()*************F*-F]dm6&F_dmF`dmFcdmF` dm-%+AXESLABELSG6$Q\"x6\"Q!Fdcv-%%VIEWG6$;$!+aEfTJ!\"*$\"+izxC%*F\\dv% (DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 22 "1st convergence result" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 34 " be a periodic function of period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 18 " and suppose that " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" } {TEXT -1 48 " is absolutely integral over a period, that is, " } {XPPEDIT 18 0 "Int(abs(phi(x)),x = -L .. L);" "6#-%$IntG6$-%$absG6#-%$ phiG6#%\"xG/F,;,$%\"LG!\"\"F0" }{TEXT -1 26 " is a finite real number ." }}{PARA 0 "" 0 "" {TEXT -1 3 "Let" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "phi[n](x) = c+``;" "6#/-&%$phiG6#%\"nG6#%\"xG,&%\" cG\"\"\"%!GF-" }{XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/L)+b[k]*sin(k*Pi*x/ L),k = 1 .. n);" "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#P iGF,%\"xGF,%\"LG!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F, /F+;F,%\"nG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "c,a[k]" "6$%\"cG&%\"aG6#%\"kG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" }{TEXT -1 29 " are Fourier coef ficients of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 10 ", that is," }}{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(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(p hi(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 " a nd " }{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 18 "Then at any point " }{XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6#\"\"!" } {TEXT -1 7 " where " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" } {TEXT -1 34 " is continuous and differentiable " }{XPPEDIT 18 0 "Limit (phi[n](x[0]),n = infinity) = phi(x[0]);" "6#/-%&LimitG6$-&%$phiG6#%\" nG6#&%\"xG6#\"\"!/F+%)infinityG-F)6#&F.6#F0" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 17 "_________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Explanation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 36 "In the previous section we saw that " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n](x[0]) = 1/(2*L);" "6#/-&%$phiG 6#%\"nG6#&%\"xG6#\"\"!*&\"\"\"F/*&\"\"#F/%\"LGF/!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(phi(x[0])*Delta[n](Pi*(x[0]-x)/L),x = -L .. L); " "6#-%$IntG6$*&-%$phiG6#&%\"xG6#\"\"!\"\"\"-&%&DeltaG6#%\"nG6#*(%#PiG F.,&&F+6#F-F.F+!\"\"F.%\"LGF:F./F+;,$F;F:F;" }{TEXT -1 14 " ------- ( i)," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Delta[n](x) = sin((2*n+1)*x/2)/sin(x/2);" "6#/-&%&DeltaG6#%\"nG6#%\"xG*&-%$sinG6# *(,&*&\"\"#\"\"\"F(F3F3F3F3F3F*F3F2!\"\"F3-F-6#*&F*F3F2F4F4" }{TEXT -1 34 " is the Dirichlet kernel of order " }{TEXT 278 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Le t " }{XPPEDIT 18 0 "u = x-x[0];" "6#/%\"uG,&%\"xG\"\"\"&F&6#\"\"!!\"\" " }{TEXT -1 8 " in (i)." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n](x[0]) = 1/(2* L);" "6#/-&%$phiG6#%\"nG6#&%\"xG6#\"\"!*&\"\"\"F/*&\"\"#F/%\"LGF/!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0]+u)*Delta[n](Pi*u/L),u = -L-x[0] .. L-x[0]);" "6#-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"%\"u GF/F/-&%&DeltaG6#%\"nG6#*(%#PiGF/F0F/%\"LG!\"\"F//F0;,&F9F:&F,6#F.F:,& F9F/&F,6#F.F:" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "Since bo th " }{XPPEDIT 18 0 "phi(x[0]+u);" "6#-%$phiG6#,&&%\"xG6#\"\"!\"\"\"% \"uGF+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Delta[n](Pi*u/L);" "6#-&%& DeltaG6#%\"nG6#*(%#PiG\"\"\"%\"uGF+%\"LG!\"\"" }{TEXT -1 27 " are peri odic functions of " }{TEXT 269 1 "u" }{TEXT -1 13 " with period " } {XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 37 ", we can in tegrate over the interval " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\"\"F %" }{TEXT -1 30 " instead of over the interval " }{XPPEDIT 18 0 "[-L-x [0], L-x[0]];" "6#7$,&%\"LG!\"\"&%\"xG6#\"\"!F&,&F%\"\"\"&F(6#F*F&" } {TEXT -1 8 ". Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n](x[0]) = 1/(2*L);" "6#/-&%$phiG6#%\"nG6#&%\"xG6#\"\"!*&\"\"\" F/*&\"\"#F/%\"LGF/!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0]+ u)*Delta[n](Pi*u/L),u = -L .. L);" "6#-%$IntG6$*&-%$phiG6#,&&%\"xG6#\" \"!\"\"\"%\"uGF/F/-&%&DeltaG6#%\"nG6#*(%#PiGF/F0F/%\"LG!\"\"F//F0;,$F9 F:F9" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Replacing the va riable " }{TEXT 270 1 "u" }{TEXT -1 20 " in the integral by " }{TEXT 271 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "phi[n](x[0]) = 1/(2*L);" "6#/-&%$phiG6#%\"nG6#&%\"xG6# \"\"!*&\"\"\"F/*&\"\"#F/%\"LGF/!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Int(phi(x[0]+x)*Delta[n](Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6 #,&&%\"xG6#\"\"!\"\"\"F,F/F/-&%&DeltaG6#%\"nG6#*(%#PiGF/F,F/%\"LG!\"\" F//F,;,$F8F9F8" }{TEXT -1 15 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Delta[n](Pi*x/L) = 1+2;" "6#/-&%&D eltaG6#%\"nG6#*(%#PiG\"\"\"%\"xGF,%\"LG!\"\",&F,F,\"\"#F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(cos(k*Pi*x/L),k = 1 .. n);" "6#-%$SumG6$-%$c osG6#**%\"kG\"\"\"%#PiGF+%\"xGF+%\"LG!\"\"/F*;F+%\"nG" }{TEXT -1 10 ", we have " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "1/(2*L)" "6#*&\"\"\"F$*&\"\"#F$%\"LGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Delta[n](Pi*x/L),y = -L .. L) = 1/(2*L);" "6 #/-%$IntG6$-&%&DeltaG6#%\"nG6#*(%#PiG\"\"\"%\"xGF/%\"LG!\"\"/%\"yG;,$F 1F2F1*&F/F/*&\"\"#F/F1F/F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1,x = \+ -L .. L)+1/L;" "6#,&-%$IntG6$\"\"\"/%\"xG;,$%\"LG!\"\"F,F'*&F'F'F,F-F' " }{XPPEDIT 18 0 "Sum(Int(cos(k*Pi*x/L),x = -L .. L),k = 1 .. n);" "6# -%$SumG6$-%$IntG6$-%$cosG6#**%\"kG\"\"\"%#PiGF.%\"xGF.%\"LG!\"\"/F0;,$ F1F2F1/F-;F.%\"nG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = \+ " }{XPPEDIT 18 0 "1+0 = 1;" "6#/,&\"\"\"F%\"\"!F%F%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "for each " }{TEXT 268 1 "n" }{TEXT -1 5 ", so " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "phi(x[0]) \+ = 1/(2*L);" "6#/-%$phiG6#&%\"xG6#\"\"!*&\"\"\"F,*&\"\"#F,%\"LGF,!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0])*Delta[n](Pi*x/L),x = - L .. L);" "6#-%$IntG6$*&-%$phiG6#&%\"xG6#\"\"!\"\"\"-&%&DeltaG6#%\"nG6 #*(%#PiGF.F+F.%\"LG!\"\"F./F+;,$F7F8F7" }{TEXT -1 15 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 27 "Then, from (ii) and (iii), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(phi[n](x[0]),n = infi nity);" "6#-%&LimitG6$-&%$phiG6#%\"nG6#&%\"xG6#\"\"!/F*%)infinityG" } {XPPEDIT 18 0 "``-phi(x[0]) = Limit(1/(2*L),n = infinity);" "6#/,&%!G \"\"\"-%$phiG6#&%\"xG6#\"\"!!\"\"-%&LimitG6$*&F&F&*&\"\"#F&%\"LGF&F./% \"nG%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(Int(phi(x[0]+x)*De lta[n](Pi*x/L),x = -L .. L)-Int(phi(x[0])*Delta[n](Pi*x/L),x = -L .. L ));" "6#-%!G6#,&-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F0F3F3-&%&Del taG6#%\"nG6#*(%#PiGF3F0F3%\"LG!\"\"F3/F0;,$F0" "6#2\"\"!%&deltaG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "delt a <= abs(x);" "6#1%&deltaG-%$absG6#%\"xG" }{XPPEDIT 18 0 "``<=L" "6#1% !G%\"LG" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "abs(sin(Pi*delta/(2 *L))) <= abs(sin(Pi*x/(2*L)));" "6#1-%$absG6#-%$sinG6#*(%#PiG\"\"\"%&d eltaGF,*&\"\"#F,%\"LGF,!\"\"-F%6#-F(6#*(F+F,%\"xGF,*&F/F,F0F,F1" } {XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 22 ", which implies tha t " }{XPPEDIT 18 0 "abs(1/sin(Pi*x/(2*L))) <= abs(1/sin(Pi*delta/(2*L )));" "6#1-%$absG6#*&\"\"\"F(-%$sinG6#*(%#PiGF(%\"xGF(*&\"\"#F(%\"LGF( !\"\"F2-F%6#*&F(F(-F*6#*(F-F(%&deltaGF(*&F0F(F1F(F2F2" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "This \+ means that the function " }{XPPEDIT 18 0 "g[[x[0]]](x);" "6#-&%\"gG6#7 #&%\"xG6#\"\"!6#F)" }{TEXT -1 42 " is absolutely integrable in the int erval " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\"\"F%" }{TEXT -1 20 ", a nd (iv) now gives" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Limit(phi[n](x[0]),n = infinity);" "6#-%&LimitG6$-&%$phiG6#%\"nG6#&%\" xG6#\"\"!/F*%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-``*phi(x[0]) = limit(1/(2*L),n = infinity);" "6#/,$*&%!G\"\"\"-%$phiG6#&%\"xG6#\" \"!F'!\"\"-%&limitG6$*&F'F'*&\"\"#F'%\"LGF'F//%\"nG%)infinityG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(g[[x[0]]](x)*sin((2*n+1)*Pi*x/(2*L) ),x = -L .. L);" "6#-%$IntG6$*&-&%\"gG6#7#&%\"xG6#\"\"!6#F-\"\"\"-%$si nG6#**,&*&\"\"#F1%\"nGF1F1F1F1F1%#PiGF1F-F1*&F8F1%\"LGF1!\"\"F1/F-;,$F F5/F1;,$F=F>F=F5-F(6$*(-&F-6#7#&F16#F36#F1F5-F@6#*(F:F5F1F5*& FF5-F76#**FCF5F:F5F1F5F=F>F5/F1;,$F=F>F=F5" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 38 " = 0, \+ by the Riemann - Lebesgue lemma." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "2nd con vergence result " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6# %\"xG" }{TEXT -1 34 " be a periodic function of period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 18 " and suppose that " } {XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 64 " is absolutely integral over a period and piecewise continuous. " }}{PARA 0 "" 0 "" {TEXT -1 3 "Let" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p hi[n](x) = c+``;" "6#/-&%$phiG6#%\"nG6#%\"xG,&%\"cG\"\"\"%!GF-" } {XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/L)+b[k]*sin(k*Pi*x/L),k = 1 .. n); " "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#PiGF,%\"xGF,%\"L G!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F,%\"nG" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 " c,a[k]" "6$%\"cG&%\"aG6#%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[k ]" "6#&%\"bG6#%\"kG" }{TEXT -1 29 " are Fourier coefficients of " } {XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 10 ", that is," }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1/(2*L)" "6#/%\"c G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{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*;,$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 4 "Let " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 18 " be a point where " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 26 " is discontinuous and let " }{XPPEDIT 18 0 "phi(x[0]+``) = Limit(phi(x[0]+h),h = 0,right);" "6#/-%$phiG6#,&&%\"xG6#\"\"!\"\"\" %!GF,-%&LimitG6%-F%6#,&&F)6#F+F,%\"hGF,/F6F+%&rightG" }{TEXT -1 7 " a nd " }{XPPEDIT 18 0 "phi(x[0]-``) = Limit(phi(x[0]+h),h = 0,left);" " 6#/-%$phiG6#,&&%\"xG6#\"\"!\"\"\"%!G!\"\"-%&LimitG6%-F%6#,&&F)6#F+F,% \"hGF,/F7F+%%leftG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 27 "Suppose that, at the point " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi(x) " "6#-%$phiG6#%\"xG" }{TEXT -1 29 " has both a right derivative " } {XPPEDIT 18 0 "Limit((phi(x[0]+h)-phi(x[0]+``))/h,h = 0,right);" "6#-% &LimitG6%*&,&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"%\"hGF0F0-F)6#,&&F-6#F/F0%! GF0!\"\"F0F1F8/F1F/%&rightG" }{TEXT -1 23 " and a left derivative " } {XPPEDIT 18 0 "Limit((phi(x[0]+h)-phi(x[0]-``))/h,h = 0,left);" "6#-%& LimitG6%*&,&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"%\"hGF0F0-F)6#,&&F-6#F/F0%!G !\"\"F8F0F1F8/F1F/%%leftG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Then the Fourier series of " } {XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 14 " converges to \+ " }{XPPEDIT 18 0 "(phi(x[0]+``)+phi(x[0]-``))/2;" "6#*&,&-%$phiG6#,&&% \"xG6#\"\"!\"\"\"%!GF-F--F&6#,&&F*6#F,F-F.!\"\"F-F-\"\"#F4" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "Limit(phi[n](x),n = infinity) = (phi( x[0]+``)+phi(x[0]-``))/2;" "6#/-%&LimitG6$-&%$phiG6#%\"nG6#%\"xG/F+%)i nfinityG*&,&-F)6#,&&F-6#\"\"!\"\"\"%!GF8F8-F)6#,&&F-6#F7F8F9!\"\"F8F8 \"\"#F?" }{TEXT -1 2 ". 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L);" "6#-%$IntG6$*&-%$phiG6#,&&%\"xG 6#\"\"!\"\"\"F,F/F/-&%&DeltaG6#%\"nG6#*(%#PiGF/F,F/%\"LG!\"\"F//F,;,$F 8F9F8" }{TEXT -1 14 " ------- (i)," }}{PARA 0 "" 0 "" {TEXT -1 6 "whe re " }{XPPEDIT 18 0 "Delta[n](x) = sin((2*n+1)*x/2)/sin(x/2);" "6#/-&% &DeltaG6#%\"nG6#%\"xG*&-%$sinG6#*(,&*&\"\"#\"\"\"F(F3F3F3F3F3F*F3F2!\" \"F3-F-6#*&F*F3F2F4F4" }{TEXT -1 34 " is the Dirichlet kernel of order " }{TEXT 279 1 "n" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Le t " }{XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 35 " is be \+ a point of discontinuity of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"x G" }{TEXT -1 29 ", and rewrite (i) in the form" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n](x[0]) = 1/(2*L);" "6#/-&%$phiG6# %\"nG6#&%\"xG6#\"\"!*&\"\"\"F/*&\"\"#F/%\"LGF/!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x[0]+x)*Delta[n](Pi*x/L),x = -L .. 0)+1/(2*L);" " 6#,&-%$IntG6$*&-%\"fG6#,&&%\"xG6#\"\"!\"\"\"F-F0F0-&%&DeltaG6#%\"nG6#* (%#PiGF0F-F0%\"LG!\"\"F0/F-;,$F9F:F/F0*&F0F0*&\"\"#F0F9F0F:F0" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x[0]+x)*Delta[n](Pi*x/L),x = 0 .. L); " "6#-%$IntG6$*&-%\"fG6#,&&%\"xG6#\"\"!\"\"\"F,F/F/-&%&DeltaG6#%\"nG6# *(%#PiGF/F,F/%\"LG!\"\"F//F,;F.F8" }{TEXT -1 15 " ------- (ii)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We saw in the previous section that" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "1/(2*L);" "6#*&\"\"\"F$*&\"\"#F$%\"LGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Delta[n](Pi*x/L),x = -L .. L) = 1;" "6#/- %$IntG6$-&%&DeltaG6#%\"nG6#*(%#PiG\"\"\"%\"xGF/%\"LG!\"\"/F0;,$F1F2F1F /" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Since the function \+ " }{XPPEDIT 18 0 "Delta[n](x);" "6#-&%&DeltaG6#%\"nG6#%\"xG" }{TEXT -1 25 " is even, it follows that" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/(2*L);" "6#*&\"\"\"F$*&\"\"#F$%\"LGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Delta[n](Pi*x/L),x = -L .. 0) = 1/(2*L); " "6#/-%$IntG6$-&%&DeltaG6#%\"nG6#*(%#PiG\"\"\"%\"xGF/%\"LG!\"\"/F0;,$ F1F2\"\"!*&F/F/*&\"\"#F/F1F/F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(De lta[n](Pi*x/L),x = 0 .. L) = 1/2;" "6#/-%$IntG6$-&%&DeltaG6#%\"nG6#*(% #PiG\"\"\"%\"xGF/%\"LG!\"\"/F0;\"\"!F1*&F/F/\"\"#F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 " Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "phi(x[0]+``) = 1/(2*L);" "6#/-%$phiG6#,&&%\"xG6#\"\"!\" \"\"%!GF,*&F,F,*&\"\"#F,%\"LGF,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Int(phi(x[0]+``)*Delta[n](Pi*x/L),x = 0 .. L);" "6#-%$IntG6$*&-%$phiG6 #,&&%\"xG6#\"\"!\"\"\"%!GF/F/-&%&DeltaG6#%\"nG6#*(%#PiGF/F,F/%\"LG!\" \"F//F,;F.F9" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Now cons ider the function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g[[x[0]+``]](x) = [phi(x[0]+x)-phi(x[0]+``)]/sin(Pi*x/(2*L));" "6#/-& %\"gG6#7#,&&%\"xG6#\"\"!\"\"\"%!GF.6#F+*&7#,&-%$phiG6#,&&F+6#F-F.F+F.F .-F56#,&&F+6#F-F.F/F.!\"\"F.-%$sinG6#*(%#PiGF.F+F.*&\"\"#F.%\"LGF.F?F? " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "We need to make sure that " }{XPPEDIT 18 0 "g[[x[0]+``]](x);" "6#-&%\"gG6#7#,&&%\"xG6#\"\" !\"\"\"%!GF-6#F*" }{TEXT -1 42 " is absolutely integrable on the inter val " }{XPPEDIT 18 0 "[0, L];" "6#7$\"\"!%\"LG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 16 "The denominator " }{XPPEDIT 18 0 "sin(Pi* x/(2*L));" "6#-%$sinG6#*(%#PiG\"\"\"%\"xGF(*&\"\"#F(%\"LGF(!\"\"" } {TEXT -1 15 " tends to 0 as " }{XPPEDIT 18 0 "proc (x) options operato r, arrow; 0 end proc;" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F*F *F*" }{TEXT -1 30 ". However, we shall show that " }{XPPEDIT 18 0 "g[[ x[0]+``]](x);" "6#-&%\"gG6#7#,&&%\"xG6#\"\"!\"\"\"%!GF-6#F*" }{TEXT -1 28 " does not tend to infinity. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g[[x[0]+``]](x) \+ = ``((phi(x[0]+x)-phi(x[0]+``))/x)*``(x/sin(Pi*x/(2*L)));" "6#/-&%\"gG 6#7#,&&%\"xG6#\"\"!\"\"\"%!GF.6#F+*&-F/6#*&,&-%$phiG6#,&&F+6#F-F.F+F.F .-F76#,&&F+6#F-F.F/F.!\"\"F.F+FAF.-F/6#*&F+F.-%$sinG6#*(%#PiGF.F+F.*& \"\"#F.%\"LGF.FAFAF." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "S ince " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 32 " has a right-hand derivative at " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "(phi(x[0]+x)-phi(x[0]+``))/x;" "6#*&,& -%$phiG6#,&&%\"xG6#\"\"!\"\"\"F*F-F--F&6#,&&F*6#F,F-%!GF-!\"\"F-F*F4" }{TEXT -1 30 " tends to a finite number as " }{XPPEDIT 18 0 "proc (x) options operator, arrow; 0 end proc;" "6#f*6#%\"xG7\"6$%)operatorG%&a rrowG6\"\"\"!F*F*F*" }{TEXT -1 16 " from the right." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "On the other hand, " } {XPPEDIT 18 0 "x/sin(Pi*x/(2*L)) = 2*L/Pi;" "6#/*&%\"xG\"\"\"-%$sinG6# *(%#PiGF&F%F&*&\"\"#F&%\"LGF&!\"\"F/*(F-F&F.F&F+F/" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(``(Pi*x/(2*L))/sin(Pi*x/(2*L)));" "6#-%!G6#*&-F$6#*( %#PiG\"\"\"%\"xGF+*&\"\"#F+%\"LGF+!\"\"F+-%$sinG6#*(F*F+F,F+*&F.F+F/F+ F0F0" }{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "2*L/Pi;" "6#*(\"\"#\" \"\"%\"LGF%%#PiG!\"\"" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "proc (x) opt ions operator, arrow; 0 end proc;" "6#f*6#%\"xG7\"6$%)operatorG%&arrow G6\"\"\"!F*F*F*" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 16 "It f ollows that " }{XPPEDIT 18 0 "g[[x[0]+``]](y);" "6#-&%\"gG6#7#,&&%\"xG 6#\"\"!\"\"\"%!GF-6#%\"yG" }{TEXT -1 29 " is bounded in some interval \+ " }{XPPEDIT 18 0 "[0, delta];" "6#7$\"\"!%&deltaG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "delta >0" "6#2\"\"!%&deltaG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "delta <= x;" "6#1%&d eltaG%\"xG" }{XPPEDIT 18 0 "``<=L" "6#1%!G%\"LG" }{TEXT -1 10 ", we ha ve " }{XPPEDIT 18 0 "sin(Pi*delta/(2*L)) <= sin(Pi*x/(2*L));" "6#1-%$s inG6#*(%#PiG\"\"\"%&deltaGF)*&\"\"#F)%\"LGF)!\"\"-F%6#*(F(F)%\"xGF)*&F ,F)F-F)F." }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 20 ",which \+ implies that " }{XPPEDIT 18 0 "1/sin(Pi*y/(2*L)) <= 1/sin(Pi*delta/(2* L));" "6#1*&\"\"\"F%-%$sinG6#*(%#PiGF%%\"yGF%*&\"\"#F%%\"LGF%!\"\"F/*& F%F%-F'6#*(F*F%%&deltaGF%*&F-F%F.F%F/F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "This means that the \+ function " }{XPPEDIT 18 0 "g[[x[0]+``]](x);" "6#-&%\"gG6#7#,&&%\"xG6# \"\"!\"\"\"%!GF-6#F*" }{TEXT -1 42 " is absolutely integrable on the i nterval " }{XPPEDIT 18 0 "[0,L]" "6#7$\"\"!%\"LG" }{TEXT -1 4 ", so" } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "limit(1/(2*L),n = i nfinity);" "6#-%&limitG6$*&\"\"\"F'*&\"\"#F'%\"LGF'!\"\"/%\"nG%)infini tyG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0]+x)*Delta[n](Pi*x/L), x = 0 .. L)-1/2;" "6#,&-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F-F0F0 -&%&DeltaG6#%\"nG6#*(%#PiGF0F-F0%\"LG!\"\"F0/F-;F/F9F0*&F0F0\"\"#F:F: " }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi(x[0]+``)" "6#-%$phiG6#,&&%\"xG6# \"\"!\"\"\"%!GF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= limit(1/(2*L),n = infinity)" "6#/%!G-%&limitG6$*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"/%\"nG%)i nfinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(Int(phi(x[0]+x)*Delta[n]( Pi*x/L),x = 0 .. L)-Int(phi(x[0]+``)*Delta[n](Pi*x/L),x = 0 .. L));" " 6#-%!G6#,&-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F0F3F3-&%&DeltaG6#% \"nG6#*(%#PiGF3F0F3%\"LG!\"\"F3/F0;F2F" }{TEXT -1 14 " \+ ------- (iii)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 39 " = 0, by the Riemann - Lebesgue lemma. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "To see this, note that, if " }{TEXT 274 1 "n" }{TEXT -1 13 " is even and " }{XPPEDIT 18 0 "n= 2*m" "6#/%\"nG*&\"\"#\"\"\"%\"mGF'" }{TEXT -1 24 " for a positive inte ger " }{TEXT 275 1 "m" }{TEXT -1 31 ", the integral in (iii) becomes" }}{PARA 256 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "Int(g[[x[0]+``] ](x)*sin(Pi*x/(2*L))*cos(2*m*Pi*x/L),x = 0 .. L)+Int(g[[x[0]+``]](x)*c os(Pi*x/(2*L))*sin(2*m*Pi*x/L),x = 0 .. L);" "6#,&-%$IntG6$*(-&%\"gG6# 7#,&&%\"xG6#\"\"!\"\"\"%!GF26#F/F2-%$sinG6#*(%#PiGF2F/F2*&\"\"#F2%\"LG F2!\"\"F2-%$cosG6#*,F;F2%\"mGF2F9F2F/F2FF2/F/;F1F=F2-F%6$*(-&F*6#7#,&&F/6#F1F2F3F26#F/F2-F@6 #**F9F2F:F2F/F2*&FF2-F66#*,FF2/F/;F1F=F2" }{TEXT 265 1 "." }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 27 "Both of these tend to 0 as " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7 \"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "limit(1/(2*L),n = in finity);" "6#-%&limitG6$*&\"\"\"F'*&\"\"#F'%\"LGF'!\"\"/%\"nG%)infinit yG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0]+x)*Delta[n](Pi*x/L),x = 0 .. L) = 1/2;" "6#/-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F-F0F0 -&%&DeltaG6#%\"nG6#*(%#PiGF0F-F0%\"LG!\"\"F0/F-;F/F9*&F0F0\"\"#F:" } {TEXT -1 1 " " }{XPPEDIT 18 0 "phi(x[0]+``)" "6#-%$phiG6#,&&%\"xG6#\" \"!\"\"\"%!GF+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "A similar arument shows that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "limit(1/(2*L),n = infinity );" "6#-%&limitG6$*&\"\"\"F'*&\"\"#F'%\"LGF'!\"\"/%\"nG%)infinityG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0]+x)*Delta[n](Pi*x/L),x = -L .. 0) = 1/2;" "6#/-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F-F0F0-&%& DeltaG6#%\"nG6#*(%#PiGF0F-F0%\"LG!\"\"F0/F-;,$F9F:F/*&F0F0\"\"#F:" } {TEXT -1 1 " " }{XPPEDIT 18 0 "phi(x[0]-``)" "6#-%$phiG6#,&&%\"xG6#\" \"!\"\"\"%!G!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Henc e " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(phi[n](x[ 0]),n = infinity) = Limit(1/(2*L),n = infinity);" "6#/-%&LimitG6$-&%$p hiG6#%\"nG6#&%\"xG6#\"\"!/F+%)infinityG-F%6$*&\"\"\"F6*&\"\"#F6%\"LGF6 !\"\"/F+F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0]+x)*Delta[n](P i*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F,F/ F/-&%&DeltaG6#%\"nG6#*(%#PiGF/F,F/%\"LG!\"\"F//F,;,$F8F9F8" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(1/ (2*L),n = infinity);" "6#/%!G-%&LimitG6$*&\"\"\"F)*&\"\"#F)%\"LGF)!\" \"/%\"nG%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0]+x)*D elta[n](Pi*x/L),x = -L .. 0);" "6#-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"! \"\"\"F,F/F/-&%&DeltaG6#%\"nG6#*(%#PiGF/F,F/%\"LG!\"\"F//F,;,$F8F9F." }{TEXT -1 4 " + " }{XPPEDIT 18 0 "Limit(1/(2*L),n = infinity);" "6#-% &LimitG6$*&\"\"\"F'*&\"\"#F'%\"LGF'!\"\"/%\"nG%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0]+x)*Delta[n](Pi*x/L),x = 0 .. L);" " 6#-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F,F/F/-&%&DeltaG6#%\"nG6#*( %#PiGF/F,F/%\"LG!\"\"F//F,;F.F8" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "phi(x[0]-``)+1/2;" "6#,&-%$phiG6#,&&%\" xG6#\"\"!\"\"\"%!G!\"\"F,*&F,F,\"\"#F.F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi(x[0]+``);" "6#-%$phiG6#,&&%\"xG6#\"\"!\"\"\"%!GF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Putting the results together - D irichlet's theorem" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 95 "Dirichlet's theorem on the convergence of Fourier series follows from the two previous results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 " phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 20 " and the derivative " } {XPPEDIT 18 0 "phi*`'`(x)" "6#*&%$phiG\"\"\"-%\"'G6#%\"xGF%" }{TEXT -1 5 " are " }{TEXT 261 20 "piecewise continuous" }{TEXT -1 14 ", that is, if " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi*`'`(x)" "6#*&%$phiG\"\"\"-%\"'G6#%\"xGF%" } {TEXT -1 154 " are continuous except at only a finite number points in any period, and do not have infinite discontinuities at these points, then the Fourier series of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"x G" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#% \"xG" }{TEXT -1 22 " at every point where " }{XPPEDIT 18 0 "phi(x)" "6 #-%$phiG6#%\"xG" }{TEXT -1 15 " is continuous." }}{PARA 0 "" 0 "" {TEXT -1 30 "At any point of discontinuity " }{XPPEDIT 18 0 "x = x[0] " "6#/%\"xG&F$6#\"\"!" }{TEXT -1 74 ", the Fourier series converges to the average of the left and right limits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(Limit(phi(x), x = x[0],left)+Limit(phi(x),x = x[0],right))/2 = (phi(x[0]-``)+phi(x[0 ]+``))/2;" "6#/*&,&-%&LimitG6%-%$phiG6#%\"xG/F,&F,6#\"\"!%%leftG\"\"\" -F'6%-F*6#F,/F,&F,6#F0%&rightGF2F2\"\"#!\"\"*&,&-F*6#,&&F,6#F0F2%!GF " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Example " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 115 "It is instructive \+ to see what happens when the left and right derivatives at a point of \+ discontinuity do not exist." }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider t he 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) = \+ PIECEWISE([-1+sqrt(-x), -1 <= x and x < 0],[0, x = 0],[1-sqrt(x), 0 < \+ x and x < 1]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$,&\"\"\"!\"\"-%%sqrt G6#,$F'F.F-31,$F-F.F'2F'\"\"!7$F7/F'F77$,&F-F--F06#F'F.32F7F'2F'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 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 21 " is discontinuous at " }{XPPEDIT 18 0 "x=0" "6#/% \"xG\"\"!" }{TEXT -1 39 " and the left and right derivatives at " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 15 " do not exist. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "f := x -> piecewise(x<0,-1+sqrt(-x),x>0,1-sqrt(x),0):\n'f(x)'=f(x );\nf_ := x -> f(x-2*floor((x+1)/2));\nplot(f_(x),x=-1.2..5.2,y=-1.1.. 1.1,color=COLOR(RGB,.4,0,1),thickness=2,ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$,&\"\"\"!\"\"*$,$F' F.#F-\"\"#F-2F'\"\"!7$,&F-F-*$F'F1F.2F4F'7$F4%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f_Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%\"fG 6#,&9$\"\"\"*&\"\"#F1-%&floorG6#,&*&#F1F3F1F0F1F1F9F1F1!\"\"F(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 523 174 174 {PLOTDATA 2 "6(-%'CURVESG6#7gp7 $$!3%**************>\"!#<$\"3WT3+!4Gd0\"!#=7$$!3QLLL8#)\\g5F*$\"3I&z** yk*4sI!#>7$$!3cmmmYX=\"R*F-$!3;@AxWd(=4$F37$$!3mKLL8A9E!)F-$!3-PVyng7T 5F-7$$!3-LLL$\\^?l'F-$!3H=zdI!*)R%=F-7$$!3Smmm'e\"\\%G&F-$!3-uT#e8\\0t #F-7$$!3!HLLL\\)e;SF-$!3#e\"Ry#*QMiOF-7$$!37+++!G[Pq#F-$!3kWZ4W?C+[F-7 $$!3!ommmmx[-#F-$!3'3')yz#f8+bF-7$$!3[LLL`q+Y8F-$!3Zg8#=7.7L'F-7$$!30+ ++?.m25F-$!311Y^- 9+&F3$!3%\\rFI]=Ow(F-7$$!3+KLLL&o'4LF3$!3'[F$pd0v!=)F-7$$!3W)******p,Q Y#F3$!3W/=/;)\\.V)F-7$$!3'[mmm'[$zh\"F3$!3A;Qc8$>!G()F-7$$!33)*****\\9 +&>\"F3$!3Cn=L\"yQo!*)F-7$$!3%HJLLL!o?x!#?$!3q-50!)fK@\"*F-7$$!35Xmmmh M\"\\$F\\q$!3zZgucQ74%*F-7$$\"3QF-+++))zt!#@$\"3At`\\a1MG(*F-7$$\"3WSL LL36aNF3$\"3)>#*)eX]w9\")F-7$$\"3gymmmGUMqF3$\"3IdHbc8vZtF-7$$\"3o,++! \\t90\"F-$\"3G^.Z.oNdnF-7$$\"3]NLL$p/&*R\"F-$\"3c\\D:`X+fiF-7$$\"3h,++ Sof7?F-$\"3P(eAkX-Q^&F-7$$\"3snmm')*)oDEF-$\"33sj**yA&e([F-7$$\"3)=+++ 3#31SF-$\"3-`'G'*>Q1n$F-7$$\"37*******>V@R&F-$\"3'f%)fGfyol#F-7$$\"3G+ ++?*pys'F-$\"3%Hy5@jSwz\"F-7$$\"3'ommmuP3%zF-$\"3aFlnn@'))3\"F-7$$\"3+ KLL8Q<$Q*F-$\"31ATc]h@LJF37$$\"3_LLLl@]g5F*$!3m>Z)4^.B2$F37$$\"3(***** *f6KE?\"F*$!3?D>[)=a/2\"F-7$$\"3*HLL`?T%G8F*$!3zD*og!480=F-7$$\"35+++; PZm9F*$!3'3*4:V;r&p#F-7$$\"33+++3L\"zf\"F*$!3#G0g(*Qp*eOF-7$$\"3bLLLLn 0N#priF-7$$\"3!ommO;e\\*=F*$!33B#[k;%)*enF-7$$\"3/+++=\">*G> F*$!3!*e_T*=+RL(F-7$$\"3GLLLs+)G'>F*$!3%H,oXuYL2)F-7$$\"3ummmE5%o*>F*$ !3%H5=Cb_zV*F-7$$\"3km;a70D,?F*$\"3/Gy\\KTPY'*F-7$$\"3wmmT)**fc+#F*$\" 3gDlF!RqwC*F-7$$\"3(om\"H%[p+,#F*$\"3i\"eB'*zJl**)F-7$$\"3bmm;q*yW,#F* $\"3YjWEvSr'z)F-7$$\"3zmm\">%zHB?F*$\"37?C+kOjt%)F-7$$\"3emmm8p6K?F*$ \"3!4^N5w!)y?)F-7$$\"3imm;d[v\\?F*$\"337JC=.UpxF-7$$\"3kmmm+GRn?F*$\"3 )[mK0m()RS(F-7$$\"3Emmm(ooE5#F*$\"3))zza0X!ez'F-7$$\"3KmmmuX%z8#F*$\"3 /\"yF1&4\"fG'F-7$$\"3CLLL,,O*>#F*$\"3l.\\!Q$*[F-7$$\"3]mmmIlV$R#F*$\"3O!Hd<3[vs$F-7$$\"3&******RA)[IDF*$ \"3VQOPYx`;FF-7$$\"3o*****zcmXm#F*$\"3=]X7gZ!z%=F-7$$\"3-+++gUH%z#F*$ \"3#**yQ?A\"o(3\"F-7$$\"3h*****RuM$QHF*$\"3j>g1T*>B8$F37$$\"3OKLLD:wnI F*$!3cuC*)QE]ZMF37$$\"3B+++SA&f?$F*$!3#HfomCk!*3\"F-7$$\"3?mmmmN=#=F-7$$\"3!******>bt!oMF*$!3f9mn77n1FF-7$$\"3[KLLpd)of$F*$ !3uRMQfA(3l$F-7$$\"3:+++_;`JPF*$!3o4X(o'og=[F-7$$\"3gmmmK@N(z$F*$!3eL \"[%HtN)\\&F-7$$\"31LLL8E?#*4kQR(F-7$$\"3s****\\UvI\\RF*$! 3SzD[V9][xF-7$$\"3)GLL$GX`mRF*$!3QC*)ejHkq\")F-7$$\"3W***\\7-[^(RF*$!3 W$*p1i*[NU)F-7$$\"3-mm;9:w$)RF*$!3VbUSCopD()F-7$$\"33**\\ig#o!))RF*$!3 _?&4a)\\n2*)F-7$$\"3gKL32]P#*RF*$!3Y*)=wXzyE\"*F-7$$\"35m;a`65bF-7$$\"3KmmmA.YpUF*$\"3juNUAK/4[F-7$$\"3fLLL\"3vSS%F*$\"34ZyKq*4Lk $F-7$$\"3l*****R1sx_%F*$\"3Z4VX_x?NFF-7$$\"3YLLLlFapYF*$\"3_eI152W<=F- 7$$\"31mmmqTM'z%F*$\"3y/yg)>)=w5F-7$$\"3L+++;4aJ\\F*$\"3IeHO$oKO[$F37$ $\"3y*****>XT41&F*$!3AP2>%pm\\4$F37$$\"3;+++++++_F*$!3cU3+!4Gd0\"F--%* AXESTICKSG6$%(DEFAULTG\"\"$-%+AXESLABELSG6$Q\"x6\"Q\"yFfhl-%&COLORG6&% $RGBG$\"\"%!\"\"\"\"!\"\"\"-%*THICKNESSG6#\"\"#-%%VIEWG6$;$!#7F^il$\"# _F^il;$!#6F^il$\"#6F^il" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 68 " is an odd fu nction its Fourier series is a sine series of the form " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[k]*sin*k*Pi*x,k = 1 .. inf inity);" "6#-%$SumG6$*,&%\"bG6#%\"kG\"\"\"%$sinGF+F*F+%#PiGF+%\"xGF+/F *;F+%)infinityG" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{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*%$sinGF*F'F*%#PiGF*F2F*/F2;\"\"!F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Int((1-sqrt( x))*sin*k*Pi*x,x = 0 .. 1);" "6#/%!G*&\"\"#\"\"\"-%$IntG6$*,,&F'F'-%%s qrtG6#%\"xG!\"\"F'%$sinGF'%\"kGF'%#PiGF'F0F'/F0;\"\"!F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 " Maple can express this integral in terms of a special function called \+ the " }{TEXT 261 23 "Fresnel cosine integral" }{TEXT -1 12 " defined b y " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "C(x)=Int(cos(Pi *t^2/2),t=0..x)" "6#/-%\"CG6#%\"xG-%$IntG6$-%$cosG6#*(%#PiG\"\"\"*$%\" tG\"\"#F0F3!\"\"/F2;\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "This function is given by " }{TEXT 280 11 "FresnelC(x)" }{TEXT -1 11 " in Maple. 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F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 334 "FS := (x,n) -> Sum((2/(Pi*k)-sqrt(2)*Fresn elC(sqrt(2*k))/\n (k*sqrt(k)*Pi))*sin(k*Pi*x),k=1..n);\nf := x -> piecewise(x<0,-1+sqrt(-x),1-sqrt(x)):\nf_ := x -> f(x-2*floor ((x+1)/2)):\nplot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5)],x=-1 .2..5.2,y=-1.1..1.1,\n color=[black,red,blue,green,magenta,coral],li nestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG% \"nG6\"6$%)operatorG%&arrowGF)-%$SumG6$*&,&*(\"\"#\"\"\"%\"kG!\"\"%#Pi GF6F4*,-%%sqrtG6#F3F4-%)FresnelCG6#-F:6#,$*&F3F4F5F4F4F4F5F6-F:6#F5F6F 7F6F6F4-%$sinG6#*(F5F4F7F49$F4F4/F5;F49%F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 787 266 266 {PLOTDATA 2 "6*-%'CURVESG6%7gp7$$!3%************ **>\"!#<$\"3WT3+!4Gd0\"!#=7$$!3QLLL8#)\\g5F*$\"3I&z**yk*4sI!#>7$$!3cmm mYX=\"R*F-$!3;@AxWd(=4$F37$$!3mKLL8A9E!)F-$!3-PVyng7T5F-7$$!3-LLL$\\^? l'F-$!3H=zdI!*)R%=F-7$$!3Smmm'e\"\\%G&F-$!3-uT#e8\\0t#F-7$$!3!HLLL\\)e 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