{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 12 102 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 "" -1 263 "" 0 1 0 0 0 0 1 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 "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 266 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{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 "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 76 "The Fourier transform of the Dira c delta function and the unit step function" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{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 49 "The Fouri er transform of the Dirac delta function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 50 "The Fourier transfo rm of the Dirac delta function " }{XPPEDIT 18 0 "delta(x)" "6#-%&delta G6#%\"xG" }{TEXT -1 12 " is given by" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Fr*[delta(x)] = Int(delta(x)*exp(-i*omega*x),x=-infinity..infinity) " "6#/*&%#FrG\"\"\"7#-%&deltaG6#%\"xGF&-%$IntG6$*&-F)6#F+F&-%$expG6#,$ *(%\"iGF&%&omegaGF&F+F&!\"\"F&/F+;,$%)infinityGF9F=" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = eval(exp(-i*ome ga*x),x = 0);" "6#/%!G-%%evalG6$-%$expG6#,$*(%\"iG\"\"\"%&omegaGF.%\"x GF.!\"\"/F0\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1" "6#/%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "alias(delt a=Dirac):\nInt(delta(x)*exp(-I*omega*x),x=-infinity..infinity);\nvalue (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%&deltaG6#%\"xG\" \"\"-%$expG6#*(^#!\"\"F+F*F+%&omegaGF+F+/F*;,$%)infinityGF1F6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "delta(x);\n`Fourier tra nsform`=inttrans[fourier](%,x,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&deltaG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~tran sformG\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "We investigate the limits of transforms of the sequences of functions used in the initial description of " }{XPPEDIT 18 0 "del ta(x)" "6#-%&deltaG6#%\"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 19 "1st description of " }{XPPEDIT 18 0 "Fr*[ delta(x)];" "6#*&%#FrG\"\"\"7#-%&deltaG6#%\"xGF%" }{TEXT -1 25 " as a \+ limit of transforms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 9 "Consider " }{XPPEDIT 18 0 "delta(x)" "6#-% &deltaG6#%\"xG" }{TEXT -1 42 " to be the limit of the sequence of puls es" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d[n](x) = PIEC EWISE([n/2, abs(x) < 1/n],[0, 1/n <= abs(x)]);" "6#/-&%\"dG6#%\"nG6#% \"xG-%*PIECEWISEG6$7$*&F(\"\"\"\"\"#!\"\"2-%$absG6#F**&F0F0F(F27$\"\"! 1*&F0F0F(F2-F56#F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 " Th e Fourier transform of " }{XPPEDIT 18 0 "d[n](x)" "6#-&%\"dG6#%\"nG6#% \"xG" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "Fr*[d[n](x)] = sin(omega/n)/ ``(omega/n);" "6#/*&%#FrG\"\"\"7#-&%\"dG6#%\"nG6#%\"xGF&*&-%$sinG6#*&% &omegaGF&F,!\"\"F&-%!G6#*&F4F&F,F5F5" }{TEXT -1 22 ", which tends to 1 as " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arr owG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "alias(H=Heaviside):\n d := (n,x) -> n/2*(H(x+1/n)-H(x-1/n)):\n'd(n,x)'=d(n,x);\n`Fourier tra nsform`=inttrans[fourier](rhs(%),x,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"dG6$%\"nG%\"xG,$*&#\"\"\"\"\"#F,*&F'F,,&-%\"HG6#,&F(F,*&F,F ,F'!\"\"F,F,-F16#,&F(F,F4F5F5F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%2Fourier~transformG*(%\"nG\"\"\"%\"wG!\"\"-%$sinG6#*&F&F)F(F'F'" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "F := (n,w) -> if w<>0 then n*sin(w/n)/w else 1 fi:\nplot(['F(1, w)','F(2,w)','F(4,w)','F(8,w)','F(16,w)',\n 'F(32,w)','F(64,w)'],w =-40..40,ytickmarks=3,\n color=[red,green,blue,magenta,coral,c yan,brown]);" }}{PARA 13 "" 1 "" {GLPLOT2D 552 259 259 {PLOTDATA 2 "6, -%'CURVESG6$7[z7$$!#S\"\"!$\"1s$)>,Hyi=!#<7$$!1LL$e%G?yR!#9$\"15VS1#47 >#F-7$$!1nmm\"p0k&RF1$\"1<0aNn)*=CF-7$$!1++]P&3Y$RF1$\"1L&*e.#yT`#F-7$ $!1LLL$Q6G\"RF1$\"1o%fR5e+`#F-7$$!1nm;HU,\"*QF1$\"1[6h1#*[0CF-7$$!1+++ vq@pQF1$\"1a#e>mQ];#F-7$$!1LL$3#*>u%QF1$\"1t!*e\\`#)==F-7$$!1nmmmFiDQF 1$\"1\\ac3F5#Q\"F-7$$!1++]Ti*QMF-7$$!1mm;p3\"[g#F1$\"1h$pRoZN/$F-7$$! 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" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "alia s(H=Heaviside):\nt := (n,x)->H(x+1/n)*(n+n^2*x)-2*H(x)*n^2*x-H(x-1/n)* (n-n^2*x):\n't(n,x)'=t(n,x);\n`Fourier transform`=inttrans[fourier](rh s(%),x,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"tG6$%\"nG%\"xG, (*&-%\"HG6#,&F(\"\"\"*&F/F/F'!\"\"F/F/,&F'F/*&)F'\"\"#F/F(F/F/F/F/**F5 F/-F,6#F(F/F4F/F(F/F1*&-F,6#,&F(F/F0F1F/,&F'F/F3F1F/F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,$**\"\"%\"\"\"%\"nG\"\"#%&om egaG!\"#-%$sinG6#,$*(F*!\"\"F)F2F+F(F(F*F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "F := (n,w) -> if \+ w<>0 then 4*n^2/w^2*sin(w/(2*n))^2 else 1 end if:\nplot(['F(1,w)','F(2 ,w)','F(4,w)','F(8,w)','F(16,w)',\n 'F(32,w)','F(64,w)'],w=-40..40 ,ytickmarks=3,\n color=[red,green,blue,magenta,coral,cyan,brow n]);" }}{PARA 13 "" 1 "" {GLPLOT2D 572 210 210 {PLOTDATA 2 "6,-%'CURVE SG6$7es7$$!#S\"\"!$\"3yF`1xDn$3#!#?7$$!3xmmmmFiDQ!#;$\"3k)oOTJ[k1#!#@7 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}{XPPEDIT 18 0 "h[n](x) = n/(Pi *(1+n^2*x^2))" "6#/-&%\"hG6#%\"nG6#%\"xG*&F(\"\"\"*&%#PiGF,,&F,F,*&F( \"\"#F*F1F,F,!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Sinc e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[1/(a^2+x^2)] = Pi/a;" "6#/*&%#FrG\"\"\"7#*&F&F&,&* $%\"aG\"\"#F&*$%\"xGF,F&!\"\"F&*&%#PiGF&F+F/" }{TEXT -1 1 " " } {XPPEDIT 18 0 "exp(-a*abs(omega))" "6#-%$expG6#,$*&%\"aG\"\"\"-%$absG6 #%&omegaGF)!\"\"" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "wher e " }{TEXT 265 1 "a" }{TEXT -1 13 " is positive," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[n/(Pi*(1+n^2*x^2))] = 1/(n*Pi)" "6# /*&%#FrG\"\"\"7#*&%\"nGF&*&%#PiGF&,&F&F&*&F)\"\"#%\"xGF.F&F&!\"\"F&*&F &F&*&F)F&F+F&F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[1/(1/(n^2)+x^2)] \+ = 1/(n*Pi);" "6#/*&%#FrG\"\"\"7#*&F&F&,&*&F&F&*$%\"nG\"\"#!\"\"F&*$%\" xGF-F&F.F&*&F&F&*&F,F&%#PiGF&F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "n*Pi* exp(-abs(omega)/n) = exp(-abs(omega)/n)" 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"6#*&%#FrG\"\"\"7#-%&deltaG6#%\"xGF%" }{TEXT -1 25 " as \+ a limit of transforms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" } {TEXT -1 59 " can be realised as the limit of the sequence of function s " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[n](x) = n/sqr t(2*Pi);" "6#/-&%\"bG6#%\"nG6#%\"xG*&F(\"\"\"-%%sqrtG6#*&\"\"#F,%#PiGF ,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-n^2*x^2/2);" "6#-%$expG6# ,$*(%\"nG\"\"#%\"xGF)F)!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Fourier transform of \+ " }{XPPEDIT 18 0 "b[n](x)" "6#-&%\"bG6#%\"nG6#%\"xG" }{TEXT -1 4 " is \+ " }{XPPEDIT 18 0 "F(omega)=exp(-omega^2/(2*n^2))" "6#/-%\"FG6#%&omegaG -%$expG6#,$*&F'\"\"#*&F-\"\"\"*$%\"nGF-F/!\"\"F2" }{TEXT -1 22 ", whic h tends to 1 as " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)op eratorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "interface (showassumed=0): assume(n>0):\nb := (n,x) -> n/sqrt(2*Pi)*exp(-n^2*x^2 /2):\n'b(n,x)'=b(n,x);\n`Fourier transform`=inttrans[fourier](rhs(%),x ,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"bG6$%\"nG%\"xG,$*&#\" \"\"\"\"#F,**%#n|irGF,F-F+%#PiG#!\"\"F--%$expG6#,$*(F-F2F/F-F(F-F2F,F, F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG-%$expG6#,$ *(\"\"#!\"\"%&omegaGF*%#n|irG!\"#F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "F := (n,w) -> exp(-w^2/(2 *n^2)):\nplot([F(1,w),F(3,w),F(9,w),F(27,w),F(81,w),F(243,w),F(729,w)] ,\n w=-50..50,color=[red,green,blue,magenta,coral,cyan,brown]);" }} {PARA 13 "" 1 "" {GLPLOT2D 581 254 254 {PLOTDATA 2 "6+-%'CURVESG6$7cp7 $$!#]\"\"!$F*F*7$$!3GLLLe%G?y%!#;F+7$$!3!pm;aesBf%F/F+7$$!3ALL$3s%3zVF /F+7$$!3MLL$e/$QkTF/F+7$$!3'om;/\"=q]RF/F+7$$!3QLL3_>f_PF/$\"3\\$*Q,(e +%Q;!$B$7$$!3)****\\(o1YZNF/$\"3u4vqDk^*Q&!$\"H7$$!3DLL3-OJNLF/$\"3i%p 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$F[el$\"3ClV\"H!za*)**Fhu7$F`el$\"3#Q,'\\ul<))**Fhu7$Feel$\"3-(oiHILn) **Fhu7$F[fl$\"3EZ3%R)RL&)**Fhu7$F^fl$\"3ow+%49WO)**Fhu7$Fafl$\"37h3d$) [0#)**Fhu7$Fdfl$\"3Iv)G4[z-)**Fhu7$Fgfl$\"3%>d=E,-&y**Fhu7$FjflFjcp-F] gl6&F_gl$\")#)eqkFbgl$\"))eqk\"FbglF^]q-%+AXESLABELSG6$Q\"w6\"Q!Fd]q-% %VIEWG6$;F(Fjfl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "The Fourier transform of th e unit step function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 20 "First note that, if " }{XPPEDIT 18 0 "Fr *[f(x)]=F(omega)" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" }{TEXT -1 7 ", then " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Fr*[f(-x)] = F(-omega)" "6#/*&%#FrG\"\"\"7#-%\"fG6#,$%\"xG!\"\"F &-%\"FG6#,$%&omegaGF-" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 11 "___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "To see why this formula h olds first note that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[f(-x)] = Int(f(-x)*exp(-i*omega*x),x=-infinity..infinity)" "6#/ *&%#FrG\"\"\"7#-%\"fG6#,$%\"xG!\"\"F&-%$IntG6$*&-F)6#,$F,F-F&-%$expG6# ,$*(%\"iGF&%&omegaGF&F,F&F-F&/F,;,$%)infinityGF-F?" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "u = -x" "6#/%\" uG,$%\"xG!\"\"" }{TEXT -1 26 " in the integral, we have " }}{PARA 256 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "Fr*[f(-x)] = -Int(f( u)*exp(i*omega*u),u=infinity..-infinity)" "6#/*&%#FrG\"\"\"7#-%\"fG6#, $%\"xG!\"\"F&,$-%$IntG6$*&-F)6#%\"uGF&-%$expG6#*(%\"iGF&%&omegaGF&F5F& F&/F5;%)infinityG,$F>F-F-" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f(u)*exp(i*omega*u),u = -infinity .. infinity);" "6#/%!G-%$IntG6$*&-%\"fG6#%\"uG\"\"\"-%$expG6#*(%\"iGF-%& omegaGF-F,F-F-/F,;,$%)infinityG!\"\"F7" }{TEXT -1 2 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f(u)*exp(-i*(-omega)*u) ,u = -infinity .. infinity);" "6#/%!G-%$IntG6$*&-%\"fG6#%\"uG\"\"\"-%$ expG6#,$*(%\"iGF-,$%&omegaG!\"\"F-F,F-F6F-/F,;,$%)infinityGF6F:" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = F(-omega);" "6#/%!G-%\"FG6#,$%&omegaG!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Now consider the function " }{XPPEDIT 18 0 "s( x)=PIECEWISE([1 ,x>0 ],[-1 , x<0])" "6#/-%\"sG6#%\"xG-%*PIECEWISEG6$7$ \"\"\"2\"\"!F'7$,$F,!\"\"2F'F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(signum( x),x=-5..5,color=red,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 394 145 145 {PLOTDATA 2 "6'-%'CURVESG6#7hn7$$!\"&\"\"!$!\"\"F*7$$!1LLLe%G? y%!#:F+7$$!1mmT&esBf%F0F+7$$!1LL$3s%3zVF0F+7$$!1ML$e/$QkTF0F+7$$!1nmT5 =q]RF0F+7$$!1LL3_>f_PF0F+7$$!1++vo1YZNF0F+7$$!1LL3-OJNLF0F+7$$!1++v$*o %Q7$F0F+7$$!1mmm\"RFj!HF0F+7$$!1LL$e4OZr#F0F+7$$!1+++v'\\!*\\#F0F+7$$! 1+++DwZ#G#F0F+7$$!1+++D.xt?F0F+7$$!1LL3-TC%)=F0F+7$$!1mmm\"4z)e;F0F+7$ $!1mmmm`'zY\"F0F+7$$!1++v=t)eC\"F0F+7$$!1nmm;1J\\5F0F+7$$!1$***\\(=[jL )!#;F+7$$!1'***\\iXg#G'FdoF+7$$!1emmT&Q(RTFdoF+7$$!1lm;/'=><#FdoF+7$$! 1++D\"yQ16\"FdoF+7$$!1EMLLe*e$\\!#=F+7$$\"1$4aj%e#R&>Fdp$\"\"\"F*7$$\" 17;/EvuV))FdpFhp7$$\"18Hd?pNt:!#uxF`qFhp7$$\"1pmT5D,`5FdoFh p7$$\"1q;zW#)>/;FdoFhp7$$\"1sm;zRQb@FdoFhp7$$\"1PLL$e,]6$FdoFhp7$$\"1- +](=>Y2%FdoFhp7$$\"1vmm\"zXu9'FdoFhp7$$\"1,+++&y))G)FdoFhp7$$\"1++]i_Q Q5F0Fhp7$$\"1,+D\"y%3T7F0Fhp7$$\"1++]P![hY\"F0Fhp7$$\"1LLL$Qx$o;F0Fhp7 $$\"1+++v.I%)=F0Fhp7$$\"1mm\"zpe*z?F0Fhp7$$\"1,++D\\'QH#F0Fhp7$$\"1LLe 9S8&\\#F0Fhp7$$\"1,+D1#=bq#F0Fhp7$$\"1LLL3s?6HF0Fhp7$$\"1++DJXaEJF0Fhp 7$$\"1ommm*RRL$F0Fhp7$$\"1om;a<.YNF0Fhp7$$\"1NLe9tOcPF0Fhp7$$\"1,++]Qk \\RF0Fhp7$$\"1NL$3dg6<%F0Fhp7$$\"1ommmxGpVF0Fhp7$$\"1++D\"oK0e%F0Fhp7$ $\"1,+v=5s#y%F0Fhp7$$\"\"&F*Fhp-%+AXESLABELSG6$Q\"x6\"%!G-%*THICKNESSG 6#\"\"#-%'COLOURG6&%$RGBG$\"*++++\"!\")F*F*-%%VIEWG6$;F(F\\w%(DEFAULTG " 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 16 "We can consider " } {XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 34 " to be the limit \+ of the functions " }{XPPEDIT 18 0 "s[n](x)=PIECEWISE([exp(-x/n) , x>0] ,[-exp(x/n), x<0])" "6#/-&%\"sG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$-%$expG6 #,$*&F*\"\"\"F(!\"\"F52\"\"!F*7$,$-F06#*&F*F4F(F5F52F*F7" }{TEXT -1 6 ", as " }{XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%)operatorG%&ar rowG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "r[n](x)=PIECEWISE([exp(-x/n), x>0],[ 0, x<0]) " "6#/-&%\"rG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$-%$expG6#,$*&F*\"\"\"F(!\" \"F52\"\"!F*7$F72F*F7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "s[n](x) = r[n] (x) - r[n](-x)" "6#/-&%\"sG6#%\"nG6#%\"xG,&-&%\"rG6#F(6#F*\"\"\"-&F.6# F(6#,$F*!\"\"F7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since \+ " }{XPPEDIT 18 0 "Fr*[r[n](x)]=1/(1/n+i*omega)" "6#/*&%#FrG\"\"\"7#-&% \"rG6#%\"nG6#%\"xGF&*&F&F&,&*&F&F&F,!\"\"F&*&%\"iGF&%&omegaGF&F&F2" } {TEXT -1 11 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Fr*[s[n](x)]=Fr*[r[n](x)-Fr*[r[n](-x)]" "6#/*&%#FrG\"\" \"7#-&%\"sG6#%\"nG6#%\"xGF&*&F%F&7#,&-&%\"rG6#F,6#F.F&*&F%F&7#-&F46#F, 6#,$F.!\"\"F&F>F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(1/n+i*omega)-1/(1/n-i*omega);" "6#/%!G,&*&\"\" \"F',&*&F'F'%\"nG!\"\"F'*&%\"iGF'%&omegaGF'F'F+F'*&F'F',&*&F'F'F*F+F'* &F-F'F.F'F+F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (``(1/n-i*omega)-``(1/n+i*omega))/(1/(n^2)+omega^2 );" "6#/%!G*&,&-F$6#,&*&\"\"\"F+%\"nG!\"\"F+*&%\"iGF+%&omegaGF+F-F+-F$ 6#,&*&F+F+F,F-F+*&F/F+F0F+F+F-F+,&*&F+F+*$F,\"\"#F-F+*$F0F9F+F-" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = -2*i*omega/(1/(n^2)+omega^2);" "6#/%!G,$**\"\"#\"\"\"%\"iGF(%&omega GF(,&*&F(F(*$%\"nGF'!\"\"F(*$F*F'F(F/F/" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 20 "Taking the limit as " }{XPPEDIT 18 0 "n->infinity " "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 15 " suggests that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Fr*[s(x)]=-2*i/omega" "6#/*&%#FrG\"\"\"7#-%\"sG6#%\"xGF&,$*(\"\"#F& %\"iGF&%&omegaG!\"\"F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 98 "The fact that this expression represe nts a pure imaginary number is consistent with the fact that " } {XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 20 " is an odd functi on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "n := 'n': assume(n,posint):\nInt(-exp(x/n)*exp(-I*om ega*x),x=-infinity..0)+\n Int(exp(-x/n)*exp(-I*omega*x),x=0 ..infinity);\n``=value(%);\n``=simplify(rhs(%));\nLimit(rhs(%),n=infin ity);\n``=value(%);\nn := 'n':\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& -%$IntG6$,$*&-%$expG6#*&%\"xG\"\"\"%#n|irG!\"\"F.-F*6#*(^#F0F.%&omegaG F.F-F.F.F0/F-;,$%)infinityGF0\"\"!F.-F%6$*&-F*6#,$F,F0F.F1F./F-;F:F9F. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%#n|irG\"\"\",&F(!\"\"*(%& omegaGF(F'F(^#F(F(F(F*F(*&F'F(,&F(F(F+F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G**^#\"\"#\"\"\"%#n|irGF'%&omegaGF(,&F(!\"\"*&)F*F'F ()F)F'F(F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$**^#\"\"#\" \"\"%#n|irGF(%&omegaGF),&F)!\"\"*&)F+F(F))F*F(F)F-F-/F*%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&^#!\"#\"\"\"%&omegaG!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The step function, or Heaviside function " }{XPPEDIT 18 0 "H(x)" "6#-%\"HG6#%\"xG" }{TEXT -1 10 " giv en by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "H(x)=PIECEW ISE([1 , x>0],[0 , x<0])" "6#/-%\"HG6#%\"xG-%*PIECEWISEG6$7$\"\"\"2\" \"!F'7$F.2F'F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "can be expressed in terms of the function " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 10 " above as " } {XPPEDIT 18 0 "H(x) = 1/2+s(x)/2;" "6#/-%\"HG6#%\"xG,&*&\"\"\"F*\"\"#! \"\"F**&-%\"sG6#F'F*F+F,F*" }{TEXT -1 34 ", so by linearity of the ope rator " }{TEXT 263 2 "Fr" }{TEXT -1 23 ", and using the result " } {XPPEDIT 18 0 "Fr*[1]=2*Pi*delta(omega)" "6#/*&%#FrG\"\"\"7#F&F&*(\"\" #F&%#PiGF&-%&deltaG6#%&omegaGF&" }{TEXT -1 9 ", we have" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[H(x)]=Pi*delta(omega)-i/omeg a" "6#/*&%#FrG\"\"\"7#-%\"HG6#%\"xGF&,&*&%#PiGF&-%&deltaG6#%&omegaGF&F &*&%\"iGF&F2!\"\"F5" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 264 14 "______________" }{TEXT 266 1 " " }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "alias(H=Heaviside,delta=Dira c):\nH(x);\n`Fourier transform`=inttrans[fourier](%,x,omega);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"HG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,&*&%#PiG\"\"\"-%&deltaG6#%&omegaG F(F(*&^#!\"\"F(F,F/F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 281 "p1 := op(1,[plottools[arrow]([0,0] ,[0,Pi],.15,.3,.2,\n color=COLOR(RGB,.4,0,.9))] ):\np2 := plot(0,w=-5..5,color=purple,thickness=2):\np3 := plot(-1/w,w =-5..5,-5..5,color=COLOR(RGB,.6,.4,0)):\nplots[display]([p1,p2,p3],tit le=`The Fourier transform F(w) of H(x)`);" }}{PARA 13 "" 1 "" {GLPLOT2D 438 334 334 {PLOTDATA 2 "6(-%)POLYGONSG6&7&7$$\"+++++v!#6$\" \"!F,7$$!+++++vF*F+7$F.$\"+BTF8D!\"*7$F(F17%7$$!+++++:!#5F17$F+$\"+aEf TJF37$$\"+++++:F9F1-%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&%$RGBG$\"\"%!\" \"F,$\"\"*FJ-%'CURVESG6%7S7$$!\"&F,F+7$$!3YLLLe%G?y%!#f_PFWF+7$$!3K++vo1YZNFWF+7$$!3;LL3-OJNLFWF+7$$!3p***\\P*o%Q7$FWF+7$ $!3Kmmm\"RFj!HFWF+7$$!33LL$e4OZr#FWF+7$$!3u*****\\n\\!*\\#FWF+7$$!3%)* ****\\ixCG#FWF+7$$!3#******\\KqP2#FWF+7$$!39LL3-TC%)=FWF+7$$!3[mmm\"4z )e;FWF+7$$!3Mmmmm`'zY\"FWF+7$$!3#****\\(=t)eC\"FWF+7$$!3!ommmh5$\\5FWF +7$$!3S$***\\(=[jL)!#=F+7$$!3)f***\\iXg#G'F[rF+7$$!3ndmmT&Q(RTF[rF+7$$ !3%\\mmTg=><#F[rF+7$$!3vDMLLe*e$\\!#?F+7$$\"3!=nm\"zRQb@F[rF+7$$\"3_,+ ](=>Y2%F[rF+7$$\"3summ\"zXu9'F[rF+7$$\"3#4+++]y))G)F[rF+7$$\"3H++]i_QQ 5FWF+7$$\"3b++D\"y%3T7FWF+7$$\"3+++]P![hY\"FWF+7$$\"3iKLL$Qx$o;FWF+7$$ \"3Y+++v.I%)=FWF+7$$\"3?mm\"zpe*z?FWF+7$$\"3;,++D\\'QH#FWF+7$$\"3%HL$e 9S8&\\#FWF+7$$\"3s++D1#=bq#FWF+7$$\"3\"HLL$3s?6HFWF+7$$\"3a***\\7`Wl7$ FWF+7$$\"3enmmm*RRL$FWF+7$$\"3%zmmTvJga$FWF+7$$\"3]MLe9tOcPFWF+7$$\"31 ,++]Qk\\RFWF+7$$\"3![LL3dg6<%FWF+7$$\"3%ymmmw(GpVFWF+7$$\"3C++D\"oK0e% FWF+7$$\"35,+v=5s#y%FWF+7$$\"\"&F,F+FD-%*THICKNESSG6#\"\"#-FN6$7ao7$FR $\"35+++++++?F[r7$FU$\"3?G0p&zi64#F[r7$FY$\"3q)3\\'oO_x@F[r7$Ffn$\"3-Z *)3=Ae$G#F[r7$Fin$\"3bMk!p3;8S#F[r7$F\\o$\"3'p(o8Ve>JDF[r7$F_o$\"3yJN$ QxC[m#F[r7$Fbo$\"3*Rr^Kz<*=GF[r7$Feo$\"3eI_xM@#[F[r7$Fjp$\"35cB`)*z;2`F[ r7$F]q$\"3]s!>5)o;GgF[r7$F`q$\"341GM9)\\@\"oF[r7$Fcq$\"3t*\\,j/3k-)F[r 7$Ffq$\"32\\C%>am+`*F[r7$Fiq$\"3G\"42!=gc*>\"FW7$F]r$\"3m_E')[lp\"f\"F W7$F`r$\"3aQv<$\\6cT#FW7$$!3Ihm\"HdGe:$F[r$\"3KA'Rk#*R(oJFW7$Fcr$\"3>g 'e*3TA/YFW7$$!3FK$3Fpy7k\"F[r$\"3=)G)4o/\"G4'FW7$$!3g***\\7yQ16\"F[r$ \"3HRG$*)\\FQ+*FW7$$!3iK$3_D)=`%)!#>$\"3G-fO-c)H=\"!#;7$$!3Epm\"zp))** z&Fb]l$\"3EXmv!HTTs\"Fe]l7$$!3gP3F>*)QtWFb]l$\"3PIt$3nTaB#Fe]l7$$!3#f+ D19*yYJFb]l$\"3#p3?VTUy<$Fe]l7$$!35!4-8D*[$[#Fb]l$\"3/oi*[#GfESFe]l7$$ !3Du\"z>O*=?=Fb]l$\"3/oh\"z(Q$R\\&Fe]l7$$!3Ueils%*)o:\"Fb]l$\"3<,Z:$4o Qk)Fe]l7$Ffr$\"3Kwzv$ouf-#!#:7$$\"3=$4aj%e#R&>Fhr$!3@hEz*[,z6&Fg_l7$$ \"337;/EvuV))Fhr$!3pE[ylBuI6Fg_l7$$\"358Hd?pNt:Fb]l$!3/#o%QF*>Fe]l7$$\"37!o\"HKk>uxFb]l$!3!zJvnd 1jG\"Fe]l7$$\"3womT5D,`5F[r$!39Of;PLc'\\*FW7$$\"3Gq;zW#)>/;F[r$!3il')3 CNkLiFW7$Fjr$!3D#*G#HbW&RYFW7$$\"3mOLL$e,]6$F[r$!3'*f^VT7F5KFW7$F]s$!3 gae!*yq@aCFW7$F`s$!3GpqEb=pE;FW7$Fcs$!3H)oT%GeV17FW7$Ffs$!3w&G)z')pLI' *F[r7$Fis$!3;v@gmsYd!)F[r7$F\\t$!3e&=HPl$f?oF[r7$F_t$!3s+&p[tZQ*fF[r7$ Fbt$!3=%o(y/&4qI&F[r7$Fet$!3&)oB$[x(y2[F[r7$Fht$!3]oaS+YXfVF[r7$F[u$!3 \"oY?+w+y+%F[r7$F^u$!3[S!fne\\hp$F[r7$Fau$!3atcs76+NMF[r7$Fdu$!3KL'>8= >%)>$F[r7$Fgu$!3#***HMHTX**HF[r7$Fju$!3@,%R)3P0?GF[r7$F]v$!3QvGetk9iEF [r7$F`v$!3xjY!\\&Q(=`#F[r7$Fcv$!3-\\J(**H9uR#F[r7$Ffv$!3c(y6y`-()G#F[r 7$Fiv$!3'\\)p%**>_J=#F[r7$F\\w$!3?Xz^&)*f34#F[r7$F_w$!35+++++++?F[r-FE 6&FG$\"\"'FJFHF,-%+AXESLABELSG6%Q\"w6\"Q!Fjgl-%%FONTG6#%(DEFAULTG-%&TI TLEG6#%CThe~Fourier~transform~F(w)~of~H(x)G-%%VIEWG6$;FRF_wFghl" 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 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 }