{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 "" 260 265 "" 1 18 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 "" -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 }{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 O utput" -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 "Introduction to Fourier transform s" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canad a" }}{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 59 "Explanation of the Fourier transform via a limiting \+ process" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 36 " be a periodic function with period " }{XPPEDIT 18 0 "T=2*L" "6 #/%\"TG*&\"\"#\"\"\"%\"LGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 64 " can be expanded in a complex Fourier series which converges to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 22 " for all real numbers " }{TEXT 268 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)= S um(c[k]*exp(i*k*omega[0]*x),k=-infinity..infinity)" "6#/-%\"fG6#%\"xG- %$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#**%\"iGF0F/F0&%&omegaG6#\"\"!F0F 'F0F0/F/;,$%)infinityG!\"\"F=" }{TEXT -1 16 " ------- (i), " }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "omega[0]" "6#&%&om egaG6#\"\"!" }{TEXT -1 38 " is the fundamental angular frequency " } {XPPEDIT 18 0 "Pi/L=2*Pi/T" "6#/*&%#PiG\"\"\"%\"LG!\"\"*(\"\"#F&F%F&% \"TGF(" }{TEXT -1 6 " , and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[k]=1/(2*L)" "6#/&%\"cG6#%\"kG*&\"\"\"F)*&\"\"#F)%\"LG F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*exp(-i*k*omega[0]*x) ,x=-L..L)" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$**%\"iGF+%\"kGF +&%&omegaG6#\"\"!F+F*F+!\"\"F+/F*;,$%\"LGF7F;" }{TEXT -1 15 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 38 "Then, substituting (ii) into (i ) gives" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = Sum (1/(2*L),k = -infinity .. infinity);" "6#/-%\"fG6#%\"xG-%$SumG6$*&\"\" \"F,*&\"\"#F,%\"LGF,!\"\"/%\"kG;,$%)infinityGF0F5" }{XPPEDIT 18 0 " `` (Int(f(t)*exp(-i*k*omega[0]*t),t = -L .. L))*exp(i*k*omega[0]*x)" "6#* &-%!G6#-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$**%\"iGF/%\"kGF/&%&ome gaG6#\"\"!F/F.F/!\"\"F//F.;,$%\"LGF;F?F/-F16#**F5F/F6F/&F86#F:F/%\"xGF /F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 56 "Here the dummy integration variable in the integral for " }{XPPEDIT 18 0 "c[k]" "6#&%\"cG6#%\"kG" }{TEXT -1 15 " is changed t o " }{TEXT 266 1 "t" }{TEXT -1 38 " to avoid confusion with the variab le " }{TEXT 267 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "T his formula can be rewritten in the form" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=1/(2*Pi)" "6#/-%\"fG6#%\"xG*&\"\"\"F)*& \"\"#F)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``(Int(f(t)*e xp(-i*k*omega[0]*t),t = -L .. L))*exp(i*k*omega[0]*x)*omega[0],k = -in finity .. infinity);" "6#-%$SumG6$*(-%!G6#-%$IntG6$*&-%\"fG6#%\"tG\"\" \"-%$expG6#,$**%\"iGF2%\"kGF2&%&omegaG6#\"\"!F2F1F2!\"\"F2/F1;,$%\"LGF >FBF2-F46#**F8F2F9F2&F;6#F=F2%\"xGF2F2&F;6#F=F2/F9;,$%)infinityGF>FN" }{TEXT -1 16 " ------- (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "Now consider what happens " }{TEXT 261 33 "as the period T tends to infinity" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "The periodic function " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 72 " approaches a non-periodic function, which we \+ may continue to denote by " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "The fundamental frequenc y " }{XPPEDIT 18 0 "omega[0]=2*Pi/T" "6#/&%&omegaG6#\"\"!*(\"\"#\"\"\" %#PiGF*%\"TG!\"\"" }{TEXT -1 47 " tends to 0, and we can consider the \+ frequency " }{XPPEDIT 18 0 "k*omega[0]" "6#*&%\"kG\"\"\"&%&omegaG6#\" \"!F%" }{TEXT -1 62 " of an arbitary \"harmonic\" to approach a conti nuous variable " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 46 "This suggests that in the limit as the p eriod " }{TEXT 269 1 "T" }{TEXT -1 108 " tends to infinity, the sum in the formula (iii) becomes an integral with respect to the frequency v ariable " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 8 " to give" } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x) = 1/(2*Pi);" "6#/-%\"fG6#%\"xG*&\"\"\"F)*&\"\"#F)% #PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(Int(f(t)*exp(-i*om ega*t),t = -infinity .. infinity))*exp(i*omega*x)*omega[0],omega = -in finity .. infinity);" "6#-%$IntG6$*(-%!G6#-F$6$*&-%\"fG6#%\"tG\"\"\"-% $expG6#,$*(%\"iGF1%&omegaGF1F0F1!\"\"F1/F0;,$%)infinityGF9F=F1-F36#*(F 7F1F8F1%\"xGF1F1&F86#\"\"!F1/F8;,$F=F9F=" }{TEXT -1 15 " ------- (iv) ." }}{PARA 0 "" 0 "" {TEXT -1 10 "Now define" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(omega) = Int(f(x)*exp(-i*omega*x),x = -infinity .. infinity);" "6#/-%\"FG6#%&omegaG-%$IntG6$*&-%\"fG6#%\"xG \"\"\"-%$expG6#,$*(%\"iGF0F'F0F/F0!\"\"F0/F/;,$%)infinityGF7F;" } {TEXT -1 14 " ------- (v) " }}{PARA 0 "" 0 "" {TEXT -1 20 "so that (i v) becomes" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=1/ (2*Pi)" "6#/-%\"fG6#%\"xG*&\"\"\"F)*&\"\"#F)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(F(omega)*exp(i*omega*x),omega=-infinity..infin ity)" "6#-%$IntG6$*&-%\"FG6#%&omegaG\"\"\"-%$expG6#*(%\"iGF+F*F+%\"xGF +F+/F*;,$%)infinityG!\"\"F5" }{TEXT -1 15 " ------- (vi)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The formula (v) c an be used to define the function " }{XPPEDIT 18 0 "F(omega)" "6#-%\"F G6#%&omegaG" }{TEXT -1 30 " associated with the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 57 " and the formula (vi) serv es to reconstruct the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 19 " from the function " }{XPPEDIT 18 0 "F(omega)" "6#-% \"FG6#%&omegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The fun ction " }{XPPEDIT 18 0 "F(omega)" "6#-%\"FG6#%&omegaG" }{TEXT -1 15 " \+ is called the " }{TEXT 261 18 "Fourier transform " }{TEXT -1 3 "of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " is then the " }{TEXT 261 25 "inverse Fourier transform" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "F(om ega)" "6#-%\"FG6#%&omegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "We use the symbol " }{TEXT 263 2 "Fr" }{TEXT -1 45 " for the Fouri er transform operator, so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Fr*[f(x)]=F(omega)" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&- %\"FG6#%&omegaG" }{XPPEDIT 18 0 "`` = Int(f(x)*exp(-i*omega*x),x = -in finity .. infinity)" "6#/%!G-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$* (%\"iGF-%&omegaGF-F,F-!\"\"F-/F,;,$%)infinityGF5F9" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 262 23 "_____________________ __" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "a nd " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr^(-1)*[F(ome ga)] = f(x);" "6#/*&)%#FrG,$\"\"\"!\"\"F(7#-%\"FG6#%&omegaGF(-%\"fG6#% \"xG" }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF &!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(F(omega)*exp(i*omega*x),om ega=-infinity..infinity)" "6#-%$IntG6$*&-%\"FG6#%&omegaG\"\"\"-%$expG6 #*(%\"iGF+F*F+%\"xGF+F+/F*;,$%)infinityG!\"\"F5" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 264 27 "_____________________ ______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "Fr;" "6#%#FrG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "Fr^(-1);" "6#)%#FrG,$\"\"\"!\"\"" }{TEXT -1 5 " are " } {TEXT 261 16 "linear operators" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{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 46 "We find the Fourier transform of 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([1, abs(x) < a],[0, \+ a <= abs(x)]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"\"2-%$absG6#F'% \"aG7$\"\"!1F1-F/6#F'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "w here " }{TEXT 270 1 "a" }{TEXT -1 14 " is positive. " }}{PARA 0 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " is a \"pulse\" of width " }{TEXT 271 1 "a" }{TEXT -1 25 ", centred a t the origin. " }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " is drawn for the case " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "f := x -> if abs(x)<1 then 1 else 0 end if:\nplot('f(x)',x=-3..3, 0..1.2,color=red,thickness=2,ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 548 142 142 {PLOTDATA 2 "6(-%'CURVESG6#7co7$$!\"$\"\"!$F*F*7 $$!3!******\\2<#pG!#iUC\"F/F+7$$!3'***\\7GVS(=\"F/F+7$$!3-++DhkaI6F/F+7 $$!31+vo4#z2))FbpFcp7$$!3S++]7RK vuFbpFcp7$$!3s,+++P'eH'FbpFcp7$$!3q)***\\7*3=+&FbpFcp7$$!3[)***\\PFcpP FbpFcp7$$!3;)****\\7VQ[#FbpFcp7$$!32)***\\i6:.8FbpFcp7$$!3Wb+++v`hH!#? 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39/]oq\"fS&QF-7$$\"3nLL$3A_1?&F1$!37'4/x^+jR$F-7$$\"3X,+]7k.6aF1$!3cu, s!=*=IGF-7$$\"3#emmmT9C#eF1$!3:&HIQaMt_\"F-7$$\"33****\\i!*3`iF1$!3xhK >2veC'*F_v7$$\"3;NLLL*zym'F1$\"3Doa]XMiD6F-7$$\"3_NL$3sr*zoF1$\"3#*H+O a1pL;F-7$$\"3'eLL$3N1#4(F1$\"35_&ol>Y./#F-7$$\"3W-+vo!*R-tF1$\"3ZCG1Ly kKBF-7$$\"3,pm;HYt7vF1$\"3k3DF-7$$\"3H-](o*GP4wF1$\"3yb+pOv4]DF -7$$\"3dNLek6,1xF1$\"3Q'Rwq/:qc#F-7$$\"3&)o;HK%\\E!yF1$\"35_#))*)fb)fD F-7$$\"37-+++xG**yF1$\"3k%Gmqyv#HDF-7$$\"3(eLL3U/37)F1$\"3!)[#3Y&*fcP# F-7$$\"3gpmmT6KU$)F1$\"3%[*pW/!)=<@F-7$$\"3qNLLLbdQ()F1$\"3O&['**z/8]9 F-7$$\"3[++]i`1h\"*F1$\"3w))RR(Q[2p&F47$$\"3A-+]P?Wl&*F1$!3z#eooc*RJHF 47$$\"#5F*F+-%&TITLEG6#%?Fourier~transform~F(w)~of~f(x)G-%+AXESLABELSG 6$Q\"w6\"Q!F^]m-%&COLORG6&%$RGBG$\"\"%!\"\"F*$\"\"*Ff]m-%%VIEWG6$;F(Fd \\m%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Note that the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 63 " in this example can be described using the Heaviside \+ function " }{XPPEDIT 18 0 "H(x) = PIECEWISE([0, x < a],[1, a < x]);" " 6#/-%\"HG6#%\"xG-%*PIECEWISEG6$7$\"\"!2F'%\"aG7$\"\"\"2F.F'" }{TEXT -1 7 " as " }{XPPEDIT 18 0 "f(x) = H(x+a)-H(x-a);" "6#/-%\"fG6#%\"x G,&-%\"HG6#,&F'\"\"\"%\"aGF-F--F*6#,&F'F-F.!\"\"F2" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "alias(H=Heaviside):\nplot(H(x+1)-H(x-1),x=-3..3,y=0..1.2,color=red ,thickness=2,ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 539 161 161 {PLOTDATA 2 "6(-%'CURVESG6#7co7$$!\"$\"\"!$F*F*7$$!3!******\\2<#pG!#iUC \"F/F+7$$!3'***\\7GVS(=\"F/F+7$$!3-++DhkaI6F/F+7$$!31+vo4#z2))FbpFcp7$$!3S++]7RKvuFbpFcp7$$!3s,+++P'eH 'FbpFcp7$$!3q)***\\7*3=+&FbpFcp7$$!3[)***\\PFcpPFbpFcp7$$!3;)****\\7VQ [#FbpFcp7$$!32)***\\i6:.8FbpFcp7$$!3Wb+++v`hH!#?Fcp7$$\"3]****\\(QIKH \"FbpFcp7$$\"38****\\7:xWCFbpFcp7$$\"3E,++vuY)o$FbpFcp7$$\"3!z******4F L(\\FbpFcp7$$\"3A)****\\d6.B'FbpFcp7$$\"3s****\\(o3lW(FbpFcp7$$\"3U*** \\iX)p@\")FbpFcp7$$\"35*****\\A))oz)FbpFcp7$$\"3G)**\\PCK-5*FbpFcp7$$ \"3X(***\\iid.%*FbpFcp7$$\"3['*\\(=F[_b*FbpFcp7$$\"3i'**\\7G?pq*FbpFcp 7$$\"3C(\\PfGcFy*FbpFcp7$$\"3w'*\\i!H#fe)*FbpFcp7$$\"3_Y(oHH5l*)*FbpFc p7$$\"3F'\\7`HGW$**FbpFcp7$$\"3-Yil(HYB(**FbpFcp7$$\"3e******Hk-,5F/F+ 7$$\"3%)****\\FL!e1\"F/F+7$$\"36+++D-eI6F/F+7$$\"3u***\\(=_(zC\"F/F+7$ $\"3M+++b*=jP\"F/F+7$$\"3g***\\(3/3(\\\"F/F+7$$\"33++vB4JB;F/F+7$$\"3u *****\\KCnu\"F/F+7$$\"3s***\\(=n#f(=F/F+7$$\"3P+++!)RO+?F/F+7$$\"30++] _!>w7#F/F+7$$\"3O++v)Q?QD#F/F+7$$\"3G+++5jypBF/F+7$$\"3<++]Ujp-DF/F+7$ $\"3++++gEd@EF/F+7$$\"39++v3'>$[FF/F+7$$\"37++D6EjpGF/F+7$$\"\"$F*F+-% *AXESTICKSG6$%(DEFAULTGF\\y-%+AXESLABELSG6$Q\"x6\"Q\"yFey-%'COLOURG6&% $RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(F[y;F+$\"#7!\" \"" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Fouri er transforms are available in Maple through the integral transform pa ckage " }{TEXT 0 8 "inttrans" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "H(x+a)-H(x-a );\n`Fourier transform`=inttrans[fourier](%,x,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%\"HG6#,&%\"xG\"\"\"%#a|irGF)F)-F%6#,&F(F)F*!\" \"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,$*(\"\"# \"\"\"%&omegaG!\"\"-%$sinG6#*&%#a|irGF(F)F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Fourier transform of the function " }{XPPEDIT 18 0 "f( x) = exp(-a*x^2);" "6#/-%\"fG6#%\"xG-%$expG6#,$*&%\"aG\"\"\"*$F'\"\"#F .!\"\"" }{TEXT -1 8 ", where " }{TEXT 272 1 "a" }{TEXT -1 13 " is posi tive." }}{PARA 0 "" 0 "" {TEXT -1 32 "The graph is drawn for the case \+ " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "f := \+ x -> exp(-x^2);\nplot(f(x),x=-3..3,y=0..1.2,color=red,thickness=2,ytic kmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)op eratorG%&arrowGF(-%$expG6#,$*$)9$\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 13 " " 1 "" {GLPLOT2D 523 181 181 {PLOTDATA 2 "6(-%'CURVESG6#7co7$$!\"$\"\" !$\"3i&zm3/)4M7!#@7$$!3!******\\2<#pG!#<$\"3!47G\"*[c*eEF-7$$!3#)***\\ 7bBav#F1$\"3KXFgj0!H/&F-7$$!36++]K3XFEF1$\"3]uqQxkE/5!#?7$$!3%)****\\F )H')\\#F1$\"3yZ*)))o6sV>F>7$$!3#****\\i3@/P#F1$\"3qohtkWkGOF>7$$!3;++D r^b^AF1$\"3MliS/yb&G'F>7$$!3$****\\7Sw%G@F1$\"37waZgPix5!#>7$$!3***** \\7;)=,?F1$\"3IHW7&>xG#=FS7$$!3/++DO\"3V(=F1$\"31`<6RMk!)HFS7$$!3#**** **\\V'zViUC\"F1$\"3f/*ykGGj7#Fgo7$$!3-++DhkaI6F1$\"3ujZ***Riby# Fgo7$$!3s******\\XF`**Fgo$\"3Z>\"eG9`Kr$Fgo7$$!3s*****\\PL0Q*Fgo$\"3u_ bHO61[TFgo7$$!3u*******>#z2))Fgo$\"3H$GA5R#[.YFgo7$$!31++Dc!e:9)Fgo$\" 3MHfFBd!Q:&Fgo7$$!3S++]7RKvuFgo$\"3:wU/?D#*=dFgo7$$!31,+D1Qf&)oFgo$\"3 mtaMzFOCiFgo7$$!3s,+++P'eH'Fgo$\"3uY]_ln^FnFgo7$$!3A++D1j$)[cFgo$\"3C] 4$)yd2osFgo7$$!3q)***\\7*3=+&Fgo$\"3=YF'3#*)f'y(Fgo7$$!3g)****\\#eo&Q% Fgo$\"3eu;40LC]#)Fgo7$$!3[)***\\PFcpPFgo$\"3;Cy/3%z`n)Fgo7$$!3;)****\\ 7VQ[#Fgo$\"3F(ov89)p,%*Fgo7$$!35)**\\P9(\\$*=Fgo$\"3MJclF!=yk*Fgo7$$!3 2)***\\i6:.8Fgo$\"3AOQaBNhJ)*Fgo7$$!3%p)*\\i:sw%)*FS$\"3MO*f-1#\\.**Fg o7$$!39$***\\(oKQm'FS$\"39!=Gdy\"pb**Fgo7$$!3C'*\\7`H\">2&FS$\"3E\\fid (3V(**Fgo7$$!3O***\\(=K**zMFS$\"3/9/Fxp*y)**Fgo7$$!3W-]P%[t!))=FS$\"3o 9(RQ\"eV'***Fgo7$$!3Wb+++v`hHF>$\"3k&)4M$H7*****Fgo7$$\"3c%*\\7`MSd8FS $\"3KkMgDw:)***Fgo7$$\"3l%**\\ilg4,$FS$\"3c=w>C#Q4***Fgo7$$\"3w%*\\Pfy ^kYFS$\"3C7g\"f#fEy**Fgo7$$\"3%[***\\i]2=jFS$\"3IELA%\\h,'**Fgo7$$\"3/ &**\\(o%*=D'*FS$\"3K:Pa`Ny2**Fgo7$$\"3]****\\(QIKH\"Fgo$\"30YtD)GYT$)* Fgo7$$\"3K******\\4+p=Fgo$\"3Q'H#y>Urc'*Fgo7$$\"38****\\7:xWCFgo$\"3kk HKL.#)>%*Fgo7$$\"3E,++vuY)o$Fgo$\"3>f;IT!3!G()Fgo7$$\"3e****\\(G(*3L%F go$\"3o!QB)G\"Q(*G)Fgo7$$\"3!z******4FL(\\Fgo$\"3u@,MzGv3yFgo7$$\"30)* **\\P$>=g&Fgo$\"3.&Rz&3Pi1tFgo7$$\"3A)****\\d6.B'Fgo$\"3i(3hZ*\\)Hy'Fg o7$$\"3(*)**\\785%QoFgo$\"3qjGlc$*zkiFgo7$$\"3s****\\(o3lW(Fgo$\"3Nzpk `acVdFgo7$$\"3U***\\iX)p@\")Fgo$\"3A4Hn'[y/<&Fgo7$$\"35*****\\A))oz)Fg o$\"35HqvoxK7YFgo7$$\"3X(***\\iid.%*Fgo$\"3yU1Oka9ITFgo7$$\"3e******Hk -,5F1$\"31bCHSiCrOFgo7$$\"36+++D-eI6F1$\"3=gG\"ou\\`y#Fgo7$$\"3u***\\( =_(zC\"F1$\"32#f!okDu1@Fgo7$$\"3M+++b*=jP\"F1$\"3C*>f`M0V]\"Fgo7$$\"3g ***\\(3/3(\\\"F1$\"3')HDQ4aDj5Fgo7$$\"33++vB4JB;F1$\"3#GL10u/4<(FS7$$ \"3u*****\\KCnu\"F1$\"3*oLc@\">%4t%FS7$$\"3s***\\(=n#f(=F1$\"3qKt9Ubgi HFS7$$\"3P+++!)RO+?F1$\"3Z!>Dj(*)*)G=FS7$$\"30++]_!>w7#F1$\"3?29SZJc\" 3\"FS7$$\"3O++v)Q?QD#F1$\"3f&p7$$\"3G+++5jypBF1$\"3B`c;%Rz&RO F>7$$\"3<++]Ujp-DF1$\"3)pqtA6*e/>F>7$$\"3++++gEd@EF1$\"3_W6VudtN5F>7$$ \"39++v3'>$[FF1$\"3w!H!*R[uRC&F-7$$\"37++D6EjpGF1$\"3mSX%)HKi_EF-7$$\" \"$F*F+-%*AXESTICKSG6$%(DEFAULTGFgal-%+AXESLABELSG6$Q\"x6\"Q\"yF`bl-%' COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fibl-%*THICKNESSG6#\"\"#-%%VIEWG6$;F (Ffal;Fibl$\"#7!\"\"" 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 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(omega) = Int( f(x)*exp(-i*omega*x),x = -infinity .. infinity);" "6#/-%\"FG6#%&omegaG -%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*(%\"iGF0F'F0F/F0!\"\"F0/F/;, $%)infinityGF7F;" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int(exp(-a*x^2)*exp(-i*omega*x),x = -infinity .. i nfinity);" "6#/%!G-%$IntG6$*&-%$expG6#,$*&%\"aG\"\"\"*$%\"xG\"\"#F/!\" \"F/-F*6#,$*(%\"iGF/%&omegaGF/F1F/F3F//F1;,$%)infinityGF3F=" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(exp (-a*x^2-i*omega*x),x = -infinity .. infinity)" "6#/%!G-%$IntG6$-%$expG 6#,&*&%\"aG\"\"\"*$%\"xG\"\"#F.!\"\"*(%\"iGF.%&omegaGF.F0F.F2/F0;,$%)i nfinityGF2F9" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "u = sqrt(a)*x+i*omega/(2*sqrt(a));" "6#/%\"uG,&*&-%%sqr tG6#%\"aG\"\"\"%\"xGF+F+*(%\"iGF+%&omegaGF+*&\"\"#F+-F(6#F*F+!\"\"F+" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "u^2 = a*x^2+i*omega*x-omega ^2/(4*a);" "6#/*$%\"uG\"\"#,(*&%\"aG\"\"\"*$%\"xGF&F*F**(%\"iGF*%&omeg aGF*F,F*F**&F/F&*&\"\"%F*F)F*!\"\"F3" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "du/dx=sqrt(a)" "6#/*&%#duG\"\"\"%#dxG!\"\"-%%sqrtG6#%\"aG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "Then, since " }{XPPEDIT 18 0 "-a*x^2-i*omega*x = (-omega^2/(4*a))-u^2;" "6#/,&*&%\"aG\"\"\"*$% \"xG\"\"#F'!\"\"*(%\"iGF'%&omegaGF'F)F'F+,&*&F.F**&\"\"%F'F&F'F+F+*$% \"uGF*F+" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "F(omega)=1/sqrt(a)" "6#/-%\"FG6#%&omegaG*&\"\"\"F)-%%sq rtG6#%\"aG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-omega^2/(4*a))*I nt(exp(-u^2),u = -infinity .. infinity);" "6#*&-%$expG6#,$*&%&omegaG\" \"#*&\"\"%\"\"\"%\"aGF-!\"\"F/F--%$IntG6$-F%6#,$*$%\"uGF*F//F7;,$%)inf inityGF/F;F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "Now the result " }{XPPEDIT 18 0 "Int(exp(-u^2),u = -infinity .. infinity) = \+ sqrt(Pi);" "6#/-%$IntG6$-%$expG6#,$*$%\"uG\"\"#!\"\"/F,;,$%)infinityGF .F2-%%sqrtG6#%#PiG" }{TEXT -1 119 ", is well-known, and is equivalent \+ to the fact that the total area under the normal probabilty distributi on curve is 1." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "F(omega) = sqrt(Pi/a)*exp(-omega^2/(4 *a));" "6#/-%\"FG6#%&omegaG*&-%%sqrtG6#*&%#PiG\"\"\"%\"aG!\"\"F.-%$exp G6#,$*&F'\"\"#*&\"\"%F.F/F.F0F0F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "interface(showassumed=0): assume(a>0):\nInt(exp(- a*x^2)*exp(-I*omega*x),x=-infinity..infinity);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#,$*&%#a|irG\"\"\")%\"xG \"\"#F-!\"\"F--F(6#*(^#F1F-%&omegaGF-F/F-F-/F/;,$%)infinityGF1F:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(-%$expG6#,$*(\"\"%!\"\"%#a|irGF, %&omegaG\"\"#F,\"\"\"F-#F,F/%#PiG#F0F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "In particular, when " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "F(omega) = sqrt(Pi)*exp(-omega^2/4);" "6#/-%\"FG6#%&omegaG*&-%%sqrtG6#%#PiG\"\" \"-%$expG6#,$*&F'\"\"#\"\"%!\"\"F5F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "plot(sqrt (Pi)*exp(-w^2/4),w=-5..5,color=COLOR(RGB,.4,0,.9),\n \+ title=`Fourier transform F(w) of f(x)`);" }}{PARA 13 "" 1 "" {GLPLOT2D 555 192 192 {PLOTDATA 2 "6'-%'CURVESG6#7en7$$!\"&\"\"!$\"38& G`x'3k@M!#?7$$!3YLLLe%G?y%!#<$\"39Vv#R#*)*3$eF-7$$!3OmmT&esBf%F1$\"3W! et123V4*F-7$$!3ALL$3s%3zVF1$\"3G)R4>gAtY\"!#>7$$!3_LL$e/$QkTF1$\"3ypX4 \"*e.@BF>7$$!3ommT5=q]RF1$\"3!=:M6*od!e$F>7$$!3ILL3_>f_PF1$\"3cSXRz5\" QC&F>7$$!3K++vo1YZNF1$\"3oyO[WU([i(F>7$$!3;LL3-OJNLF1$\"3!*RL'R&yT)4\" !#=7$$!3p***\\P*o%Q7$F1$\"3eRWZWk\\X:FX7$$!3Kmmm\"RFj!HF1$\"3WBz(\\wp_ 9#FX7$$!33LL$e4OZr#F1$\"3\\QFyFH33GFX7$$!3u*****\\n\\!*\\#F1$\"3OP()os -o>PFX7$$!3%)*****\\ixCG#F1$\"3n*R)[Rbz=[FX7$$!3#******\\KqP2#F1$\"3]v 30hPb[gFX7$$!39LL3-TC%)=F1$\"3))H\"*y*f\")p\"F17$$!3Ihm\"HdGe:$FX$\"3\"4kS;\"z')G<#FX$\"3wz5@QRn^$\"3A/sl\\Q'4x\"F17$$!3vDML Le*e$\\F-$\"3Q6^LbIWs$\"3G,m'*3$Q8x\"F17$$\"3womT5 D,`5FX$\"3!HJ-hDZvw\"F17$$\"3Gq;zW#)>/;FX$\"3%[K/C9(3hv\"F17$$\"3mOLL$e,]6$FX$\"3cIO$R#[(*HY2%FX$\"3=kK\"[q\"R+:E\\()4zt$FX7$$\"3s++D1#=bq#F1$\"3#[7G(y\"yL%GF X7$$\"3\"HLL$3s?6HF1$\"3I,zA;%)4I@FX7$$\"3a***\\7`Wl7$F1$\"3HF/2!R&**Q :FX7$$\"3enmmm*RRL$F1$\"3FMo@Ak$45\"FX7$$\"3%zmmTvJga$F1$\"3i#Qon;?Uk( F>7$$\"3]MLe9tOcPF1$\"3dk(o%3#yn?&F>7$$\"31,++]Qk\\RF1$\"3QIhMbl1)e$F> 7$$\"3![LL3dg6<%F1$\"3%\\FRP%\\[)G#F>7$$\"3%ymmmw(GpVF1$\"3uPTma>5*\\ \"F>7$$\"3C++D\"oK0e%F1$\"3e6%\\![chW$*F-7$$\"35,+v=5s#y%F1$\"3\"H%G<. 2D@eF-7$$\"\"&F*F+-%&TITLEG6#%?Fourier~transform~F(w)~of~f(x)G-%+AXESL ABELSG6$Q\"w6\"Q!Fi]l-%&COLORG6&%$RGBG$\"\"%!\"\"F*$\"\"*Fa^l-%%VIEWG6 $;F(F_]l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "interface(showassumed=0): assume(a >0):\nexp(-a*x^2);\n`Fourier transform`=inttrans[fourier](%,x,omega); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$*&%#a|irG\"\"\")%\"xG\" \"#F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG*&- %$expG6#,$*(\"\"%!\"\"%&omegaG\"\"#%#a|irGF,F,\"\"\"*&%#PiGF0F/F,#F0F. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Fourier transform of the fu nction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 10 " given \+ by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWI SE([1, abs(x) <= 1],[2, 1 < abs(x) and abs(x) <= 2],[0, 2 < abs(x)]); " "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$\"\"\"1-%$absG6#F'F,7$\"\"#32F,-F/ 6#F'1-F/6#F'F27$\"\"!2F2-F/6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "f := x -> i f abs(x)<=1 then 1 elif abs(x)<=2 then 2 else 0 end if:\nplot('f(x)',x =-4..4,y=0..2.2,color=red,thickness=2,ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 543 211 211 {PLOTDATA 2 "6(-%'CURVESG6#7ip7$$!\"%\"\"!$F* F*7$$!3ommmmFiDQ!#1DBF/F+7$$!3%******\\RD% [AF/F+7$$!3kmmmw))yr@F/F+7$$!3!)****\\#f^'G@F/F+7$$!3SLLL3V^&3#F/F+7$$ !3U++Dmc%R1#F/F+7$$!3+nm;CqPU?F/F+7$$!3H+]7.FfJ?F/F+7$$!3eLL3#Q33-#F/F +7$$!3++Dc@iT:?F/F+7$$!3(omT51C+,#F/F+7$$!3tL3_+>j/?F/F+7$$!3;+++S(R#* *>F/$\"\"#F*7$$!35+++?4h7>F/Fio7$$!30++++@)f#=F/Fio7$$!3-+++gi,f;F/Fio 7$$!3qmmm\"G&R2:F/Fio7$$!3XLLLtK5F8F/Fio7$$!3_LLL$yP2D\"F/Fio7$$!3eLLL $HsV<\"F/Fio7$$!39++v$oc*H6F/Fio7$$!3!pmmT2Tb3\"F/Fio7$$!3;+]PpKLj5F/F io7$$!3mLLeka7T5F/Fio7$$!3=+v=i:-I5F/Fio7$$!3$pm\"zfw\"*=5F/Fio7$$!3>] Pf3dO85F/Fio7$$!3oLeRdP\"y+\"F/Fio7$$!3%p\"z>1=E-5F/Fio7$$!3+-++]&)4n* *!#=$\"\"\"F*7$$!3cpmmT(******z-6j'F^sF_s7$$\"3q\"******4# 32$)F^sF_s7$$\"3q#****\\qM8F/Fio7$$\"3% )*******HSu]\"F/Fio7$$\"3'HLL$ep'Rm\"F/Fio7$$\"3')******R>4N=F/Fio7$$ \"3&GLLed*f:>F/Fio7$$\"3#emm;@2h*>F/Fio7$$\"3[K$ek\"oO,?F/F+7$$\"38*** \\7UEm+#F/F+7$$\"3yl;/Eg)=,#F/F+7$$\"3VKL$3jXr,#F/F+7$$\"3tlmTS[mF?F/F +7$$\"3.******\\S=Q?F/F+7$$\"3ilm;pCAf?F/F+7$$\"3mKLL))3E!3#F/F+7$$\"3 JmmmExLA@F/F+7$$\"3]*****\\c9W;#F/F+7$$\"3#HLLe;!pYAF/F+7$$\"3Lmmmmd'* GBF/F+7$$\"3j*****\\iN7]#F/F+7$$\"3aLLLt>:nEF/F+7$$\"35LLL.a#o$GF/F+7$ $\"3ammm^Q40IF/F+7$$\"3y******z]rfJF/F+7$$\"3gmmmc%GpL$F/F+7$$\"3/LLL8 -V&\\$F/F+7$$\"3=+++XhUkOF/F+7$$\"3=+++:o " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(omega) = Int(f(x)*exp(-i*omega*x),x \+ = -infinity .. infinity);" "6#/-%\"FG6#%&omegaG-%$IntG6$*&-%\"fG6#%\"x G\"\"\"-%$expG6#,$*(%\"iGF0F'F0F/F0!\"\"F0/F/;,$%)infinityGF7F;" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(2*exp(-i*omega*x),x = -2 .. -1)+Int(exp(-i*omega*x),x = -1 .. 1)+ Int(2*exp(-i*omega*x),x = 1 .. 2);" "6#/%!G,(-%$IntG6$*&\"\"#\"\"\"-%$ expG6#,$*(%\"iGF+%&omegaGF+%\"xGF+!\"\"F+/F3;,$F*F4,$F+F4F+-F'6$-F-6#, $*(F1F+F2F+F3F+F4/F3;,$F+F4F+F+-F'6$*&F*F+-F-6#,$*(F1F+F2F+F3F+F4F+/F3 ;F+F*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*exp(-i*omega*x)/(-i*omega );" "6#/%!G*(\"\"#\"\"\"-%$expG6#,$*(%\"iGF'%&omegaGF'%\"xGF'!\"\"F',$ *&F-F'F.F'F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([-1, ``],[`` , ``],[-2, ``]);" "6#-%*PIECEWISEG6%7$,$\"\"\"!\"\"%!G7$F*F*7$,$\"\"#F )F*" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "exp(-i*omega*x)/(-i*omega);" "6 #*&-%$expG6#,$*(%\"iG\"\"\"%&omegaGF*%\"xGF*!\"\"F*,$*&F)F*F+F*F-F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[-1, ``]);" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7$,$F'!\"\"F(" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "2*exp(-i*omega*x)/(-i*omega);" "6#*(\"\"#\"\"\"-%$expG6 #,$*(%\"iGF%%&omegaGF%%\"xGF%!\"\"F%,$*&F+F%F,F%F.F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2, ``],[``, ``],[1, ``]);" "6#-%*PIECEWISEG 6%7$\"\"#%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "2*``(exp( i*omega)-exp(2*i*omega))/(-i*omega)+(exp(-i*omega)-exp(i*omega))/(-i*o mega)+2*``(exp(-2*i*omega)-exp(-i*omega))/(-i*omega);" "6#,(*(\"\"#\" \"\"-%!G6#,&-%$expG6#*&%\"iGF&%&omegaGF&F&-F,6#*(F%F&F/F&F0F&!\"\"F&,$ *&F/F&F0F&F4F4F&*&,&-F,6#,$*&F/F&F0F&F4F&-F,6#*&F/F&F0F&F4F&,$*&F/F&F0 F&F4F4F&*(F%F&-F(6#,&-F,6#,$*(F%F&F/F&F0F&F4F&-F,6#,$*&F/F&F0F&F4F4F&, $*&F/F&F0F&F4F4F&" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*``(exp(2*i*om ega)-exp(-2*i*omega))/(i*omega)-(exp(i*omega)-exp(-i*omega))/(i*omega) ;" "6#/%!G,&*(\"\"#\"\"\"-F$6#,&-%$expG6#*(F'F(%\"iGF(%&omegaGF(F(-F-6 #,$*(F'F(F0F(F1F(!\"\"F6F(*&F0F(F1F(F6F(*&,&-F-6#*&F0F(F1F(F(-F-6#,$*& F0F(F1F(F6F6F(*&F0F(F1F(F6F6" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4*``((exp(2*i*om ega)-exp(-2*i*omega))/(2*i))/omega-2*``((exp(i*omega)-exp(-i*omega))/( 2*i))/omega;" "6#/%!G,&*(\"\"%\"\"\"-F$6#*&,&-%$expG6#*(\"\"#F(%\"iGF( %&omegaGF(F(-F.6#,$*(F1F(F2F(F3F(!\"\"F8F(*&F1F(F2F(F8F(F3F8F(*(F1F(-F $6#*&,&-F.6#*&F2F(F3F(F(-F.6#,$*&F2F(F3F(F8F8F(*&F1F(F2F(F8F(F3F8F8" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (4*sin*2*omega-2*sin*omega)/omega;" "6#/%!G*&,&** \"\"%\"\"\"%$sinGF)\"\"#F)%&omegaGF)F)*(F+F)F*F)F,F)!\"\"F)F,F." } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "plot((4*sin(2*w)-2*sin(w))/w,w=-10..10,color=CO LOR(RGB,.4,0,.9),\n title=`Fourier transform F(w) of f(x)`);" }} {PARA 13 "" 1 "" {GLPLOT2D 542 261 261 {PLOTDATA 2 "6'-%'CURVESG6#7at7 $$!#5\"\"!$\"3x]#*oCK#)RZ!#=7$$!3!pmmm\"p0k&*!#<$\"3L1nnIfKS9F-7$$!3uK L$32'F-7$$! 3;mm;/'=3l)F1$!34b=GcxfQiF-7$$!3+mmT&=Vrf)F1$!3%f4!3P&*3\\jF-7$$!3&emm mwnMa)F1$!3&3gX$)3$H-kF-7$$!3qlm\"zM#z*[)F1$!3%euq*GUo(R'F-7$$!3clm;Hp 6O%)F1$!3`@@!\\ed`L'F-7$$!3SlmT5:W#Q)F1$!3pq6xf0.;iF-7$$!3/nmm\"4m(G$) F1$!3'3z'=4//TgF-7$$!3?++Dc[3:\")F1$!3Uect!G7'R[F-7$$!3OLL$3i.9!zF1$!3 rlWfu!ox+$F-7$$!3fmm;/R=0vF1$\"3)3'**)z0CG>*!#>7$$!31ML$3i_+I(F1$\"3ap zO)4]Ed#F-7$$!3k++]P8#\\4(F1$\"3si*[!4&Q\\e$F-7$$!3(RLL3d%)=/(F1$\"3*Q 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LORG6&%$RGBG$\"\"%!\"\"F*$\"\"*F][n-%%VIEWG6$;F(F[jm%(DEFAULTG" 1 2 0 1 10 0 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 23 "Note that the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 63 " in this example \+ can be described using the Heaviside function " }{XPPEDIT 18 0 "H(x)=P IECEWISE([ 0, x<1],[1 , x>1])" "6#/-%\"HG6#%\"xG-%*PIECEWISEG6$7$\"\"! 2F'\"\"\"7$F.2F.F'" }{TEXT -1 6 " as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = 2*H(x+2)- H(x+1)+H(x-1)-2*H(x-2);" "6#/-%\"fG6#%\"xG,**&\"\"#\"\"\"-%\"HG6#,&F'F +F*F+F+F+-F-6#,&F'F+F+F+!\"\"-F-6#,&F'F+F+F3F+*&F*F+-F-6#,&F'F+F*F3F+F 3" }{TEXT -1 2 ". 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=combine(rhs(%));\n``=frontend(expand,[rhs(%)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"#\"\"\"-%\"HG6#,&%\"xGF&F%F&F&F&-F(6#,&F+F&F&F& !\"\"-F(6#,&F+F&F&F/F&*&F%F&-F(6#,&F+F&F%F/F&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,$**\"\"#\"\"\"%&omegaG!\"\"-%$sin G6#F)F(,&*&\"\"%F(-%$cosGF-F(F(F(F*F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&*&\"\"%\"\"\"-%$sinG6#,$*&\"\"#F)%&omegaGF)F)F)F)*&F/F)- F+6#F0F)!\"\"F)F0F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(\"\"%\" \"\"%&omegaG!\"\"-%$sinG6#,$*&\"\"#F(F)F(F(F(F(*(F0F(-F,6#F)F(F)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 58 ": We \+ have seen in example 1 that the Fourier transform of " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "phi[a](x) = PIECEWISE([1, abs(x ) <= a],[0, a < abs(x)]);" "6#/-&%$phiG6#%\"aG6#%\"xG-%*PIECEWISEG6$7$ \"\"\"1-%$absG6#F*F(7$\"\"!2F(-F26#F*" }{TEXT -1 4 " " }}{PARA 0 " " 0 "" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "F[a](omega) = 2*sin*a*omega/omega;" "6#/-&%\"FG6#%\"aG6 #%&omegaG*,\"\"#\"\"\"%$sinGF-F(F-F*F-F*!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The funct ion of this example is " }{XPPEDIT 18 0 "2*phi[2](x)-phi[1](x);" "6#,& *&\"\"#\"\"\"-&%$phiG6#F%6#%\"xGF&F&-&F)6#F&6#F,!\"\"" }{TEXT -1 91 ", so its transform could be obtained using the linearity of the Fourier transform operator." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Fourier transfo rm of the function " }{XPPEDIT 18 0 "f(x) = exp(-a*abs(x));" "6#/-%\"f G6#%\"xG-%$expG6#,$*&%\"aG\"\"\"-%$absG6#F'F.!\"\"" }{TEXT -1 8 ", whe re " }{TEXT 273 1 "a" }{TEXT -1 13 " is positive." }}{PARA 0 "" 0 "" {TEXT -1 32 "The graph is drawn for the case " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "f := x -> exp(-abs(x));\nplo t(f(x),x=-3..3,y=0..1.2,color=red,thickness=2,ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$ expG6#,$-%$absG6#9$!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 558 161 161 {PLOTDATA 2 "6(-%'CURVESG6#7Z7$$!\"$\"\"!$\"3W%R'yOoqy\\!#>7$$!3!* *****\\2<#pG!#<$\"3M-!*)o\\LVn&F-7$$!3#)***\\7bBav#F1$\"3aWlPp$3#ejF-7 $$!36++]K3XFEF1$\"3S4#R.nViA(F-7$$!3%)****\\F)H')\\#F1$\"3SKY*=ja(>#)F -7$$!3#****\\i3@/P#F1$\"3(yI7;6PTM*F-7$$!3;++Dr^b^AF1$\"3#Hb!4\"QaB0\" !#=7$$!3$****\\7Sw%G@F1$\"3it^1A\\=!>\"FM7$$!3*****\\7;)=,?F1$\"3[tt(e yXFM7$$!3!******\\!)H%*\\ \"F1$\"3]/HHTUdKAFM7$$!3/+++vl[p8F1$\"3pJ?;zXPUDFM7$$!3\"******\\>iUC \"F1$\"3'Q$)HwIM:)GFM7$$!3-++DhkaI6F1$\"3G9X)[zn&GKFM7$$!3s******\\XF` **FM$\"3K!*[tlR-'p$FM7$$!3u*******>#z2))FM$\"3rM`.[(\\Lb$*FM7$$!3 Wb+++v`hH!#?$\"3GoDEN%G/(**FM7$$\"3%[***\\i]2=jF-$\"31@^1Wxt(Q*FM7$$\" 3]****\\(QIKH\"FM$\"3cBx^(o+py)FM7$$\"3K******\\4+p=FM$\"3!>o851m_H)FM 7$$\"38****\\7:xWCFM$\"3*>WUL\")Q6$yFM7$$\"3?++v$\\>m1$FM$\"3i5$=4I$** etFM7$$\"3E,++vuY)o$FM$\"3T0'HC\"QJ:pFM7$$\"3!z******4FL(\\FM$\"3AFIdB h]\"3'FM7$$\"3A)****\\d6.B'FM$\"3IF5x?J;j`FM7$$\"3s****\\(o3lW(FM$\"3O >9cM2+\\ZFM7$$\"35*****\\A))oz)FM$\"3))p&\\4%\\&3(GFM7$$\"3M+++b*=jP\"F1$\"3P,B%eJj]_#FM7$$\"3g***\\ (3/3(\\\"F1$\"3y(z*>4c#yB#FM7$$\"33++vB4JB;F1$\"3U6u)3bXC(>FM7$$\"3u** ***\\KCnu\"F1$\"3G\"GFt.TMu\"FM7$$\"3s***\\(=n#f(=F1$\"3]#>Ev:H@`\"FM7 $$\"3P+++!)RO+?F1$\"3y#RQ'G.'GN\"FM7$$\"30++]_!>w7#F1$\"3Ali;jd?\">\"F M7$$\"3O++v)Q?QD#F1$\"3W]'=uEt*\\5FM7$$\"3G+++5jypBF1$\"3y]L*3Vq+N*F-7 $$\"3<++]Ujp-DF1$\"3yb\\_ZnR'=)F-7$$\"3++++gEd@EF1$\"3S[*=g'e%)osF-7$$ \"39++v3'>$[FF1$\"3d))4HLv`.kF-7$$\"37++D6EjpGF1$\"3G]KX&4w>n&F-7$$\" \"$F*F+-%*AXESTICKSG6$%(DEFAULTGF[]l-%+AXESLABELSG6$Q\"x6\"Q\"yFd]l-%' COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F]^l-%*THICKNESSG6#\"\"#-%%VIEWG6$;F (Fj\\l;F]^l$\"#7!\"\"" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(omega) = Int( exp(-a*abs(x))*exp(-i*omega*x),x = -infinity .. infinity);" "6#/-%\"FG 6#%&omegaG-%$IntG6$*&-%$expG6#,$*&%\"aG\"\"\"-%$absG6#%\"xGF2!\"\"F2-F -6#,$*(%\"iGF2F'F2F6F2F7F2/F6;,$%)infinityGF7F@" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(exp(a*x)*exp (-i*omega*x),x = -infinity .. 0)+Int(exp(-a*x)*exp(-i*omega*x),x = 0 . . infinity);" "6#/%!G,&-%$IntG6$*&-%$expG6#*&%\"aG\"\"\"%\"xGF/F/-F+6# ,$*(%\"iGF/%&omegaGF/F0F/!\"\"F//F0;,$%)infinityGF7\"\"!F/-F'6$*&-F+6# ,$*&F.F/F0F/F7F/-F+6#,$*(F5F/F6F/F0F/F7F//F0;F " 0 "" {MPLTEXT 1 0 101 "plot(2/(1+w^2),w=-5..5,color=COLOR (RGB,.4,0,.9),\n title=`Fourier transform F(w) of f(x)`);" }}{PARA 13 "" 1 "" {GLPLOT2D 585 175 175 {PLOTDATA 2 "6'-%'CURVESG6#7a o7$$!\"&\"\"!$\"3u#p2Bp2Bp(!#>7$$!3YLLLe%G?y%!#<$\"3a#)\\\"=q\"\\z$)F- 7$$!3OmmT&esBf%F1$\"3o%)fIps\"R0*F-7$$!3ALL$3s%3zVF1$\"36A*R&y%zD\"**F -7$$!3_LL$e/$QkTF1$\"3SvLa5&)Q!4\"!#=7$$!3ommT5=q]RF1$\"3X2p&HnNU?\"FC 7$$!3ILL3_>f_PF1$\"3LGK9!z)3E8FC7$$!3K++vo1YZNF1$\"3]uc:d'oAZ\"FC7$$!3 ;LL3-OJNLF1$\"3#zLB0[x&\\;FC7$$!3p***\\P*o%Q7$F1$\"3;AiiP\"4!f=FC7$$!3 Kmmm\"RFj!HF1$\"3PJr#frKr6#FC7$$!33LL$e4OZr#F1$\"3oFGCAca*Q#FC7$$!3u** ***\\n\\!*\\#F1$\"3/$*Ry@&H/w#FC7$$!3%)*****\\ixCG#F1$\"3Te&*)yPl2A$FC 7$$!3#******\\KqP2#F1$\"3E)*z,6B@tPFC7$$!39LL3-TC%)=F1$\"3QldkU8C&R%FC 7$$!3[mmm\"4z)e;F1$\"3%[.t:7h1L&FC7$$!3Mmmmm`'zY\"F1$\"3K+g\\,.IRjFC7$ $!3#****\\(=t)eC\"F1$\"3x#4$pZ$oi$yFC7$$!3OL$3x'*)fZ6F1$\"3]uLl4M\">j) FC7$$!3!ommmh5$\\5F1$\"34%[\"*yWP!>&*FC7$$!3tIL3xrs9%*FC$\"3WkNc$*oBg5 F17$$!3S$***\\(=[jL)FC$\"3/@NL\"*y(*z6F17$$!3q%****\\Pw%4tFC$\"3;N(R-D RNI\"F17$$!3)f***\\iXg#G'FC$\"3aNT)pi))RV\"F17$$!3$oK$3_:<6_FC$\"3YK_T (HnGd\"F17$$!3ndmmT&Q(RTFC$\"3Tqx))*4'R2<#FC$\"36!HcJO0*4>F17$$!3FK$3Fpy7k\"FC$\"3aJVVAt` Z>F17$$!3g***\\7yQ16\"FC$\"3s-d]N-jv>F17$$!3iK$3_D)=`%)F-$\"3;t\\W<,\" e)>F17$$!3Epm\"zp))**z&F-$\"3;j^:$e%H$*>F17$$!3#f+D19*yYJF-$\"3m/Yy-:- )*>F17$$!3vDMLLe*e$\\!#?$\"3KM<0v7&***>F17$$\"3+l;a)3RBE#F-$\"3?Jm,))o (*)*>F17$$\"3bsmTgxE=]F-$\"3mPqb\\g(\\*>F17$$\"37!o\"HKk>uxF-$\"3c=wV! *\\)z)>F17$$\"3womT5D,`5FC$\"3)z!\\Y*\\m!y>F17$$\"3Gq;zW#)>/;FC$\"37-* )))fA#)\\>F17$$\"3!=nm\"zRQb@FC$\"3G6#3)R7@6>F17$$\"3mOLL$e,]6$FC$\"3P 1#GkH+J#=F17$$\"3_,+](=>Y2%FC$\"34G)yOxG_r\"F17$$\"36QLe*[K56&FC$\"3QQ +C^zv&e\"F17$$\"3summ\"zXu9'FC$\"3#)G_/jEZ^9F17$$\"3#yLLe9i\"=sFC$\"3% =U?4F3\\J\"F17$$\"3#4+++]y))G)FC$\"3+8%\\wv(\\&=\"F17$$\"3%>++DcljL*FC $\"3IhEf:.co5F17$$\"3H++]i_QQ5F1$\"3&3^ZPH4Ni*FC7$$\"3U+](=-N(R6F1$\"3 Eq3?v*\\%*p)FC7$$\"3b++D\"y%3T7F1$\"3;0jp.C6tyFC7$$\"3+++]P![hY\"F1$\" 3#o6%fCF.]jFC7$$\"3iKLL$Qx$o;F1$\"3M'z2aOMhG&FC7$$\"3Y+++v.I%)=F1$\"3m bO8:l.&R%FC7$$\"3?mm\"zpe*z?F1$\"3ykPwrC+bPFC7$$\"3;,++D\\'QH#F1$\"3' \\wxQZhR>$FC7$$\"3%HL$e9S8&\\#F1$\"3co-z8.!zw#FC7$$\"3s++D1#=bq#F1$\"3 9x=#*Gc*QS#FC7$$\"3\"HLL$3s?6HF1$\"3U)pq#e&*y5@FC7$$\"3a***\\7`Wl7$F1$ \"3yd$=v8+h&=FC7$$\"3enmmm*RRL$F1$\"3mK " 0 "" {MPLTEXT 1 0 202 "interface(showassumed=0): a ssume(a>0):\nInt(exp(a*t)*exp(-I*omega*t),t =-infinity..0) +\n I nt(exp(-a*t)*exp(-I*omega*t),t =0..infinity);\n``=value(%);\n``= norma l(rhs(%));\n'F(omega)'=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$*&-%$expG6#*&%#a|irG\"\"\"%\"tGF-F--F)6#*(^#!\"\"F-%& omegaGF-F.F-F-/F.;,$%)infinityGF3\"\"!F--F%6$*&-F)6#,$F+F3F-F/F-/F.;F9 F8F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"\"F',&%#a|irGF'*&^ #!\"\"F'%&omegaGF'F'F,F'*&F'F',&F)F'*&F-F'^#F'F'F'F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\"%#a|irGF(,&F)F(*&^#!\"\"F(%&ome gaGF(F(F-,&F)F(*&F.F(^#F(F(F(F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"FG6#%&omegaG,$*(\"\"#\"\"\"%#a|irGF+,&*$)F,F*F+F+*$)F'F*F+F+!\"\"F +" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "interface(showassumed=0): assume(a>0):\nexp(-a*abs(x ));\n`Fourier transform`=inttrans[fourier](%,x,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$*&%#a|irG\"\"\"-%$absG6#%\"xGF)!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,$*(\"\"#\"\"\"% #a|irGF(,&*$)F)F'F(F(*$)%&omegaGF'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 5" }}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Fourier transform of the function " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 10 " given by " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([1-abs(x)/a, abs(x) < a],[0, a <= abs(x)]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&\"\"\"F-*& -%$absG6#F'F-%\"aG!\"\"F32-F06#F'F27$\"\"!1F2-F06#F'" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 20 "where a is positive." }}{PARA 0 "" 0 "" {TEXT -1 32 "The graph is drawn for the case " }{XPPEDIT 18 0 "a = \+ 2" "6#/%\"aG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "f := x -> if abs(x)<2 then \+ 1-abs(x)/2 else 0 end if:\nplot('f(x)',x=-3..3,y=0..1.2,color=red,thic kness=2,ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 616 174 174 {PLOTDATA 2 "6(-%'CURVESG6#7Y7$$!\"$\"\"!$F*F*7$$!3!******\\2<#pG!#F/$\"300+]iDf7J!#>7$$ !3/++DO\"3V(=F/$\"3I)***\\(=$f%G'FM7$$!3#******\\V'zViUC\"F /$\"3[+++D!*oyPFX7$$!3-++DhkaI6F/$\"3))***\\PpnsM%FX7$$!3s******\\XF`* *FX$\"39+++DFOB]FX7$$!3u*******>#z2))FX$\"39++++R5'f&FX7$$!3S++]7RKvuF X$\"3!)***\\P/QBE'FX7$$!3s,+++P'eH'FX$\"39******\\\"o?&oFX7$$!3q)***\\ 7*3=+&FX$\"3k++vVb4*\\(FX7$$!3[)***\\PFcpPFX$\"3w++DJ'=_6)FX7$$!3;)*** *\\7VQ[#FX$\"3#4++vVy!e()FX7$$!32)***\\i6:.8FX$\"3'4+](=WU[$*FX7$$!39$ ***\\(oKQm'FM$\"3M+]il$3om*FX7$$!3Wb+++v`hH!#?$\"3s****\\7B>&)**FX7$$ \"3%[***\\i]2=jFM$\"3E+](oC'4%o*FX7$$\"3]****\\(QIKH\"FX$\"3C++D1[Q`$* FX7$$\"38****\\7:xWCFX$\"3W++vVUhx()FX7$$\"3E,++vuY)o$FX$\"3O****\\iiw b\")FX7$$\"3!z******4FL(\\FX$\"30,++]kL8vFX7$$\"3A)****\\d6.B'FX$\"3)3 ++D@W[)oFX7$$\"3s****\\(o3lW(FX$\"39++DccuwiFX7$$\"35*****\\A))oz)FX$ \"3W++]()eb,cFX7$$\"3e******Hk-,5F/$\"35-++]y'[*\\FX7$$\"36+++D-eI6F/$ \"3[*****\\())4ZVFX7$$\"3u***\\(=_(zC\"F/$\"3Q,+D1R7gPFX7$$\"3M+++b*=j P\"F/$\"3G)****\\A0%=JFX7$$\"3g***\\(3/3(\\\"F/$\"3/-+Dczf9DFX7$$\"33+ +vB4JB;F/$\"3g***\\7QXM)=FX7$$\"3u*****\\KCnu\"F/$\"3G,++v$yjE\"FX7$$ \"3s***\\(=n#f(=F/$\"3!R,+D1kO?'FM7$$\"30+]P\\`9Q>F/$\"3g(**\\7`KF4$FM 7$$\"3P+++!)RO+?F/F+7$$\"3A++D;:*R1#F/F+7$$\"30++]_!>w7#F/F+7$$\"3O++v )Q?QD#F/F+7$$\"3G+++5jypBF/F+7$$\"3<++]Ujp-DF/F+7$$\"3++++gEd@EF/F+7$$ \"39++v3'>$[FF/F+7$$\"37++D6EjpGF/F+7$$\"\"$F*F+-%*AXESTICKSG6$%(DEFAU LTGFaz-%+AXESLABELSG6$Q\"x6\"Q\"yFjz-%'COLOURG6&%$RGBG$\"*++++\"!\")F+ F+-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(F`z;F+$\"#7!\"\"" 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 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "F(omega) = Int(f(x)*exp(-i*omega*x),x = -infinity .. in finity);" "6#/-%\"FG6#%&omegaG-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#, $*(%\"iGF0F'F0F/F0!\"\"F0/F/;,$%)infinityGF7F;" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Int((1+x/a)*exp (-i*omega*x),x = -a .. 0)+Int((1-x/a)*exp(-i*omega*x),x = 0 .. a);" "6 #/%!G,&-%$IntG6$*&,&\"\"\"F+*&%\"xGF+%\"aG!\"\"F+F+-%$expG6#,$*(%\"iGF +%&omegaGF+F-F+F/F+/F-;,$F.F/\"\"!F+-F'6$*&,&F+F+*&F-F+F.F/F/F+-F16#,$ *(F5F+F6F+F-F+F/F+/F-;F:F.F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int( (1-x/a)*exp(-i*omega*x),x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*&,&\" \"\"F)*&%\"xGF)%\"aG!\"\"F-F)-%$expG6#,$*(%\"iGF)%&omegaGF)F+F)F-F)F+- F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxGF-F)F+" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "u=1-x/a" "6#/%\"uG,&\"\"\"F&*&%\"xGF&%\"aG!\"\"F*" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "v = exp(-i*omega*x)/(-i*omega)" " 6#/%\"vG*&-%$expG6#,$*(%\"iG\"\"\"%&omegaGF,%\"xGF,!\"\"F,,$*&F+F,F-F, F/F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 41 "so, by the integ ration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-In t(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\" F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Int((1-x/a)*exp(-i*omega*x),x = 0 .. a) = (1-x/a)*``(exp(-i*omega*x)/( -i*omega));" "6#/-%$IntG6$*&,&\"\"\"F)*&%\"xGF)%\"aG!\"\"F-F)-%$expG6# ,$*(%\"iGF)%&omegaGF)F+F)F-F)/F+;\"\"!F,*&,&F)F)*&F+F)F,F-F-F)-%!G6#*& -F/6#,$*(F3F)F4F)F+F)F-F),$*&F3F)F4F)F-F-F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([a, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6 %7$%\"aG%!G7$F(F(7$\"\"!F(" }{XPPEDIT 18 0 "-Int(``(-1/a)*``(exp(-i*om ega*x)/(-i*omega)),x = 0 .. a);" "6#,$-%$IntG6$*&-%!G6#,$*&\"\"\"F-%\" aG!\"\"F/F--F)6#*&-%$expG6#,$*(%\"iGF-%&omegaGF-%\"xGF-F/F-,$*&F8F-F9F -F/F/F-/F:;\"\"!F.F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(i*omega)-1/(i*a*omega);" "6#/%!G,&*&\"\"\"F '*&%\"iGF'%&omegaGF'!\"\"F'*&F'F'*(F)F'%\"aGF'F*F'F+F+" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(exp(-i*omega*x),x = 0 .. a);" "6#-%$IntG6$-%$exp G6#,$*(%\"iG\"\"\"%&omegaGF,%\"xGF,!\"\"/F.;\"\"!%\"aG" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "-i/omega-1/(i*a*omega);" "6#,&*&%\"iG\"\"\"%&omegaG!\" \"F(*&F&F&*(F%F&%\"aGF&F'F&F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(ex p(-i*omega*x)/(-i*omega));" "6#-%!G6#*&-%$expG6#,$*(%\"iG\"\"\"%&omega GF-%\"xGF-!\"\"F-,$*&F,F-F.F-F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIE CEWISE([a, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%\"aG%!G7$F(F(7 $\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -i/omega-exp(-i*omega*x)/( a*omega^2);" "6#/%!G,&*&%\"iG\"\"\"%&omegaG!\"\"F**&-%$expG6#,$*(F'F(F )F(%\"xGF(F*F(*&%\"aGF(*$F)\"\"#F(F*F*" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([a, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%\"aG%!G7 $F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -i/omega-1/(a*om ega^2);" "6#/%!G,&*&%\"iG\"\"\"%&omegaG!\"\"F**&F(F(*&%\"aGF(*$F)\"\"# F(F*F*" }{XPPEDIT 18 0 "``(exp(-i*a*omega)-1);" "6#-%!G6#,&-%$expG6#,$ *(%\"iG\"\"\"%\"aGF-%&omegaGF-!\"\"F-F-F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 " so that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Int((1-x/a)*exp(-i*omega*x),x = 0 .. a) = -i/omega+1/(a *omega^2);" "6#/-%$IntG6$*&,&\"\"\"F)*&%\"xGF)%\"aG!\"\"F-F)-%$expG6#, $*(%\"iGF)%&omegaGF)F+F)F-F)/F+;\"\"!F,,&*&F3F)F4F-F-*&F)F)*&F,F)*$F4 \"\"#F)F-F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1-exp(-i*a*omega));" " 6#-%!G6#,&\"\"\"F'-%$expG6#,$*(%\"iGF'%\"aGF'%&omegaGF'!\"\"F0" } {TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 11 "Similarly, " }{XPPEDIT 18 0 "Int((1+x/a)* exp(-i*omega*x),x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*&,&\"\"\"F)*&% \"xGF)%\"aG!\"\"F)F)-%$expG6#,$*(%\"iGF)%&omegaGF)F+F)F-F)F+-F%6$*&%\" uGF)-%!G6#*&%#dvGF)%#dxGF-F)F+" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = 1+x/a;" "6#/%\"uG,&\"\"\"F&*&%\"xGF&%\"aG!\"\"F&" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "v = exp(-i*omega*x)/(-i*omega)" "6#/%\"vG*&-%$e xpG6#,$*(%\"iG\"\"\"%&omegaGF,%\"xGF,!\"\"F,,$*&F+F,F-F,F/F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "so, by the integration by parts formula: " }{XPPEDIT 18 0 "Int(u*` `(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6 #*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F /F0F)F1F0" }{TEXT -1 11 ", we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((1+x/a)*exp(-i*omega*x),x = -a .. 0) = (1+x/a)* ``(exp(-i*omega*x)/(-i*omega));" "6#/-%$IntG6$*&,&\"\"\"F)*&%\"xGF)%\" aG!\"\"F)F)-%$expG6#,$*(%\"iGF)%&omegaGF)F+F)F-F)/F+;,$F,F-\"\"!*&,&F) F)*&F+F)F,F-F)F)-%!G6#*&-F/6#,$*(F3F)F4F)F+F)F-F),$*&F3F)F4F)F-F-F)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([0, ``],[``, ``],[-a, ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$%\"aG!\"\"F(" }{XPPEDIT 18 0 "-I nt(``(1/a)*``(exp(-i*omega*x)/(-i*omega)),x = -a .. 0);" "6#,$-%$IntG6 $*&-%!G6#*&\"\"\"F,%\"aG!\"\"F,-F)6#*&-%$expG6#,$*(%\"iGF,%&omegaGF,% \"xGF,F.F,,$*&F7F,F8F,F.F.F,/F9;,$F-F.\"\"!F." }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(-i*omega)+1/( i*a*omega);" "6#/%!G,&*&\"\"\"F',$*&%\"iGF'%&omegaGF'!\"\"F,F'*&F'F'*( F*F'%\"aGF'F+F'F,F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-i*omega* x),x = -a .. 0);" "6#-%$IntG6$-%$expG6#,$*(%\"iG\"\"\"%&omegaGF,%\"xGF ,!\"\"/F.;,$%\"aGF/\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = i/omega+1 /(i*a*omega);" "6#/%!G,&*&%\"iG\"\"\"%&omegaG!\"\"F(*&F(F(*(F'F(%\"aGF (F)F(F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(exp(-i*omega*x)/(-i*omeg a));" "6#-%!G6#*&-%$expG6#,$*(%\"iG\"\"\"%&omegaGF-%\"xGF-!\"\"F-,$*&F ,F-F.F-F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([0, ``],[``, `` ],[-a, ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$%\"aG!\"\"F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = i/omega+exp(-i*omega*x)/(a*omega^2 );" "6#/%!G,&*&%\"iG\"\"\"%&omegaG!\"\"F(*&-%$expG6#,$*(F'F(F)F(%\"xGF (F*F(*&%\"aGF(*$F)\"\"#F(F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWI SE([0, ``],[``, ``],[-a, ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$ %\"aG!\"\"F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int((1+x/a)*exp(-i*omega*x),x = -a .. 0) = i/omega+1/( a*omega^2);" "6#/-%$IntG6$*&,&\"\"\"F)*&%\"xGF)%\"aG!\"\"F)F)-%$expG6# ,$*(%\"iGF)%&omegaGF)F+F)F-F)/F+;,$F,F-\"\"!,&*&F3F)F4F-F)*&F)F)*&F,F) *$F4\"\"#F)F-F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1-exp(i*a*omega)); " "6#-%!G6#,&\"\"\"F'-%$expG6#*(%\"iGF'%\"aGF'%&omegaGF'!\"\"" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 18 "From (i) and ( ii)," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(omega) =1/( a*omega^2)" "6#/-%\"FG6#%&omegaG*&\"\"\"F)*&%\"aGF)*$F'\"\"#F)!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[2-exp(i*a*omega)-exp(i*a*omega)]" "6#7 #,(\"\"#\"\"\"-%$expG6#*(%\"iGF&%\"aGF&%&omegaGF&!\"\"-F(6#*(F+F&F,F&F -F&F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/(a*omega^2);" "6#/%!G*&\"\"#\"\"\"*&%\"aGF'*$%&omegaGF&F' !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1-(exp(i*a*omega)+exp(i*a*om ega))/2);" "6#-%!G6#,&\"\"\"F'*&,&-%$expG6#*(%\"iGF'%\"aGF'%&omegaGF'F '-F+6#*(F.F'F/F'F0F'F'F'\"\"#!\"\"F5" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*(1-cos*a*omega)/(a*omega^2) ;" "6#/%!G*(\"\"#\"\"\",&F'F'*(%$cosGF'%\"aGF'%&omegaGF'!\"\"F'*&F+F'* $F,F&F'F-" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = a*sin(a*omega/2)^2/((a*omega/2)^2);" "6#/%!G*(%\"a G\"\"\"*$-%$sinG6#*(F&F'%&omegaGF'\"\"#!\"\"F.F'*$*(F&F'F-F'F.F/F.F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "In particular, if " }{XPPEDIT 18 0 "a = 2" "6#/%\"aG\"\"# " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "F(omega) = (1-cos*2*omega)/(omega^2 );" "6#/-%\"FG6#%&omegaG*&,&\"\"\"F**(%$cosGF*\"\"#F*F'F*!\"\"F**$F'F- F." }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*sin^2*omega/(omega^2);" "6#** \"\"#\"\"\"*$%$sinGF$F%%&omegaGF%*$F(F$!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "plot(2*sin(w)^2/w^2,w=-8..8,color=COLOR(RGB,.4,0,.9),\n \+ title=`Fourier transform F(w) of f(x)`);" }}{PARA 13 "" 1 "" {GLPLOT2D 515 234 234 {PLOTDATA 2 "6'-%'CURVESG6#7ip7$$!\")\"\"!$\"3%y G0!QH%)eI!#>7$$!3OLLLLbC^w!#<$\"3aWW+5,(yF$F-7$$!3?mmmOhzZtF1$\"3/AS'R ?xN$GF-7$$!3LLLL`b`1qF1$\"3uvAf$eu[y\"F-7$$!3#HLLL(G,jmF1$\"3w#3#fW6j# >'!#?7$$!30nmm'*G7@jF1$\"3eg.Ar`m+s!#A7$$!3XLLLBr9/gF1$\"33WfIMNv3UFC7 $$!3!)******pq$fn&F1$\"3k5C(**)=J@?F-7$$!3fLLLj<]O`F1$\"3d[XMD?gCYF-7$ $!3Q+++I]:)*\\F1$\"3=]#f%R1tptF-7$$!3(QLL$G%RT#[F1$\"3w)**z'HR-([)F-7$ $!3YmmmEQ7]YF1$\"3adiUJ&>L@*F-7$$!3))*******y]o\\%F1$\"3Q@J`H\"yzV*F-7 $$!3HLLL`xdVVF1$\"3E1=X\\L)HA*F-7$$!3!ommmhG5<%F1$\"33M*p#zCLV%)F-7$$! 3I+++![z%)*RF1$\"3[bJ:#e)*f9(F-7$$!35++++U'>l$F1$\"3/hd+wNcyNF-7$$!33+ ++g$)*\\[$F1$\"3yFy;#f6o'=F-7$$!3/+++?D.=LF1$\"3G1/6p=%pf&FC7$$!3;LL$3 .AAC$F1$\"3F31`cn6?>FC7$$!3smmmT:TmJF1$\"3U(HvU]([G7!#@7$$!3G++]_5g!4$ 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**\\7Z-]F1$\"3`w\"z9BM#QtF-7$$\"32nmmYRIM`F1$\"3'oX?>&Q0VYF-7$$\"3?mmm 13ltcF1$\"3E<8:[]EO?F-7$$\"33LLL.x=5gF1$\"3\")zw%)*Q0\\-%FC7$$\"3d**** **f,V>jF1$\"3cO![;([AwlFI7$$\"3?LLL8p&Qn'F1$\"3q7]b93m6lFC7$$\"33mmmE/ '3*pF1$\"3x?uwC%o$H " 0 "" {MPLTEXT 1 0 160 "interface(showassumed=0): a ssume(a>0):\nInt((1+x/a)*exp(-I*omega*x),x=-a..0)+\n Int((1-x/a)*e xp(-I*omega*x),x=0..a);\n``=value(%);\n'F(omega)'=simplify(rhs(%));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$*&,&\"\"\"F)*&%\"xGF)%#a|i rG!\"\"F)F)-%$expG6#*(^#F-F)%&omegaGF)F+F)F)/F+;,$F,F-\"\"!F)-F%6$*&,& F)F)F*F-F)F.F)/F+;F7F,F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(,( -%$expG6#*(%#a|irG\"\"\"%&omegaGF-^#F-F-!\"\"F+F-F-F-F-F,F0F.!\"#F-**, (**F(F-F.F-F,F-F/F-F-F(F0F-F-F--F)6#*(^#F0F-F.F-F,F-F-F.F1F,F0F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%&omegaG,$**\"\"#\"\"\",&-%$c osG6#*&F'F+%#a|irGF+F+F+!\"\"F+F1F2F'!\"#F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Using the Heaviside function, w e have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)= H(x+ a)*(1+x/a)-2*H(x)*(x/a)-H(x-a)*(1-x/a)" "6#/-%\"fG6#%\"xG,(*&-%\"HG6#, &F'\"\"\"%\"aGF.F.,&F.F.*&F'F.F/!\"\"F.F.F.*(\"\"#F.-F+6#F'F.*&F'F.F/F 2F.F2*&-F+6#,&F'F.F/F2F.,&F.F.*&F'F.F/F2F2F.F2" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "alias(H=Heaviside):\ninterface(showassumed=0);\nassume(a>0):\nH(x +a)*(1+x/a)-2*H(x)*(x/a)-H(x-a)*(1-x/a);\n`Fourier transform`=inttrans [fourier](%,x,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%\"HG6#, &%\"xG\"\"\"%#a|irGF*F*,&F*F**&F)F*F+!\"\"F*F*F***\"\"#F*-F&6#F)F*F)F* F+F.F.*&-F&6#,&F)F*F+F.F*,&F*F*F-F.F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,$**\"\"%\"\"\"%#a|irG!\"\"%&omegaG!\"#-%$s inG6#,$*(\"\"#F*F)F(F+F(F(F2F(" }}}{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 51 "The rea l and imaginary parts of a Fourier transform" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 14 " Suppose that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " is a " }{TEXT 261 4 "real" }{TEXT -1 47 " valued function and has the Fourier transf orm " }{XPPEDIT 18 0 "F(omega)" "6#-%\"FG6#%&omegaG" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 5 " Then" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "F(omega)=Int(f(x)*exp(-i*omega*x),x=-infinity..infin ity)" "6#/-%\"FG6#%&omegaG-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*(% \"iGF0F'F0F/F0!\"\"F0/F/;,$%)infinityGF7F;" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f(x)*(cos*omega*x- i*sin*omega*x),x = -infinity .. infinity);" "6#/%!G-%$IntG6$*&-%\"fG6# %\"xG\"\"\",&*(%$cosGF-%&omegaGF-F,F-F-**%\"iGF-%$sinGF-F1F-F,F-!\"\"F -/F,;,$%)infinityGF5F9" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f(x)*cos*omega*x,x = -infinity .. infin ity)-i*Int(f(x)*sin*omega*x,x = -infinity .. infinity);" "6#/%!G,&-%$I ntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF.%&omegaGF.F-F./F-;,$%)infinityG!\"\" F4F.*&%\"iGF.-F'6$**-F+6#F-F.%$sinGF.F0F.F-F./F-;,$F4F5F4F.F5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Hence the " }{TEXT 261 24 "real and imaginary parts" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "F(omega)" "6#-%\"FG6#%&omegaG" }{TEXT -1 4 " are " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Re(F(omega))" "6 #-%#ReG6#-%\"FG6#%&omegaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "A(omega) \+ = Int(f(x)*cos*omega*x,x = -infinity .. infinity);" "6#/-%\"AG6#%&omeg aG-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF0F'F0F/F0/F/;,$%)infinityG!\"\" F5" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 " " 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "Im(F(omega))" "6#-%#ImG6#-%\" FG6#%&omegaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "B(omega) = -Int(f(x)*s in*omega*x,x = -infinity .. infinity);" "6#/-%\"BG6#%&omegaG,$-%$IntG6 $**-%\"fG6#%\"xG\"\"\"%$sinGF1F'F1F0F1/F0;,$%)infinityG!\"\"F6F7" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "A(omega);" "6#-%\"AG6#%&omegaG" }{TEXT -1 7 " is an " } {TEXT 261 4 "even" }{TEXT -1 16 " function, since" }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "A(-omega) = Int(f(x)*cos(-omega*x),x = -infinity .. infinity);" "6#/-%\"AG6#,$%&omegaG!\"\"-%$IntG6$*&-%\" fG6#%\"xG\"\"\"-%$cosG6#,$*&F(F2F1F2F)F2/F1;,$%)infinityGF)F;" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(f(x)*cos*omega*x,x = -infinity .. infin ity) = A(omega);" "6#/-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF,%&omegaGF, F+F,/F+;,$%)infinityG!\"\"F2-%\"AG6#F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "B(omega);" "6#-%\" BG6#%&omegaG" }{TEXT -1 7 " is an " }{TEXT 261 3 "odd" }{TEXT -1 17 " \+ function, since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B (-omega) = -Int(f(x)*sin(-omega*x),x = -infinity .. infinity);" "6#/-% \"BG6#,$%&omegaG!\"\",$-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#,$*&F(F3 F2F3F)F3/F2;,$%)infinityGF)F " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 0 "" 0 " " {TEXT -1 46 "We find the Fourier transform of the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 10 " given by " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([e xp(-a*x), 0 < x],[0, x <= 0]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$-%$e xpG6#,$*&%\"aG\"\"\"F'F2!\"\"2\"\"!F'7$F51F'F5" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 274 1 "a" }{TEXT -1 14 " is positive. " }}{PARA 0 "" 0 "" {TEXT -1 32 "The graph is drawn for the case " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "f := x -> piecewise(x<0,0,exp(-x));\nplot(f(x),x=-3..3,y=0..1.2,th ickness=2,ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6# %\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6%29$\"\"!F1-%$expG6#,$F0! \"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 590 191 191 {PLOTDATA 2 "6'-% 'CURVESG6$7jn7$$!\"$\"\"!$F*F*7$$!3!******\\2<#pG!#iUC\"F/F+7$$!3-++Dhk aI6F/F+7$$!3s******\\XF`**!#=F+7$$!3u*******>#z2))FgnF+7$$!3S++]7RKvuF gnF+7$$!3s,+++P'eH'FgnF+7$$!3q)***\\7*3=+&FgnF+7$$!3[)***\\PFcpPFgnF+7 $$!3;)****\\7VQ[#FgnF+7$$!32)***\\i6:.8FgnF+7$$!39$***\\(oKQm'!#>F+7$$ !3Wb+++v`hH!#?F+7$$\"3![W7y]bB<\"Fdp$\"3eOVK9LG))**Fgn7$$\"3/X\\i:&[iI &Fdp$\"3i(HLv/yq%**Fgn7$$\"3GXuVB:9S%*Fdp$\"3s78JnF/1**Fgn7$$\"3c%*\\7 `MSd8F`p$\"3UZDSsno851m_H)Fgn7$$\"38****\\7:xWCFgn$\"3*>WUL\")Q6$yFgn7$$\"3?++v$ \\>m1$Fgn$\"3i5$=4I$**etFgn7$$\"3E,++vuY)o$Fgn$\"3T0'HC\"QJ:pFgn7$$\"3 e****\\(G(*3L%Fgn$\"35)HN*)*H-&['Fgn7$$\"3!z******4FL(\\Fgn$\"3AFIdBh] \"3'Fgn7$$\"3A)****\\d6.B'Fgn$\"3IF5x?J;j`Fgn7$$\"3s****\\(o3lW(Fgn$\" 3O>9cM2+\\ZFgn7$$\"35*****\\A))oz)Fgn$\"3))p&\\4%\\&3(GFgn7$$\"3M+++b*=jP\"F/$\"3P,B%eJj]_#Fgn7$ $\"3g***\\(3/3(\\\"F/$\"3y(z*>4c#yB#Fgn7$$\"33++vB4JB;F/$\"3U6u)3bXC(> Fgn7$$\"3u*****\\KCnu\"F/$\"3G\"GFt.TMu\"Fgn7$$\"3s***\\(=n#f(=F/$\"3] #>Ev:H@`\"Fgn7$$\"3P+++!)RO+?F/$\"3y#RQ'G.'GN\"Fgn7$$\"30++]_!>w7#F/$ \"3Ali;jd?\">\"Fgn7$$\"3O++v)Q?QD#F/$\"3W]'=uEt*\\5Fgn7$$\"3G+++5jypBF /$\"3y]L*3Vq+N*F`p7$$\"3<++]Ujp-DF/$\"3yb\\_ZnR'=)F`p7$$\"3++++gEd@EF/ $\"3S[*=g'e%)osF`p7$$\"39++v3'>$[FF/$\"3d))4HLv`.kF`p7$$\"37++D6EjpGF/ $\"3G]KX&4w>n&F`p7$$\"\"$F*$\"3W%R'yOoqy\\F`p-%'COLOURG6&%$RGBG$\"#5! \"\"F+F+-%*AXESTICKSG6$%(DEFAULTGFf[l-%+AXESLABELSG6$Q\"x6\"Q\"yFh\\l- %*THICKNESSG6#\"\"#-%%VIEWG6$;F(Fe[l;F+$\"#7F_\\l" 1 2 0 1 10 2 2 9 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 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(omega) = Int(f(x)*exp(-i *omega*x),x = -infinity .. infinity);" "6#/-%\"FG6#%&omegaG-%$IntG6$*& -%\"fG6#%\"xG\"\"\"-%$expG6#,$*(%\"iGF0F'F0F/F0!\"\"F0/F/;,$%)infinity GF7F;" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(exp(-a*x)*exp(-i*omega*x),x = 0 .. infinity);" "6#/%!G- %$IntG6$*&-%$expG6#,$*&%\"aG\"\"\"%\"xGF/!\"\"F/-F*6#,$*(%\"iGF/%&omeg aGF/F0F/F1F//F0;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(exp(-(a+i*omega)*x),x = 0 .. i nfinity);" "6#/%!G-%$IntG6$-%$expG6#,$*&,&%\"aG\"\"\"*&%\"iGF/%&omegaG F/F/F/%\"xGF/!\"\"/F3;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = Limit(``,R = infinity);" "6#/%!G-%&LimitG6$F$/%\"RG%)infinityG" } {XPPEDIT 18 0 "exp(-(a+i*omega)*x)/(-(a+i*omega));" "6#*&-%$expG6#,$*& ,&%\"aG\"\"\"*&%\"iGF+%&omegaGF+F+F+%\"xGF+!\"\"F+,$,&F*F+*&F-F+F.F+F+ F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``],[``, ``],[0, ` `]);" "6#-%*PIECEWISEG6%7$%\"RG%!G7$F(F(7$\"\"!F(" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(a+i*omega);" "6#/%!G*&\"\"\"F&,&%\"aGF&*&%\"iGF &%&omegaGF&F&!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = a/(a^2+omega^2) -i*omega/(a^2+omega^2);" "6#/%!G,&*&%\"aG\"\"\",&*$F'\"\"#F(*$%&omegaG F+F(!\"\"F(*(%\"iGF(F-F(,&*$F'F+F(*$F-F+F(F.F." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "In partic ular, if " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "F(omega)=1/(1+omega^2)-i*omega/(1+omega^2)" "6#/-%\"FG6 #%&omegaG,&*&\"\"\"F*,&F*F**$F'\"\"#F*!\"\"F**(%\"iGF*F'F*,&F*F**$F'F- F*F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "We can illustr ate " }{XPPEDIT 18 0 "F(omega)" "6#-%\"FG6#%&omegaG" }{TEXT -1 67 " gr aphically by plotting the graphs of the real and imaginary parts" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "A(omega) = 1/(1+omeg a^2);" "6#/-%\"AG6#%&omegaG*&\"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "B(omega) = -omega/(1+omega^2);" "6#/-% \"BG6#%&omegaG,$*&F'\"\"\",&F*F**$F'\"\"#F*!\"\"F." }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "plot([1/(1+w^2),-w/(1+w^2)],w=-5..5,color=[COLOR(RGB,.4,0,.9),COL OR(RGB,.6,.4,0)],\n title=`Fourier transform F(w)=A(w)+i B(w) of f(x )`,legend=[`A(w)`,`B(w)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 630 291 291 {PLOTDATA 2 "6'-%'CURVESG6%7ao7$$!\"&\"\"!$\"3QYQ:YQ:YQ!#>7$$!3YLLLe%G ?y%!#<$\"3G\"\\24&eu*=%F-7$$!3OmmT&esBf%F1$\"3M#*HlM'ep_%F-7$$!3ALL$3s %3zVF1$\"31h*p#R(*Gc\\F-7$$!3_LL$e/$QkTF1$\"3-xor_D%>X&F-7$$!3ommT5=q] RF1$\"3DPXyk$y6-'F-7$$!3ILL3_>f_PF1$\"3oThr]RWImF-7$$!3K++vo1YZNF1$\"3 cs$ydGV8O(F-7$$!3;LL3-OJNLF1$\"3e*o;ES()yC)F-7$$!3p***\\P*o%Q7$F1$\"3# 36J\")oX]H*F-7$$!3Kmmm\"RFj!HF1$\"3olN'zNm&e5!#=7$$!33LL$e4OZr#F1$\"3# QT@6\"Gx%>\"F[o7$$!3u*****\\n\\!*\\#F1$\"3_'*>*3w9-Q\"F[o7$$!3%)***** \\ixCG#F1$\"3?zZ%*)o#Q5;F[o7$$!3#******\\KqP2#F1$\"37***3b:1m)=F[o7$$! 39LL3-TC%)=F1$\"3q#)GKr1i(>#F[o7$$!3[mmm\"4z)e;F1$\"3U^ip<# F[o$\"3e]9y:o_\\&*F[o7$$!3FK$3Fpy7k\"F[o$\"3ld;<7moP(*F[o7$$!3g***\\7y Q16\"F[o$\"3e8&Gv<^\"y)*F[o7$$!3iK$3_D)=`%)F-$\"3yl[A(e]!H**F[o7$$!3Ep m\"zp))**z&F-$\"3%e\"ex:HZm**F[o7$$!3#f+D19*yYJF-$\"3KBI#R^2,***F[o7$$ !3vDMLLe*e$\\!#?$\"3cr'e_Pc(****F[o7$$\"3+l;a)3RBE#F-$\"3-cJ3SW)[***F[ o7$$\"3bsmTgxE=]F-$\"3M)=&yZ-)[(**F[o7$$\"37!o\"HKk>uxF-$\"3!G4)=_\\#* R**F[o7$$\"3womT5D,`5F[o$\"3!*RXK(\\K.*)*F[o7$$\"3Gq;zW#)>/;F[o$\"3g5X W*H6\"\\(*F[o7$$\"3!=nm\"zRQb@F[o$\"3Uc5/*>cgb*F[o7$$\"3mOLL$e,]6$F[o$ \"3%=.T@[,b6*F[o7$$\"3_,+](=>Y2%F[o$\"3WSTRoQ9w&)F[o7$$\"36QLe*[K56&F[ o$\"3*=>+iv*yGzF[o7$$\"3summ\"zXu9'F[o$\"33WhA:LOdsF[o7$$\"3#yLLe9i\"= sF[o$\"3?4@ga8aulF[o7$$\"3#4+++]y))G)F[o$\"3/lqC)y)[FfF[o7$$\"3%>++Dcl jL*F[o$\"3Y1L'zd,GM&F[o7$$\"3H++]i_QQ5F1$\"3VbP(oka<\"[F[o7$$\"3U+](=- N(R6F1$\"39N/g()\\s\\VF[o7$$\"3b++D\"y%3T7F1$\"3e_\"[=?cl$RF[o7$$\"3++ +]P![hY\"F1$\"3UeqHij,vJF[o7$$\"3iKLL$Qx$o;F1$\"3<)*Qq#=nIk#F[o7$$\"3Y +++v.I%)=F1$\"3%y#ocd#=v>#F[o7$$\"3?mm\"zpe*z?F1$\"3Q#)=)eB,v(=F[o7$$ \"3;,++D\\'QH#F1$\"3[#))Qpt!)pf\"F[o7$$\"3%HL$e9S8&\\#F1$\"3IM^*o:]RQ \"F[o7$$\"3s++D1#=bq#F1$\"3dQ4Y9y%>?\"F[o7$$\"3\"HLL$3s?6HF1$\"3?\\`8z ZRb5F[o7$$\"3a***\\7`Wl7$F1$\"3%*)y\"f(o+0G*F-7$$\"3enmmm*RRL$F1$\"3Gj 'e9@CTD)F-7$$\"3%zmmTvJga$F1$\"3-rNN>+%oO(F-7$$\"3]MLe9tOcPF1$\"34!oGO I/!=mF-7$$\"31,++]Qk\\RF1$\"3qw=$395U-'F-7$$\"3![LL3dg6<%F1$\"3uTG`9>? 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X%\\$F[o7$F^y$!3wt9q/-V_SF[o7$Fcy$!3m1$yqvC9Y%F[o7$Fhy$!3Iy/?`0hXZF[o7 $F]z$!3(e@*ePMA8\\F[o7$$\"3V,+DJ?i7))F[o$!3sy'>Wt@.'\\F[o7$Fbz$!3QGn(f '[B))\\F[o7$$\"3W-+v$44,')*F[o$!3]`s^rQ]**\\F[o7$Fgz$!3!3gz?6bk*\\F[o7 $F\\[l$!3uc'))GZsIF[o7$F^^l$!32(>0s')*e,HF[o7$Fc^l$!35.8%)fa(=v#F[o7$ Fh^l$!3mb$\\O'[I7EF[o7$F]_l$!3khAR0b'f[#F[o7$Fb_l$!3UEWQ`%[$zBF[o7$Fg_ l$!3)RsT?**4rE#F[o7$F\\`l$!3MJj#H*Qyu@F[o7$Fa`l$!3wm_C'\\LQ3#F[o7$Ff`l $!3CA-*=d#G.?F[o7$F[al$!3=Bp2Bp2B>F[o-F^al6&F`al$\"\"'FcalFaalF*-Fgal6 #%%B(w)G-%&TITLEG6#%KFourier~transform~F(w)=A(w)+i~B(w)~of~f(x)G-%+AXE SLABELSG6$Q\"w6\"Q!Fc`m-%%VIEWG6$;F(F[al%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "A(w)" "B(w)" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We can also plot the g raph of " }{XPPEDIT 18 0 "abs(F(omega)) = 1/(sqrt(1+omega^2));" "6#/- %$absG6#-%\"FG6#%&omegaG*&\"\"\"F,-%%sqrtG6#,&F,F,*$F*\"\"#F,!\"\"" } {TEXT -1 25 " . . . . the spectrum of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 2 ". 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\"\"F.F*-%$expG6#,$*(%\"iGF*%&omegaGF*F,F*F.F*/F,;\"\"!F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -i/omega-1/(a*omega^2);" "6#/%!G,&*&%\"iG\"\" \"%&omegaG!\"\"F**&F(F(*&%\"aGF(*$F)\"\"#F(F*F*" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(exp(-i*a*omega)-1);" "6#-%!G6#,&-%$expG6#,$*(%\"iG\" \"\"%\"aGF-%&omegaGF-!\"\"F-F-F0" }{TEXT -1 19 " (See example 5) " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -i/omega-1/(a*o mega^2);" "6#/%!G,&*&%\"iG\"\"\"%&omegaG!\"\"F**&F(F(*&%\"aGF(*$F)\"\" #F(F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(cos*a*omega-i*sin*a*omega- 1);" "6#-%!G6#,(*(%$cosG\"\"\"%\"aGF)%&omegaGF)F)**%\"iGF)%$sinGF)F*F) F+F)!\"\"F)F/" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "so that \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "F(omega) = (1-cos*a*omega)/(a*omega^2)+i*(sin*a*omeg a-a*omega)/(a*omega^2);" "6#/-%\"FG6#%&omegaG,&*&,&\"\"\"F+*(%$cosGF+% \"aGF+F'F+!\"\"F+*&F.F+*$F'\"\"#F+F/F+*(%\"iGF+,&*(%$sinGF+F.F+F'F+F+* &F.F+F'F+F/F+*&F.F+*$F'F2F+F/F+" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "In particular, if " } {XPPEDIT 18 0 "a = 2" "6#/%\"aG\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "F(omega) = (1-cos*2*omega)/(2*omega^2)+i*(sin*2*omega-2*omega)/(2*ome ga^2);" "6#/-%\"FG6#%&omegaG,&*&,&\"\"\"F+*(%$cosGF+\"\"#F+F'F+!\"\"F+ *&F.F+*$F'F.F+F/F+*(%\"iGF+,&*(%$sinGF+F.F+F'F+F+*&F.F+F'F+F/F+*&F.F+* $F'F.F+F/F+" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 58 "We can plot the graphs of the real and im aginary parts of " }{XPPEDIT 18 0 "F(omega)" "6#-%\"FG6#%&omegaG" } {TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A(om ega) = (1-cos*2*omega)/(2*omega^2);" "6#/-%\"AG6#%&omegaG*&,&\"\"\"F** (%$cosGF*\"\"#F*F'F*!\"\"F**&F-F**$F'F-F*F." }{TEXT -1 7 " and " } {XPPEDIT 18 0 "B(omega) = (sin*2*omega-2*omega)/(2*omega^2);" "6#/-%\" BG6#%&omegaG*&,&*(%$sinG\"\"\"\"\"#F,F'F,F,*&F-F,F'F,!\"\"F,*&F-F,*$F' F-F,F/" }{TEXT -1 2 ". 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8)4XX24N'Fcv7$Fc]l$!3A!eNX!>aijFcv7$$\"3Y]7`W2/6;Ffn$!3u/'RNsd?O'Fcv7$ $\"3!3](=UFcv7$Fhbl$!3PMlr&=0Ph\"Fcv7$Fbcl$!3O..8d&)*HY\"Fcv7$F\\dl$!3Qd %\\kum[N\"Fcv7$Fadl$!3G^p'o%Ht(H\"Fcv7$Ffdl$!3UDx?E.&fF\"Fcv7$F[el$!3u M>#R=+,F\"Fcv7$Feel$!3K*[#)fr!R47Fcv7$F_fl$!3]pmE5TEr5Fcv7$Fifl$!3t#p# GLX8T&*Fin7$F^gl$!3#)*\\,n@I06*Fin7$Fcgl$!3cmgsYliy!*Fin7$Fhgl$!3M:z#o `FCz)Fin7$F]hl$!3QX5T\\S4D!)Fin7$Fbhl$!3%yBf0gU>P(Fin7$Fghl$!3GC!QEDq) )3(Fin7$F\\il$!3Q&H]vZ3!pqFin7$Fail$!3=8%)z\\#Hi)oFin-Fdil6&Ffil$\"\"' FiilFgilFjil-F^jl6#%%B(w)G-%&TITLEG6#%KFourier~transform~F(w)=A(w)+i~B (w)~of~f(x)G-%+AXESLABELSG6$Q\"w6\"Q!F\\an-%%VIEWG6$;F(Fail%(DEFAULTG " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "A(w)" "B(w) " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We can also plot the grap h of " }{XPPEDIT 18 0 "abs(F(omega)) = sqrt(2-2*cos*2*w-4*sin*2*w*w+4* w^2)/(2*w^2);" "6#/-%$absG6#-%\"FG6#%&omegaG*&-%%sqrtG6#,*\"\"#\"\"\"* *F0F1%$cosGF1F0F1%\"wGF1!\"\"*,\"\"%F1%$sinGF1F0F1F4F1F4F1F5*&F7F1*$F4 F0F1F1F1*&F0F1*$F4F0F1F5" }{TEXT -1 23 " . . . the spectrum of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "plot(sqr t(2-2*cos(2*w)-4*sin(2*w)*w+4*w^2)/(2*w^2),\n w=-20..20,0..1,colo r=COLOR(RGB,0,.7,0),title=`spectrum of f(x)`);" }}{PARA 13 "" 1 "" {GLPLOT2D 564 233 233 {PLOTDATA 2 "6'-%'CURVESG6#7[r7$$!#?\"\"!$\"3a(e b^'HG6\\!#>7$$!3QLLL$Q6G\">!#;$\"39#oZyN$pb^F-7$$!3bmm;M!\\p$=F1$\"3c! 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\\w8=Fdq7$$\"3ElmmT30piF\\q$\"3_$)\\r\"zL()f\"Fdq7$$\"3_ILLL&4Nn'F\\q$ \"3V>NI*fD)>9Fdq7$$\"3O)****\\_K%*)oF\\q$\"3Wh\\kkRda8Fdq7$$\"3?mmm;bN 0rF\\q$\"3[twY#3JHJ\"Fdq7$$\"30MLL3&y7K(F\\q$\"3o[v>]@q\"H\"Fdq7$$\"3* =+++],s`(F\\q$\"3U.cub'=XG\"Fdq7$$\"3OLL$e9=&GzF\\q$\"3]S,8BY(GG\"Fdq7 $$\"3%[mm;zM)>$)F\\q$\"3a9YZy5>l7Fdq7$$\"3wML$eCZwu)F\\q$\"3j%eAWup!37 Fdq7$$\"3m/+++(fa<*F\\q$\"3(*)G!G\"4*G=6Fdq7$$\"3Aom;zy*zd*F\\q$\"3))* 4p/&4iF5Fdq7$$\"3!=LL$eg`!)**F\\q$\"3/]ly<^kt&*F-7$$\"3umm;W/8S5F1$\"3 V_w\\>bE2#*F-7$$\"3I++]#G2A3\"F1$\"3D2&)QaN$G8*F-7$$\"3tmm\"H3XL7\"F1$ \"3=Q%*=\"*>C9\"*F-7$$\"3;LLL$)G[k6F1$\"3u9K*o>w[&*)F-7$$\"3\\mm\"zM]v ?\"F1$\"3Wr7Tnmon&)F-7$$\"3#)****\\7yh]7F1$\"3Kw73P0WM!)F-7$$\"3ULLe** o4#H\"F1$\"3K9V&y7pYa(F-7$$\"3/nmm')fdL8F1$\"3zjSeB2wAsF-7$$\"37nm;WV* fP\"F1$\"3TO)Q!>[H,rF-7$$\"3=nmm,FT=9F1$\"31]u<'R134(F-7$$\"3!QLLe#pa- :F1$\"3]0V[i0UuoF-7$$\"3W+++Sv&)z:F1$\"3!>P2[@!e$H'F-7$$\"3#RLL$GUYo;F 1$\"3Hy@4[d7KeF-7$$\"39nmm1^rZF1$\"39FYzCvJa^F-7$$\"#?F*F+-%&TITLEG6#% 1spectrum~of~f(x)G-%+AXESLABELSG6$Q\"w6\"Q!Fg^m-%&COLORG6&%$RGBG$F*F*$ \"\"(!\"\"F]_m-%%VIEWG6$;F(F]^m;F]_m$\"\"\"F*" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Using the Heaviside function, w e have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = H(x )*(1-x/a)-H(x-a)*(1-x/a);" "6#/-%\"fG6#%\"xG,&*&-%\"HG6#F'\"\"\",&F-F- *&F'F-%\"aG!\"\"F1F-F-*&-F+6#,&F'F-F0F1F-,&F-F-*&F'F-F0F1F1F-F1" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "alias(H=Heaviside):\ninterface(showassumed=0): \+ assume(a>0):\nH(x)*(1-x/a)-H(x-a)*(1-x/a);\ninttrans[fourier](%,x,omeg a):\n`Fourier transform`=simplify(convert(simplify(%),trig));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%\"HG6#%\"xG\"\"\",&F)F)*&F(F)%#a |irG!\"\"F-F)F)*&-F&6#,&F(F)F,F-F)F*F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,$*(,*\"\"\"!\"\"-%$cosG6#*&%#a|irGF(%&omeg aGF(F(*&^#F)F(-%$sinGF,F(F(*(F.F(F/F(^#F(F(F(F(F.F)F/!\"#F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Fourier transform of the function " } {XPPEDIT 18 0 "f(x) = x*exp(-a*x^2);" "6#/-%\"fG6#%\"xG*&F'\"\"\"-%$ex pG6#,$*&%\"aGF)*$F'\"\"#F)!\"\"F)" }{TEXT -1 8 ", where " }{TEXT 275 1 "a" }{TEXT -1 13 " is positive." }}{PARA 0 "" 0 "" {TEXT -1 32 "The \+ graph is drawn for the case " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f := x -> x*exp(-x^2);\nplot(f(x),x=-3..3,color= red,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(*&9$\"\"\"-%$expG6#,$*$)F-\"\"#F.!\"\"F.F(F(F (" }}{PARA 13 "" 1 "" {GLPLOT2D 580 164 164 {PLOTDATA 2 "6'-%'CURVESG6 #7co7$$!\"$\"\"!$!3%oQ+E7%H-P!#@7$$!3!******\\2<#pG!#<$!3Ac%)GgL7HwF-7 $$!3#)***\\7bBav#F1$!3=!HczpK&*Q\"!#?7$$!36++]K3XFEF1$!3k6D1#zg'QEF97$ $!3%)****\\F)H')\\#F1$!3S*3H*)oRm&[F97$$!3#****\\i3@/P#F1$!3xd\\r``T,' )F97$$!3;++Dr^b^AF1$!3Ls,(y,G_T\"!#>7$$!3$****\\7Sw%G@F1$!3=L$)fVnp$H# FN7$$!3*****\\7;)=,?F1$!3/9cHi-#zk$FN7$$!3/++DO\"3V(=F1$!3cSt=\\Uk'e&F N7$$!3#******\\V'zViUC\"F1$!3GI[-,*4dk#Fbo7$$!3-++DhkaI6F1$!3? P^!Rr2#\\JFbo7$$!3s******\\XF`**Fbo$!3eB#z2)) Fbo$!3[T$['H;laSFbo7$$!31++Dc!e:9)Fbo$!3%)\\%o1&3+'>%Fbo7$$!3S++]7RKvu Fbo$!3!Q<#yE)z]F%Fbo7$$!3c+v$fQ\"*yK(Fbo$!3;x#)\\\"3KKG%Fbo7$$!3u+]Pf) e/=(Fbo$!3A&\\%3W$)z(G%Fbo7$$!3*3]7GLEI.(Fbo$!3Y&=X92&p)G%Fbo7$$!31,+D 1Qf&)oFbo$!3IB98!QVeG%Fbo7$$!3S,]7`(G2f'Fbo$!3+bH%)HTgoUFbo7$$!3s,+++P 'eH'Fbo$!3wc5_fGbNUFbo7$$!3A++D1j$)[cFbo$!3#=l.O.m1$Fbo$ \"3)36Uh([P\"z#Fbo7$$\"3E,++vuY)o$Fbo$\"3ejz=ytH>KFbo7$$\"3e****\\(G(* 3L%Fbo$\"3K]1iP/?!f$Fbo7$$\"3!z******4FL(\\Fbo$\"3OCD?J#[N)QFbo7$$\"30 )***\\P$>=g&Fbo$\"3c+cD)fQI4%Fbo7$$\"3A)****\\d6.B'Fbo$\"3WI6f#*4,EUFb o7$$\"3f)*\\7`3OMlFbo$\"3y#za3.BNE%Fbo7$$\"3(*)**\\785%QoFbo$\"3iT$y!R n7%G%Fbo7$$\"39*\\7.xM/*pFbo$\"3aQ<&>jf#)G%Fbo7$$\"3N**\\P4%fC9(Fbo$\" 3o%Fbo7$$\"35 *****\\A))oz)Fbo$\"3uA>)Q=8u0%Fbo7$$\"3e******Hk-,5F1$\"31x3d2I^i\"HE Fbo7$$\"3M+++b*=jP\"F1$\"3u(oD4'RSq?Fbo7$$\"3g***\\(3/3(\\\"F1$\"3)f/% )G%)yw7#F1$\"3,+J*)eV:,BFN7$$\"3O++v)Q?QD# F1$\"3sJWR)\\nAS\"FN7$$\"3G+++5jypBF1$\"3mf=XAa-D')F97$$\"3<++]Ujp-DF1 $\"3KGFD0#3mw%F97$$\"3++++gEd@EF1$\"3\\:EB*ec_r#F97$$\"39++v3'>$[FF1$ \"32=N@!z67W\"F97$$\"37++D6EjpGF1$\"3Z)>@d7a?h(F-7$$\"\"$F*$\"3%oQ+E7% H-PF--%+AXESLABELSG6$Q\"x6\"Q!F^bl-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F *Fgbl-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(Ffal%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "F(omega) = Int(f(x)*exp(-i*omega*x),x = -infinity .. in finity);" "6#/-%\"FG6#%&omegaG-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#, $*(%\"iGF0F'F0F/F0!\"\"F0/F/;,$%)infinityGF7F;" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(x*exp(-a*x^2 )*exp(-i*omega*x),x = -infinity .. infinity);" "6#/%!G-%$IntG6$*(%\"xG \"\"\"-%$expG6#,$*&%\"aGF**$F)\"\"#F*!\"\"F*-F,6#,$*(%\"iGF*%&omegaGF* F)F*F3F*/F);,$%)infinityGF3F=" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = sqrt(a)*x+i*omega/(2*sqrt(a));" "6#/%\"uG,&*&-%%sqrtG6#%\"aG\"\"\"%\"xGF+F+*(%\"iGF+%&omegaGF+*&\"\"#F +-F(6#F*F+!\"\"F+" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "u^2 = a* x^2+i*omega*x-omega^2/(4*a);" "6#/*$%\"uG\"\"#,(*&%\"aG\"\"\"*$%\"xGF& F*F**(%\"iGF*%&omegaGF*F,F*F**&F/F&*&\"\"%F*F)F*!\"\"F3" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x =u/sqrt(a)-i*omega/(2*a)" "6#/%\"xG,&*&%\"uG\" \"\"-%%sqrtG6#%\"aG!\"\"F(*(%\"iGF(%&omegaGF(*&\"\"#F(F,F(F-F-" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "du/dx=sqrt(a)" "6#/*&%#duG\"\"\"%# dxG!\"\"-%%sqrtG6#%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(omega)=1/s qrt(a)" "6#/-%\"FG6#%&omegaG*&\"\"\"F)-%%sqrtG6#%\"aG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-omega^2/(4*a))*Int((u/sqrt(a)-i*omega/(2*a) )*exp(-u^2),u = -infinity .. infinity);" "6#*&-%$expG6#,$*&%&omegaG\" \"#*&\"\"%\"\"\"%\"aGF-!\"\"F/F--%$IntG6$*&,&*&%\"uGF--%%sqrtG6#F.F/F- *(%\"iGF-F)F-*&F*F-F.F-F/F/F--F%6#,$*$F6F*F/F-/F6;,$%)infinityGF/FDF- " }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(omega) = exp(-omega^2/(4*a));" "6#/-%\"FG6#%&omegaG-%$expG6#,$*&F' \"\"#*&\"\"%\"\"\"%\"aGF0!\"\"F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(` `(1/a)*Int(u*exp(-u^2),u = -infinity .. infinity)-``(i*omega/(2*a*sqrt (a)))*Int(exp(-u^2),u = -infinity .. infinity));" "6#-%!G6#,&*&-F$6#*& \"\"\"F+%\"aG!\"\"F+-%$IntG6$*&%\"uGF+-%$expG6#,$*$F2\"\"#F-F+/F2;,$%) infinityGF-F " 0 "" {MPLTEXT 1 0 109 "inte rface(showassumed=0): assume(a>0):\nInt(x*exp(-a*x^2)*exp(-I*omega*x), x=-infinity..infinity);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(%\"xG\"\"\"-%$expG6#,$*&%#a|irGF()F'\"\"#F(!\"\"F(-F*6#* (^#F1F(%&omegaGF(F'F(F(/F';,$%)infinityGF1F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*,^##!\"\"\"\"#\"\"\"-%$expG6#,$*(\"\"%F(%#a|irGF(% &omegaGF)F(F*F2F*F1#!\"$F)%#PiG#F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 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