{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 0 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " 259 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 259 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {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 "Time s" 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 40 "Some properties of the Fourier tr ansform" }}{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 "res tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "Shifting and symmetry properties of the F ourier transform" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Shifting properties" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "Fr*[f(x)] = F(omega);" "6#/*&%#FrG\"\"\"7#-%\"fG6# %\"xGF&-%\"FG6#%&omegaG" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "Fr*[f(x )*exp(i*a*x)] = F(omega-a);" "6#/*&%#FrG\"\"\"7#*&-%\"fG6#%\"xGF&-%$ex pG6#*(%\"iGF&%\"aGF&F,F&F&F&-%\"FG6#,&%&omegaGF&F2!\"\"" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT 262 29 "_____________________________" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "Fr*[f(x)] = F(omega);" "6#/*&%#FrG\" \"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Fr*[f(x)*exp(i*a*x)] = Int(f(x)*exp(i*a*x)*exp(-i*omega*x),x = -inf inity .. infinity);" "6#/*&%#FrG\"\"\"7#*&-%\"fG6#%\"xGF&-%$expG6#*(% \"iGF&%\"aGF&F,F&F&F&-%$IntG6$*(-F*6#F,F&-F.6#*(F1F&F2F&F,F&F&-F.6#,$* (F1F&%&omegaGF&F,F&!\"\"F&/F,;,$%)infinityGFAFE" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f(x)*exp(-i* (omega-a)*x),x = -infinity .. infinity);" "6#/%!G-%$IntG6$*&-%\"fG6#% \"xG\"\"\"-%$expG6#,$*(%\"iGF-,&%&omegaGF-%\"aG!\"\"F-F,F-F7F-/F,;,$%) infinityGF7F;" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = F(omega-a);" "6#/%!G-%\"FG6#,&%&omegaG\"\"\"%\"aG! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 269 31 "________________ _______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "Fr*[f(x)] = F(omega) ;" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" }{TEXT -1 7 ", \+ then " }{XPPEDIT 18 0 "Fr*[f(x-a)] = F(omega)*exp(-i*omega*a);" "6#/*& %#FrG\"\"\"7#-%\"fG6#,&%\"xGF&%\"aG!\"\"F&*&-%\"FG6#%&omegaGF&-%$expG6 #,$*(%\"iGF&F3F&F-F&F.F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 263 31 "_______________________________" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "Fr *[f(x)] = F(omega);" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omega G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Fr*[f(x-a)] = Int(f(x-a)*exp(-i*o mega*x),x = -infinity .. infinity);" "6#/*&%#FrG\"\"\"7#-%\"fG6#,&%\"x GF&%\"aG!\"\"F&-%$IntG6$*&-F)6#,&F,F&F-F.F&-%$expG6#,$*(%\"iGF&%&omega GF&F,F&F.F&/F,;,$%)infinityGF.F@" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "u = x - a" "6#/%\"uG,&%\"xG\"\" \"%\"aG!\"\"" }{TEXT -1 17 " in the integral," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Fr*[f(x-a)] = Int(f(u)*exp(-i*omega*(u +a)),u = -infinity .. infinity);" "6#/*&%#FrG\"\"\"7#-%\"fG6#,&%\"xGF& %\"aG!\"\"F&-%$IntG6$*&-F)6#%\"uGF&-%$expG6#,$*(%\"iGF&%&omegaGF&,&F5F &F-F&F&F.F&/F5;,$%)infinityGF.FA" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-i*omega*a)*Int(f(u)*exp(-i*om ega*u),u = -infinity .. infinity);" "6#/%!G*&-%$expG6#,$*(%\"iG\"\"\"% &omegaGF,%\"aGF,!\"\"F,-%$IntG6$*&-%\"fG6#%\"uGF,-F'6#,$*(F+F,F-F,F7F, F/F,/F7;,$%)infinityGF/F?F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = F(omega)*exp(-i*omega*a);" "6#/%!G *&-%\"FG6#%&omegaG\"\"\"-%$expG6#,$*(%\"iGF*F)F*%\"aGF*!\"\"F*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 266 36 "_______________________ _____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "Fr*[f(x)] = F(omega) ;" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" }{TEXT -1 7 ", \+ then " }{XPPEDIT 18 0 "Fr*[f(x)*cos*a*x] = (F(omega-a)+F(omega+a))/2; " "6#/*&%#FrG\"\"\"7#**-%\"fG6#%\"xGF&%$cosGF&%\"aGF&F,F&F&*&,&-%\"FG6 #,&%&omegaGF&F.!\"\"F&-F26#,&F5F&F.F&F&F&\"\"#F6" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT 264 36 "____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "Fr*[f(x)] = F(omega);" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xG F&-%\"FG6#%&omegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[f(x)*cos*a*x] = Fr*[f(x)*``((exp(i*a*x)+exp(-i*a*x))/2)];" "6#/*&%#FrG\"\"\"7#**-% \"fG6#%\"xGF&%$cosGF&%\"aGF&F,F&F&*&F%F&7#*&-F*6#F,F&-%!G6#*&,&-%$expG 6#*(%\"iGF&F.F&F,F&F&-F:6#,$*(F=F&F.F&F,F&!\"\"F&F&\"\"#FBF&F&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" } {XPPEDIT 18 0 "``(Fr*[f(x)*exp(i*a*x)]+Fr*[f(x)*exp(-i*a*x)]);" "6#-%! G6#,&*&%#FrG\"\"\"7#*&-%\"fG6#%\"xGF)-%$expG6#*(%\"iGF)%\"aGF)F/F)F)F) F)*&F(F)7#*&-F-6#F/F)-F16#,$*(F4F)F5F)F/F)!\"\"F)F)F)" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (F(omega-a)+F(omega+a))/2;" "6#/%!G*&,&-%\"FG6#,&% &omegaG\"\"\"%\"aG!\"\"F,-F(6#,&F+F,F-F,F,F,\"\"#F." }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 38 "_______ _______________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "Fr*[ f(x)] = F(omega);" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG " }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "Fr*[f(x)*sin*a*x] = (F(omega+a )-F(omega-a))*i/2;" "6#/*&%#FrG\"\"\"7#**-%\"fG6#%\"xGF&%$sinGF&%\"aGF &F,F&F&*(,&-%\"FG6#,&%&omegaGF&F.F&F&-F26#,&F5F&F.!\"\"F9F&%\"iGF&\"\" #F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 265 38 "_________________ _____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "Fr*[f(x)] = F(om ega);" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[f(x)*sin*a*x] = Fr*[f(x)*``((exp(i*a*x)-e xp(-i*a*x))/(2*i))];" "6#/*&%#FrG\"\"\"7#**-%\"fG6#%\"xGF&%$sinGF&%\"a GF&F,F&F&*&F%F&7#*&-F*6#F,F&-%!G6#*&,&-%$expG6#*(%\"iGF&F.F&F,F&F&-F:6 #,$*(F=F&F.F&F,F&!\"\"FBF&*&\"\"#F&F=F&FBF&F&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= 1/(2*i)" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"iGF&!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``(Fr*[f(x)*exp(i*a*x)]-Fr*[f(x)*exp(-i *a*x)]);" "6#-%!G6#,&*&%#FrG\"\"\"7#*&-%\"fG6#%\"xGF)-%$expG6#*(%\"iGF )%\"aGF)F/F)F)F)F)*&F(F)7#*&-F-6#F/F)-F16#,$*(F4F)F5F)F/F)!\"\"F)F)F? " }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = (F(omega-a)-F(omega+a))/(2*i);" "6#/%!G*&,&-%\"FG6#,&%&omegaG\"\" \"%\"aG!\"\"F,-F(6#,&F+F,F-F,F.F,*&\"\"#F,%\"iGF,F." }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (F(omega+a)-F(o mega-a))*i/2;" "6#/%!G*(,&-%\"FG6#,&%&omegaG\"\"\"%\"aGF,F,-F(6#,&F+F, F-!\"\"F1F,%\"iGF,\"\"#F1" }{TEXT -1 1 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Symmetry property" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "Fr*[f(x)] = F(omega);" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" } {TEXT -1 7 ", then " }{XPPEDIT 18 0 "Fr*[F(x)] = 2*Pi*f(-omega);" "6#/ *&%#FrG\"\"\"7#-%\"FG6#%\"xGF&*(\"\"#F&%#PiGF&-%\"fG6#,$%&omegaG!\"\"F &" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 268 26 "___________________ _______" }{TEXT 260 1 " " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "Fr*[f(x)] = F (omega);" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*Pi*f(x) = Int(F(omega)*exp(i*omega*x ),omega = -infinity .. infinity);" "6#/*(\"\"#\"\"\"%#PiGF&-%\"fG6#%\" xGF&-%$IntG6$*&-%\"FG6#%&omegaGF&-%$expG6#*(%\"iGF&F3F&F+F&F&/F3;,$%)i nfinityG!\"\"F<" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*Pi*f(-x) = Int(F (omega)*exp(-i*omega*x),omega = -infinity .. infinity);" "6#/*(\"\"#\" \"\"%#PiGF&-%\"fG6#,$%\"xG!\"\"F&-%$IntG6$*&-%\"FG6#%&omegaGF&-%$expG6 #,$*(%\"iGF&F5F&F,F&F-F&/F5;,$%)infinityGF-F?" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 28 "Interchanging the variables " }{TEXT 270 1 "t" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 22 " in the integral gives" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "2*Pi*f(-omega) = Int(F(t)*exp(-i*omega*x),x = -infinity .. infinity);" "6#/*(\"\"#\"\"\"%#PiGF&-%\"fG6#,$%&omegaG!\"\"F&-%$In tG6$*&-%\"FG6#%\"tGF&-%$expG6#,$*(%\"iGF&F,F&%\"xGF&F-F&/F<;,$%)infini tyGF-F@" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Fr*[F(x)];" "6#/%!G*&%#FrG\"\"\"7#-%\"FG6#%\"xGF' " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "E xamples" }}{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 "g(x)" "6#-%\"gG6#%\"xG" } {TEXT -1 10 " given by " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "g(x) = PIECEWISE([cos*a*x, abs(x) < d],[0, d <= abs(x)] );" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$*(%$cosG\"\"\"%\"aGF.F'F.2-%$abs G6#F'%\"dG7$\"\"!1F4-F26#F'" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "where a and d are positive." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 37 " is a \"fi nite width\" cosine function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG " }{TEXT -1 47 " can be expressed as the product of the \"pulse\"" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([1, abs(x) < d],[0, d <= abs(x)]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\" \"\"2-%$absG6#F'%\"dG7$\"\"!1F1-F/6#F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "H(x+d)-H(x+d);" "6#,&-%\"HG6#,&%\"xG\"\"\"%\"dGF)F)-F%6#,&F(F)F*F)! \"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 52 "where H is the H eaviside function, and the function " }{XPPEDIT 18 0 "cos*a*x;" "6#*(% $cosG\"\"\"%\"aGF%%\"xGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "For an example, take " }{XPPEDIT 18 0 "a=2" "6#/%\"aG\"\"#" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "d=11*Pi/4" "6#/%\"dG*(\"#6\"\"\"%#P iGF'\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "alias(H=Heaviside):\nf := x -> H(x+11*Pi/4)-H(x-11*Pi/4):\n'f(x)'=f(x);\ng := x -> f(x)*cos(2*x): \n'g(x)'=g(x);\nplot(g(x),x=-12..12,color=red,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%\"HG6#,&F'\"\"\"*(\"#6F-\" \"%!\"\"%#PiGF-F-F--F*6#,&F'F-*(F/F-F0F1F2F-F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&,&-%\"HG6#,&F'\"\"\"*(\"#6F.\"\"%!\"\"% #PiGF.F.F.-F+6#,&F'F.*(F0F.F1F2F3F.F2F2F.-%$cosG6#,$*&\"\"#F.F'F.F.F. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "alias(H=Heaviside):\ninterface(sho wassumed=0): assume(a>0):\nf := x -> H(x+d)-H(x-d):\n'f(x)'=f(x);\n'f( x)*cos(a*x)'=f(x)*cos(a*x);\n`Fourier transform`=inttrans[fourier](rhs (%),x,omega);\n``=combine(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6#%\"xG,&-%\"HG6#,&F'\"\"\"%\"dGF-F--F*6#,&F'F-F.!\"\"F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"fG6#%\"xG\"\"\"-%$cosG6#*&%\"aG F)F(F)F)*&,&-%\"HG6#,&F(F)%\"dGF)F)-F26#,&F(F)F5!\"\"F9F)-F+6#*&%#a|ir GF)F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,&*& -%$sinG6#*&%\"dG\"\"\",&%&omegaG!\"\"%#a|irGF,F,F,F-F/F,*&,&F.F,F0F,F/ -F(6#*&F+F,F2F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,**&-%$si nG6#,&*&%\"dG\"\"\"%&omegaGF.!\"\"*&%#a|irGF.F-F.F.F.F/F.F.*&F(F.F2F.F .*&-F)6#,&F,F.F1F.F.F/F.F0*&F5F.F2F.F.F.,&*$)F/\"\"#F.F0*$)F2F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "For the specific case whe re " }{XPPEDIT 18 0 "a = 2" "6#/%\"aG\"\"#" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "d = 11*Pi/4" "6#/%\"dG*(\"#6\"\"\"%#PiGF'\"\"%!\"\"" } {TEXT -1 28 ", the Fourier transform is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "G(omega)= sin(11*Pi/4*(omega-2))/(omega-2)+si n(11*Pi/4*(omega+2))/(omega+2)" "6#/-%\"GG6#%&omegaG,&*&-%$sinG6#**\"# 6\"\"\"%#PiGF/\"\"%!\"\",&F'F/\"\"#F2F/F/,&F'F/F4F2F2F/*&-F+6#**F.F/F0 F/F1F2,&F'F/F4F/F/F/,&F'F/F4F/F2F/" }{TEXT -1 2 ". 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"6#/%\"dG\"$+\"" }{TEXT -1 28 ", the Fourier transform is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "G(omega) = s in(100*(omega-2))/(omega-2)+sin(100*(omega+2))/(omega+2);" "6#/-%\"GG6 #%&omegaG,&*&-%$sinG6#*&\"$+\"\"\"\",&F'F/\"\"#!\"\"F/F/,&F'F/F1F2F2F/ *&-F+6#*&F.F/,&F'F/F1F/F/F/,&F'F/F1F/F2F/" }{TEXT -1 2 ". 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" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "g(x) = exp(-b*abs(x))*cos*a*x;" "6#/-%\"gG6#%\"xG**-%$e xpG6#,$*&%\"bG\"\"\"-%$absG6#F'F/!\"\"F/%$cosGF/%\"aGF/F'F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 273 1 "a" } {TEXT -1 5 " and " }{TEXT 274 1 "b" }{TEXT -1 14 " are positive." }} {PARA 0 "" 0 "" {TEXT -1 21 "For an example, take " }{XPPEDIT 18 0 "a= 2" "6#/%\"aG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b = 1/10;" "6#/ %\"bG*&\"\"\"F&\"#5!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "g := x -> exp(-abs(x) /10)*cos(2*x);\nplot(g(x),x=-30..30,color=red);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$expG6#,$- %$absG6#9$#!\"\"\"#5\"\"\"-%$cosG6#,$F4\"\"#F8F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 522 221 221 {PLOTDATA 2 "6&-%'CURVESG6#7]al7$$!#I\"\"!$!1 Xq.<]yTZ!#<7$$!1+]PM@l$)H!#9$!1=7]zo,g]F-7$$!1++voUInHF1$!1<2b\"z%\\U[ 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&46Ffu7$$\"1]il()[\\1AF1$\"1`vs,&))))3\"Ffu7$$\"1v=$[FF1$!13XM%fF1N(Fcr7$$\"1+]PfykyFF1$\"1`[C'yeC[$F -7$$\"1+++5h(*3GF1$\"1Cd!4C'o?cF-7$$\"1+DJN-9CGF1$\"1JINOn4BfF-7$$\"1+ ]igVIRGF1$\"102fTMi#o&F-7$$\"1+v$f[oW&GF1$\"1sjR5y:P\\F-7$$\"1++D6EjpG F1$\"1#GInUu*oPF-7$$\"1+vVeWA-HF1$\"1%ecE!*RB6%FH7$$\"1+]i0j\"[$HF1$!1 L4CmGE)*GF-7$$\"1](=#HA6^HF1$!1GWB4HP/TF-7$$\"1+D\"G:3u'HF1$!1$RjsE(eX [F-7$$\"1]iSwSq$)HF1$!1&)GmQ=%)f]F-7$$\"#IF*F+-%+AXESLABELSG6$Q\"x6\"% !G-%'COLOURG6&%$RGBG$\"*++++\"!\")F*F*-%%VIEWG6$;F(Fiiq%(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 4 "Let " }{XPPEDIT 18 0 "f(x)=exp(- b*abs(x))" "6#/-%\"fG6#%\"xG-%$expG6#,$*&%\"bG\"\"\"-%$absG6#F'F.!\"\" " }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" } {TEXT -1 14 " has the form " }{XPPEDIT 18 0 "g(x) = f(x)*cos*a*x;" "6# /-%\"gG6#%\"xG**-%\"fG6#F'\"\"\"%$cosGF,%\"aGF,F'F," }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 27 "By the cosine shift formula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[g(x)] = Fr*[f(x)*cos*a* x];" "6#/*&%#FrG\"\"\"7#-%\"gG6#%\"xGF&*&F%F&7#**-%\"fG6#F+F&%$cosGF&% \"aGF&F+F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (F(omega-a)+F(omega+a) )/2;" "6#/%!G*&,&-%\"FG6#,&%&omegaG\"\"\"%\"aG!\"\"F,-F(6#,&F+F,F-F,F, F,\"\"#F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "F(omega)=Fr*[f(x)]" "6#/-%\"FG6#%&omegaG*&%#FrG\"\"\"7# -%\"fG6#%\"xGF*" }{XPPEDIT 18 0 "`` = 2*b/(b^2+omega^2);" "6#/%!G*(\" \"#\"\"\"%\"bGF',&*$F(F&F'*$%&omegaGF&F'!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Fr*[g(x)] = b/(b^2+(omega-a)^2)+b/(b^2+(omega+a)^2); " "6#/*&%#FrG\"\"\"7#-%\"gG6#%\"xGF&,&*&%\"bGF&,&*$F.\"\"#F&*$,&%&omeg aGF&%\"aG!\"\"F1F&F6F&*&F.F&,&*$F.F1F&*$,&F4F&F5F&F1F&F6F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "interface(showassumed=0 ):\nassume(a>0,b>0):\nexp(-b*abs(x))*cos(a*x);\n`Fourier transform`=in ttrans[fourier](%,x,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$ex pG6#,$*&%#b|irG\"\"\"-%$absG6#%\"xGF*!\"\"F*-%$cosG6#*&%#a|irGF*F.F*F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,$*,\"\"#\"\" \"%#b|irGF(,(*$)F)F'F(F(*$)%&omegaGF'F(F(*$)%#a|irGF'F(F(F(,*F+F(F-F(* (F'F(F/F(F2F(!\"\"F0F(F5,*F+F(F-F(*(F'F(F/F(F2F(F(F0F(F5F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "For the specific case where " } {XPPEDIT 18 0 "a = 2" "6#/%\"aG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b = 1/10;" "6#/%\"bG*&\"\"\"F&\"#5!\"\"" }{TEXT -1 26 ", the Fou rier transform is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " G(omega) = 1/(1/10+10*(omega-2)^2)+1/(1/10+10*(omega+2)^2);" "6#/-%\"G G6#%&omegaG,&*&\"\"\"F*,&*&F*F*\"#5!\"\"F**&F-F**$,&F'F*\"\"#F.F2F*F*F .F**&F*F*,&*&F*F*F-F.F**&F-F**$,&F'F*F2F*F2F*F*F.F*" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "plot(1/(1/10+10*(w-2)^2)+1/(1/10+10*(w+2)^2),w=-6..6,\ncolor=CO LOR(RGB,.4,0,.9),title=`Fourier transform G(w) of g(x)`);" }}{PARA 13 "" 1 "" {GLPLOT2D 516 330 330 {PLOTDATA 2 "6'-%'CURVESG6#7ar7$$!\"'\" \"!$\"3\"yL(R(3_$3y!#?7$$!3z******\\TVQd!#<$\"3%3bKk^(p>))F-7$$!3l**** \\-r%3^&F1$\"3!**o7+mc'y)*F-7$$!3A+++l;!\\D&F1$\"3ShSJeF'H8\"!#>7$$!3o *****\\lfs*\\F1$\"3!*e$))p(e5;8F>7$$!3%)****\\s@%3u%F1$\"3_S:H*eA%\\:F >7$$!3J++]U.6.XF1$\"3Fl\"*)e2())H=F>7$$!3')****\\-G&pD%F1$\"3\\z!y)pnn 9AF>7$$!3(*****\\AjP-SF1$\"3%34,!=dMlFF>7$$!33++]sih[PF1$\"3CTtv@kKiNF >7$$!3%)******pGf([$F1$\"3G7$$!3)******\\J$odKF1$\"3g'He*)eY Rk'F>7$$!3y******4'f))*HF1$\"3.(RlN(3MK5!#=7$$!3#*******zj\"*oGF1$\"3O IZ(ywF$\\8Ffo7$$!33+++]J(*QFF1$\"3Rr::ah!G%=Ffo7$$!3%*******p([Ph#F1$ \"3V\"\\\"\\*)G-LEFfo7$$!3#)*******QC&)[#F1$\"3QD%4f')H72%Ffo7$$!3#*** *\\il3[P#F1$\"3e!HG90cvp'Ffo7$$!3/++]AH4hAF1$\"3!=-ZvgvZG\"F17$$!3J+v$ 4<)GFAF1$\"3m%3+15#RF;F17$$!37+]P>M[$>#F1$\"3IPY6\"G%z8@F17$$!3/]PfV5e w@F1$\"3tqE%*z;/MCF17$$!3'**\\7ymy'f@F1$\"3Kvs$y_$*G#GF17$$!3')\\7.#Hw F9#F1$\"3Y9%3xF8pH$F17$$!3y***\\i\"R(e7#F1$\"31$)R0HZ?vQF17$$!3o\\(o/a r*3@F1$\"3MwrZc/RxXF17$$!3e*\\(ok\"p?4#F1$\"3:U/eI6==aF17$$!3yuozwzh$3 #F1$\"390UQUs7\"*eF17$$!3%*\\i!*)ym^2#F1$\"3+/oY_ax&R'F17$$!37Dc,,crm? 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~g(x)G-%+AXESLABELSG6$Q\"w6\"Q!Ff`m-%&COLORG6&%$RGBG$\"\"%!\"\"F*$\"\" *F^am-%%VIEWG6$;F(F\\`m%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Fourier transform of the function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = \+ exp(-b*x^2)*cos*a*x;" "6#/-%\"gG6#%\"xG**-%$expG6#,$*&%\"bG\"\"\"*$F' \"\"#F/!\"\"F/%$cosGF/%\"aGF/F'F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 271 1 "a" }{TEXT -1 5 " and " }{TEXT 272 1 "b" }{TEXT -1 14 " are positive." }}{PARA 0 "" 0 "" {TEXT -1 21 "For a n example, take " }{XPPEDIT 18 0 "a=2" "6#/%\"aG\"\"#" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "b = 1/100;" "6#/%\"bG*&\"\"\"F&\"$+\"!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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(F-7$$\"3!QLLe#pa-:F1$\"3q:F1$\"3!Q&)fV FZf\\)F-7$$\"3W+++Sv&)z:F1$\"3J'=[l=Vo5)F-7$$\"3=nm;%)3;C;F1$\"37p&H5% \\T]MF-7$$\"3#RLL$GUYo;F1$!3)4I1gHlrI#F-7$$\"3Cnm\"z%pF)o\"F1$!3Q4>96q 1iSF-7$$\"3`++]n'*33_#o]]F-7$$\"3J+](ou3yt\"F1$!3aV&3bfUUy %F-7$$\"39nmm1^rZJ \"F-7$$\"35++]sI@K=F1$\"3S*3.)p>$)=V_=F1$\"3%H(R#RxpMd#F-7$$\"31+]7B8ai=F1$\"313\"**yJxo!GF -7$$\"3G+++S2ls=F1$\"3b=mn7r%)3HF-7$$\"3]+](o:gF)=F1$\"3Wa)H=h(p%)GF-7 $$\"3O++vt&pG*=F1$\"30=3qQ?gWFF-7$$\"3A+]i!**yH!>F1$\"3tI88W=\"G]#F-7$ $\"3W++]2%)38>F1$\"3C.8**QOww@F-7$$\"3A++v.Uac>F1$\"3/MPg?IF9IF)7$$\"# ?F*F+-%+AXESLABELSG6$Q\"x6\"Q!Fi^q-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F *Fb_q-%%VIEWG6$;F(Fc^q%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = exp(-b*x^2);" "6#/-%\"fG6#%\"xG-%$expG6#,$*&%\"bG\"\"\"*$F' \"\"#F.!\"\"" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "g(x)" "6#-%\"g G6#%\"xG" }{TEXT -1 14 " has the form " }{XPPEDIT 18 0 "g(x) = f(x)*co s*a*x;" "6#/-%\"gG6#%\"xG**-%\"fG6#F'\"\"\"%$cosGF,%\"aGF,F'F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "By the cosine shift form ula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[g(x)] = Fr *[f(x)*cos*a*x];" "6#/*&%#FrG\"\"\"7#-%\"gG6#%\"xGF&*&F%F&7#**-%\"fG6# F+F&%$cosGF&%\"aGF&F+F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "(F(omega-a)+ F(omega+a))/2" "6#*&,&-%\"FG6#,&%&omegaG\"\"\"%\"aG!\"\"F*-F&6#,&F)F*F +F*F*F*\"\"#F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "F(omega)=Fr*[f(x)]" "6#/-%\"FG6#%&omegaG*&%#FrG\"\"\"7 #-%\"fG6#%\"xGF*" }{XPPEDIT 18 0 " ``=F(omega) " "6#/%!G-%\"FG6#%&omeg aG" }{XPPEDIT 18 0 "`` = sqrt(Pi/b)*exp(-w^2/(4*b));" "6#/%!G*&-%%sqrt G6#*&%#PiG\"\"\"%\"bG!\"\"F+-%$expG6#,$*&%\"wG\"\"#*&\"\"%F+F,F+F-F-F+ " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[g(x)] = 1/2" "6#/*&%#FrG\"\" \"7#-%\"gG6#%\"xGF&*&F&F&\"\"#!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 " ``(sqrt(Pi/b)*exp(-(omega-a)^2/(4*b))+sqrt(Pi/b)*exp(-(omega+a)^2/(4*b )));" "6#-%!G6#,&*&-%%sqrtG6#*&%#PiG\"\"\"%\"bG!\"\"F--%$expG6#,$*&,&% &omegaGF-%\"aGF/\"\"#*&\"\"%F-F.F-F/F/F-F-*&-F)6#*&F,F-F.F/F--F16#,$*& ,&F6F-F7F-F8*&F:F-F.F-F/F/F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ sqrt(Pi/b);" "6#/%!G-%%sqrtG6#*&%#PiG\"\"\"%\"bG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``((exp(1/(2*b*omega*a))+exp(-1/(2*b*omega*a)))/2); " "6#-%!G6#*&,&-%$expG6#*&\"\"\"F,**\"\"#F,%\"bGF,%&omegaGF,%\"aGF,!\" \"F,-F)6#,$*&F,F,**F.F,F/F,F0F,F1F,F2F2F,F,F.F2" }{TEXT -1 1 " " } {XPPEDIT 18 0 "exp(-(a^2+omega^2)/(4*b))" "6#-%$expG6#,$*&,&*$%\"aG\" \"#\"\"\"*$%&omegaGF+F,F,*&\"\"%F,%\"bGF,!\"\"F2" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = sqrt(Pi/b);" "6#/%!G-%%sqrtG6#*&%#PiG\"\"\"%\"bG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cosh(omega*a/(2*b))*exp(-(a^2+om ega^2)/(4*b));" "6#*&-%%coshG6#*(%&omegaG\"\"\"%\"aGF)*&\"\"#F)%\"bGF) !\"\"F)-%$expG6#,$*&,&*$F*F,F)*$F(F,F)F)*&\"\"%F)F-F)F.F.F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "interface(showassumed=0):\nassume(a>0,b>0):\nexp(-b* x^2)*cos(a*x);\n`Fourier transform`=inttrans[fourier](%,x,omega);\n``= map(simplify,expand(convert(rhs(%),exp)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$*&%#b|irG\"\"\")%\"xG\"\"#F*!\"\"F*-%$cosG 6#*&%#a|irGF*F,F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~tran sformG**%#b|irG#!\"\"\"\"#%#PiG#\"\"\"F)-%%coshG6#,$**F)F(F&F(%&omegaG F,%#a|irGF,F,F,-%$expG6#,$*(\"\"%F(,&*$)F3F)F,F,*$)F2F)F,F,F,F&F(F(F, " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(*(%#b|irG#! \"\"F)-%$expG6#,$*(\"\"%F-,&%&omegaGF(%#a|irGF(F)F+F-F-F(%#PiGF'F(F(*& F'F(*(F+F,-F/6#,$*(F3F-,&F5F(F6F-F)F+F-F-F(F7F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "For the specific case where a = 2 and " } {XPPEDIT 18 0 "b = 1/100;" "6#/%\"bG*&\"\"\"F&\"$+\"!\"\"" }{TEXT -1 26 ", the Fourier transform is" 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F'\"\"\"-%$expG6#,$*&%\"bGF)*$F'\"\"#F)!\"\"F)" }{TEXT -1 10 ", so tha t " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 14 " has the for m " }{XPPEDIT 18 0 "g(x) = f(x)*sin*a*x;" "6#/-%\"gG6#%\"xG**-%\"fG6#F '\"\"\"%$sinGF,%\"aGF,F'F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "By the cosine shift formula" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Fr*[g(x)] = Fr*[f(x)*sin(a*x)];" "6#/*&%#FrG\"\"\"7# -%\"gG6#%\"xGF&*&F%F&7#*&-%\"fG6#F+F&-%$sinG6#*&%\"aGF&F+F&F&F&" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "(F(omega+a)-F(omega-a))*i/2;" "6#*(,& -%\"FG6#,&%&omegaG\"\"\"%\"aGF*F*-F&6#,&F)F*F+!\"\"F/F*%\"iGF*\"\"#F/ " }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "F(omega)=Fr*[f(x)]" "6#/-%\"FG6#%&omegaG*&%#FrG\"\"\"7#-%\"fG6#%\"x GF*" }{XPPEDIT 18 0 " ``=F(omega) " "6#/%!G-%\"FG6#%&omegaG" } {XPPEDIT 18 0 "`` = (-i*omega*sqrt(Pi)/(2*b*sqrt(b)))*exp(-omega^2/(4* b));" "6#/%!G*&,$**%\"iG\"\"\"%&omegaGF)-%%sqrtG6#%#PiGF)*(\"\"#F)%\"b GF)-F,6#F1F)!\"\"F4F)-%$expG6#,$*&F*F0*&\"\"%F)F1F)F4F4F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[g(x)] = i/2;" "6#/*&%#FrG\"\"\"7#-%\"gG6# %\"xGF&*&%\"iGF&\"\"#!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "``((-i*(o mega+a)*sqrt(Pi)/(2*b*sqrt(b)))*exp(-(omega+a)^2/(4*b))-(-i*(omega-a)* sqrt(Pi)/(2*b*sqrt(b)))*exp(-(omega-a)^2/(4*b)));" "6#-%!G6#,&*&,$**% \"iG\"\"\",&%&omegaGF+%\"aGF+F+-%%sqrtG6#%#PiGF+*(\"\"#F+%\"bGF+-F06#F 5F+!\"\"F8F+-%$expG6#,$*&,&F-F+F.F+F4*&\"\"%F+F5F+F8F8F+F+*&,$**F*F+,& F-F+F.F8F+-F06#F2F+*(F4F+F5F+-F06#F5F+F8F8F+-F:6#,$*&,&F-F+F.F8F4*&F@F +F5F+F8F8F+F8" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sqrt(Pi)/(4*b*sq rt(b));" "6#/%!G*&-%%sqrtG6#%#PiG\"\"\"*(\"\"%F*%\"bGF*-F'6#F-F*!\"\" " }{XPPEDIT 18 0 "``((omega+a)*exp(-(omega+a)^2/(4*b))-(omega-a)*exp(- (omega-a)^2/(4*b)));" "6#-%!G6#,&*&,&%&omegaG\"\"\"%\"aGF*F*-%$expG6#, $*&,&F)F*F+F*\"\"#*&\"\"%F*%\"bGF*!\"\"F6F*F**&,&F)F*F+F6F*-F-6#,$*&,& F)F*F+F6F2*&F4F*F5F*F6F6F*F6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "interface(s howassumed=0):\nassume(a>0,b>0):\nx*exp(-b*x^2)*sin(a*x);\n`Fourier tr ansform`=inttrans[fourier](%,x,omega);\nmap(simplify,expand(convert(rh s(%),exp))):\nmap(factor,[selectremove(has,%,a+omega)]):\n``=op(1,%)+o p(2,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"xG\"\"\"-%$expG6#,$*&% #b|irGF%)F$\"\"#F%!\"\"F%-%$sinG6#*&%#a|irGF%F$F%F%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%2Fourier~transformG,$*&#\"\"\"\"\"#F(**%#PiGF'-%$ex pG6#,$*(\"\"%!\"\",&*$)%#a|irGF)F(F(*$)%&omegaGF)F(F(F(%#b|irGF2F2F(,& *&F6F(-%%coshG6#,$**F)F2F:F2F9F(F6F(F(F(F(*&F9F(-%%sinhGF?F(F2F(F:#!\" $F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"%F(**%#b |irG#!\"$\"\"#,&%&omegaGF(%#a|irGF(F(-%$expG6#,$*(F)!\"\"F/F.F+F7F7F(% #PiG#F(F.F(F(*&F'F(**F8F9-F36#,$*(F)F7,&F1F(F0F7F.F+F7F7F(F@F(F+F,F(F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "For \+ the specific case where " }{XPPEDIT 18 0 "a = 2" "6#/%\"aG\"\"#" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "b = 1/100;" "6#/%\"bG*&\"\"\"F&\"$+ \"!\"\"" }{TEXT -1 27 ", the Fourier transform is " }}{PARA 256 "" 0 " " {TEXT -1 3 " " }{XPPEDIT 18 0 "G(omega) = 250*sqrt(Pi)*((omega+2)* exp((-25)*(omega+2)^2)-(omega-2)*exp(-25*(omega-2)^2));" "6#/-%\"GG6#% &omegaG*(\"$]#\"\"\"-%%sqrtG6#%#PiGF*,&*&,&F'F*\"\"#F*F*-%$expG6#*&,$ \"#D!\"\"F**$,&F'F*F2F*F2F*F*F**&,&F'F*F2F9F*-F46#,$*&F8F**$,&F'F*F2F9 F2F*F9F*F9F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "x*exp(-x^2/100)*sin(2*x);\n `Fourier transform`=inttrans[fourier](%,x,omega);\nmap(simplify,expand (convert(rhs(%),exp))):\nmap(factor,[selectremove(has,%,2+omega)]):\n` `=op(1,%)+op(2,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"xG\"\"\"-%$ expG6#,$*&\"$+\"!\"\"F$\"\"#F,F%-%$sinG6#,$*&F-F%F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,$*,\"$+&\"\"\"%#PiG#F(\" \"#-%$expG6#!$+\"F(-F-6#,$*&\"#DF()%&omegaGF+F(!\"\"F(,&*&F6F(-%%sinhG 6#,$*&\"$+\"F(F6F(F(F(F7*&F+F(-%%coshGF " 0 "" {MPLTEXT 1 0 159 "G := w -> 250*sqrt(P i)*(exp(-25*(w+2)^2)*(w+2)-exp(-25*(w-2)^2)*(w-2));\nplot(G(w),w=-3.5. .3.5,color=COLOR(RGB,.4,0,.9),title=`Fourier transform G(w) of g(x)`); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGf*6#%\"wG6\"6$%)operatorG%&a rrowGF(,$*(\"$]#\"\"\"-%%sqrtG6#%#PiGF/,&*&-%$expG6#,$*&\"#DF/),&9$F/ \"\"#F/F?F/!\"\"F/F=F/F/*&-F76#,$*&F;F/),&F>F/F?F@F?F/F@F/FGF/F@F/F/F( F(F(" }}{PARA 13 "" 1 "" 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0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "W e find the Fourier transform of " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = sin*a*x/x;" "6#/-%\"fG6#%\"xG**%$sinG\"\"\"%\"a GF*F'F*F'!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where \+ " }{TEXT 277 1 "a" }{TEXT -1 13 " is positive." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We have seen that that " }{XPPEDIT 18 0 "Fr*[phi[a](x)] = 2*sin(a*omega)/omega;" "6#/*&%#FrG\" \"\"7#-&%$phiG6#%\"aG6#%\"xGF&*(\"\"#F&-%$sinG6#*&F,F&%&omegaGF&F&F5! \"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "phi[a](x) = PIECEWISE([1, abs(x) < a],[0, a <= abs(x)]);" "6#/-&%$phiG6#%\"aG6#%\"xG-%*PIECEWIS EG6$7$\"\"\"2-%$absG6#F*F(7$\"\"!1F(-F26#F*" }{TEXT -1 22 " is a puls e of width " }{TEXT 278 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "From the symmetry property," }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Fr*[2*sin*a*x/x] = 2*Pi*phi[a](-omega);" "6#/*&%#FrG \"\"\"7#*,\"\"#F&%$sinGF&%\"aGF&%\"xGF&F,!\"\"F&*(F)F&%#PiGF&-&%$phiG6 #F+6#,$%&omegaGF-F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[sin*a*x/x] = \+ Pi*phi[a](-omega);" "6#/*&%#FrG\"\"\"7#**%$sinGF&%\"aGF&%\"xGF&F+!\"\" F&*&%#PiGF&-&%$phiG6#F*6#,$%&omegaGF,F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = Pi*phi[a](omega);" "6#/%!G*&%#PiG\"\"\"-&%$phiG6#%\"aG6#%&omegaGF '" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "since " }{XPPEDIT 18 0 "phi[d](omega);" "6#-&%$phiG6#%\"dG6#%&omegaG" }{TEXT -1 21 " is \+ an even function." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 149 "interface(showassumed=0): assume(a>0):\nsin (a*x)/x;\ninttrans[fourier](%,x,omega):\n`Fourier transform`=simplify( %);\n``=convert(rhs(%),piecewise,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$sinG6#*&%#a|irG\"\"\"%\"xGF)F)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG,$*&%#PiG\"\"\",&-%\"HG6#,&%&omega GF(%#a|irG!\"\"F(-F+6#,&F.F(F/F(F0F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6'7$\"\"!2%&omegaG,$%#a|irG!\"\"7$,$*&%#PiG\"\" \"%*undefinedGF3F./F+F,7$F22F+F-7$F0/F+F-7$F)2F-F+" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 33 "We find the Fo urier transform of " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = 1/(a^2+x^2);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&*$%\"aG\"\"#F)*$F 'F-F)!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {TEXT 279 1 "a" }{TEXT -1 13 " is positive." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We have seen that that " } {XPPEDIT 18 0 "Fr*[exp(-a*abs(x))] = 2*a/(a^2+omega^2);" "6#/*&%#FrG\" \"\"7#-%$expG6#,$*&%\"aGF&-%$absG6#%\"xGF&!\"\"F&*(\"\"#F&F-F&,&*$F-F4 F&*$%&omegaGF4F&F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "Fr om the symmetry property," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Fr*[2*a/(a^2+x^2)] = 2*Pi*exp(-a*abs(-omega));" "6#/*&% #FrG\"\"\"7#*(\"\"#F&%\"aGF&,&*$F*F)F&*$%\"xGF)F&!\"\"F&*(F)F&%#PiGF&- %$expG6#,$*&F*F&-%$absG6#,$%&omegaGF/F&F/F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{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 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "interface(showassumed=0): a ssume(a>0):\n1/(a^2+x^2);\ninttrans[fourier](%,x,omega):\n`Fourier tra nsform`=simplify(%);\n``=convert(rhs(%),piecewise,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$)%#a|irG\"\"#F$F$*$)%\"xGF)F$F$! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG*(%#PiG\" \"\",(*&-%$expG6#,$*&%#a|irGF'%&omegaGF'!\"\"F'-%\"HG6#F0F'F'-F+6#F.F' *&F5F'F2F'F1F'F/F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISE G6%7$*(%#a|irG!\"\"-%$expG6#*&F*\"\"\"%&omegaGF0F0%#PiGF02F1\"\"!7$*(F *F+F2F0%*undefinedGF0/F1F47$*(F*F+-F-6#,$F/F+F0F2F02F4F1" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 }