{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 "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "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 37 "Convolution and the Fourier trans form" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Ca nada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "The convolution of two functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 261 11 "convolution" }{TEXT -1 18 " of two functions " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 17 " is the function " } {XPPEDIT 18 0 "h(x) " "6#-%\"hG6#%\"xG" }{TEXT -1 10 "defined by" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = Int(f(x-y)*g(y ),y = -infinity .. infinity);" "6#/-%\"hG6#%\"xG-%$IntG6$*&-%\"fG6#,&F '\"\"\"%\"yG!\"\"F0-%\"gG6#F1F0/F1;,$%)infinityGF2F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "We write h = f * g or " }{XPPEDIT 18 0 "h(x)=f(x)*`*`*g(x)" "6#/-%\"hG6#%\"xG*(-%\"fG6#F'\"\"\"%\"*GF,-%\"gG6 #F'F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The operation of convolution is " }{TEXT 261 12 "commu tative " }}{PARA 256 "" 0 "" {TEXT -1 10 " ( f * g )" }{XPPEDIT 18 0 " ``(x) = f(x)*`*`*g(x)" "6#/-%!G6#%\"xG*(-%\"fG6#F'\"\"\"%\"*GF,-%\"gG6 #F'F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = Int(f(x -y)*g(y),y = -infinity .. infinity);" "6#/%!G-%$IntG6$*&-%\"fG6#,&%\"x G\"\"\"%\"yG!\"\"F.-%\"gG6#F/F./F/;,$%)infinityGF0F7" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -Int(f(u)*g(x- u),u = infinity .. -infinity);" "6#/%!G,$-%$IntG6$*&-%\"fG6#%\"uG\"\" \"-%\"gG6#,&%\"xGF.F-!\"\"F./F-;%)infinityG,$F7F4F4" }{TEXT -1 8 ", wh ere " }{XPPEDIT 18 0 "u = x -y" "6#/%\"uG,&%\"xG\"\"\"%\"yG!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "du/dy = -1" "6#/*&%#duG\"\"\"%#dyG! \"\",$F&F(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int(g(x-u)*f(u),u = -infinity .. infinity);" "6#/% !G-%$IntG6$*&-%\"gG6#,&%\"xG\"\"\"%\"uG!\"\"F.-%\"fG6#F/F./F/;,$%)infi nityGF0F7" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = g(x)*`*`*f(x);" "6#/%!G*(-%\"gG6#%\"xG\"\"\"%\"*GF *-%\"fG6#F)F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 12 " = ( g * f )" }{XPPEDIT 18 0 "``(x)" "6#-%!G6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Convoluti on is also " }{TEXT 261 8 "bilinear" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 31 " [ f + g ] * h = f * h + g * h " }}{PARA 256 "" 0 " " {TEXT -1 31 " h * [ f + g ] = h * f + h * g " }}{PARA 256 "" 0 "" {TEXT -1 42 " (c f ) * g = c ( f * g ) = f * ( c g ), " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }{TEXT 261 11 "associative" }{TEXT -1 2 ": " }} {PARA 256 "" 0 "" {TEXT -1 32 " ( f * g ) * h = f * ( g * h ). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "It is int eresting to see how the Dirac delta function behaves under convolution ." }}{PARA 256 "" 0 "" {TEXT -1 6 "( f * " }{XPPEDIT 18 0 "delta" "6#% &deltaG" }{TEXT -1 2 " )" }{XPPEDIT 18 0 "``(x) = Int(f(x-y)*delta(y), y = -infinity .. infinity);" "6#/-%!G6#%\"xG-%$IntG6$*&-%\"fG6#,&F'\" \"\"%\"yG!\"\"F0-%&deltaG6#F1F0/F1;,$%)infinityGF2F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = eval(f(x-y),y = 0);" "6#/%!G-%%evalG6$-%\"fG6#,&% \"xG\"\"\"%\"yG!\"\"/F.\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = f(x); " "6#/%!G-%\"fG6#%\"xG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 5 " f * " }{XPPEDIT 18 0 "del ta" "6#%&deltaG" }{TEXT -1 6 " = f. " }}{PARA 0 "" 0 "" {TEXT -1 15 "W e also write: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 65 "Because of the commutative property of convolution, we al so have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta" "6 #%&deltaG" }{TEXT -1 10 " * f = f. " }}{PARA 0 "" 0 "" {TEXT -1 29 "Th us the delta function is a " }{TEXT 261 12 "unit element" }{TEXT -1 34 " for the operation of convolution." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "alias(delta=Dirac): f := 'f':\nInt(f(x-y)*delta(y),y=-infinity..infinity);\nvalue(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%\"fG6#,&%\"xG\"\"\"%\"yG! \"\"F,-%&deltaG6#F-F,/F-;,$%)infinityGF.F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"fG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 23 "The convolution theorem" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 82 "The main result con cerning convolution and the Fourier transform is the following " } {TEXT 261 19 "convolution theorem" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "Fr*[f (x)]=F(omega)" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "Fr*[g(x)]=G(omega)" "6#/*&%#FrG\"\" \"7#-%\"gG6#%\"xGF&-%\"GG6#%&omegaG" }{TEXT -1 7 ", then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[f(x)*`*`*g(x)] = F(omeg a)*G(omega)" "6#/*&%#FrG\"\"\"7#*(-%\"fG6#%\"xGF&%\"*GF&-%\"gG6#F,F&F& *&-%\"FG6#%&omegaGF&-%\"GG6#F5F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{TEXT 262 17 "________________ " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "In words: " }{TEXT 261 63 "the transform of a convolution is the product of the transform s" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Pro of of the convolution theorem" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Fourier transform of " }{XPPEDIT 18 0 "f(x)*`*`*g(x)" "6#*(-% \"fG6#%\"xG\"\"\"%\"*GF(-%\"gG6#F'F(" }{TEXT -1 4 " is " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[f(x)*`*`*g(x)] = Int(``(Int( f(x-y)*g(y),y = -infinity .. infinity))*exp(-i*omega*x),x = -infinity \+ .. infinity);" "6#/*&%#FrG\"\"\"7#*(-%\"fG6#%\"xGF&%\"*GF&-%\"gG6#F,F& F&-%$IntG6$*&-%!G6#-F26$*&-F*6#,&F,F&%\"yG!\"\"F&-F/6#F>F&/F>;,$%)infi nityGF?FEF&-%$expG6#,$*(%\"iGF&%&omegaGF&F,F&F?F&/F,;,$FEF?FE" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Changing the order of the integration, we have" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[f(x)*`*`*g(x)] = Int(``(Int(f(x-y )*exp(-i*omega*x),x = -infinity .. infinity))*g(y),y = -infinity .. in finity);" "6#/*&%#FrG\"\"\"7#*(-%\"fG6#%\"xGF&%\"*GF&-%\"gG6#F,F&F&-%$ IntG6$*&-%!G6#-F26$*&-F*6#,&F,F&%\"yG!\"\"F&-%$expG6#,$*(%\"iGF&%&omeg aGF&F,F&F?F&/F,;,$%)infinityGF?FJF&-F/6#F>F&/F>;,$FJF?FJ" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 49 "By the shifting property of the Fou rier transform" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (f(x-y)*exp(-i*omega*x),x = -infinity .. infinity) = F(omega)*exp(-i*o mega*y);" "6#/-%$IntG6$*&-%\"fG6#,&%\"xG\"\"\"%\"yG!\"\"F--%$expG6#,$* (%\"iGF-%&omegaGF-F,F-F/F-/F,;,$%)infinityGF/F:*&-%\"FG6#F6F--F16#,$*( F5F-F6F-F.F-F/F-" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[f(x)*`*`*g(x)] = \+ Int(F(omega)*g(y)*exp(-i*omega*y),y = -infinity .. infinity);" "6#/*&% #FrG\"\"\"7#*(-%\"fG6#%\"xGF&%\"*GF&-%\"gG6#F,F&F&-%$IntG6$*(-%\"FG6#% &omegaGF&-F/6#%\"yGF&-%$expG6#,$*(%\"iGF&F8F&F;F&!\"\"F&/F;;,$%)infini tyGFBFF" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = F(omega)*``(Int(g(y)*exp(-i*omega*y),y = -infinity .. infinity));" "6#/%!G*&-%\"FG6#%&omegaG\"\"\"-F$6#-%$IntG6$*&-%\"gG 6#%\"yGF*-%$expG6#,$*(%\"iGF*F)F*F4F*!\"\"F*/F4;,$%)infinityGF;F?F*" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = F(omega)*G(omega);" "6#/%!G*&-%\"FG6#%&omegaG\"\"\"-%\"GG6#F)F*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 54 "Conversely, under suitable conditions on f and g, \+ if " }{XPPEDIT 18 0 "Fr*[f(x)]=F(omega)" "6#/*&%#FrG\"\"\"7#-%\"fG6#% \"xGF&-%\"FG6#%&omegaG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Fr*[g(x)] =G(omega)" "6#/*&%#FrG\"\"\"7#-%\"gG6#%\"xGF&-%\"GG6#%&omegaG" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Fr*[f(x)*g(x)] = 1/(2*Pi);" "6#/*&%#FrG\"\"\"7#*&-%\"fG6#%\"xGF&-%\"gG 6#F,F&F&*&F&F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "F (omega)*`*`*G(omega)" "6#*(-%\"FG6#%&omegaG\"\"\"%\"*GF(-%\"GG6#F'F(" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 19 "___ _______________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Exam ples" }}{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 4 "Let " }{XPPEDIT 18 0 "f(x) = \+ PIECEWISE([exp(-a*x), `if `*0 < x],[0, `if `*x <= 0]);" "6#/-%\"fG 6#%\"xG-%*PIECEWISEG6$7$-%$expG6#,$*&%\"aG\"\"\"F'F2!\"\"2*&%&if~~~GF2 \"\"!F2F'7$F71*&F6F2F'F2F7" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "g(x ) = PIECEWISE([exp(-b*x), `if `*0 < x],[0, `if `*x <= 0]);" "6#/-% \"gG6#%\"xG-%*PIECEWISEG6$7$-%$expG6#,$*&%\"bG\"\"\"F'F2!\"\"2*&%&if~~ ~GF2\"\"!F2F'7$F71*&F6F2F'F2F7" }{TEXT -1 9 ", where " }{XPPEDIT 18 0 "a > 0" "6#2\"\"!%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b > 0" " 6#2\"\"!%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)*`*`*g(x) = Int(f (x-y)*g(y),y = -infinity .. infinity);" "6#/*(-%\"fG6#%\"xG\"\"\"%\"*G F)-%\"gG6#F(F)-%$IntG6$*&-F&6#,&F(F)%\"yG!\"\"F)-F,6#F5F)/F5;,$%)infin ityGF6F<" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = PIECEWISE([Int(exp(-a*(x-y))*exp(-b*y),y = 0 .. x) , `if `*0 <= x],[0, `if `*x < 0]);" "6#/%!G-%*PIECEWISEG6$7$-%$Int G6$*&-%$expG6#,$*&%\"aG\"\"\",&%\"xGF3%\"yG!\"\"F3F7F3-F.6#,$*&%\"bGF3 F6F3F7F3/F6;\"\"!F51*&%&if~~~GF3F?F3F57$F?2*&FBF3F5F3F?" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = PIECEWISE([exp(-a*x)*Int(exp((a-b)*y),y = 0 .. \+ x), `if `*0 <= x],[0, `if `*x < 0]);" "6#/%!G-%*PIECEWISEG6$7$*&-% $expG6#,$*&%\"aG\"\"\"%\"xGF0!\"\"F0-%$IntG6$-F+6#*&,&F/F0%\"bGF2F0%\" yGF0/F;;\"\"!F1F01*&%&if~~~GF0F>F0F17$F>2*&FAF0F1F0F>" }{TEXT -1 4 " \+ . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If \+ " }{XPPEDIT 18 0 "a<>b" "6#0%\"aG%\"bG" }{TEXT -1 7 ", then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)*`*`*g(x) = PIECEWISE([ exp(-a*x)*``(exp((a-b)*x)/(a-b)-1/(a-b)), `if `*0 <= x],[0, `if `* x < 0]);" "6#/*(-%\"fG6#%\"xG\"\"\"%\"*GF)-%\"gG6#F(F)-%*PIECEWISEG6$7 $*&-%$expG6#,$*&%\"aGF)F(F)!\"\"F)-%!G6#,&*&-F46#*&,&F8F)%\"bGF9F)F(F) F),&F8F)FCF9F9F)*&F)F),&F8F)FCF9F9F9F)1*&%&if~~~GF)\"\"!F)F(7$FJ2*&FIF )F(F)FJ" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([(exp(-a*x)-exp(- b*x))/(b-a), `if `*0 <= x],[0, `if `*x < 0]);" "6#/%!G-%*PIECEWISE G6$7$*&,&-%$expG6#,$*&%\"aG\"\"\"%\"xGF1!\"\"F1-F,6#,$*&%\"bGF1F2F1F3F 3F1,&F8F1F0F3F31*&%&if~~~GF1\"\"!F1F27$F=2*&F " 0 "" {MPLTEXT 1 0 157 "alias(H=Heaviside):\nf := x -> piecewise(x<0,0,x*exp (-a*x)):\n'f(x)'=f(x);\n``=convert(rhs(%),Heaviside);\n`Fourier transf orm`=inttrans[fourier](rhs(%),x,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"!2F'F,7$*&F'\"\"\"-%$expG6#,$ *&%\"aGF0F'F0!\"\"F0%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% !G*(-%\"HG6#%\"xG\"\"\"F)F*-%$expG6#,$*&%\"aGF*F)F*!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG*&\"\"\"F&*$),&%\"aGF&* &%&omegaGF&^#F&F&F&\"\"#F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 18 "For example, take " }{XPPEDIT 18 0 "a = 1 " "6#/%\"aG\"\"\"" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "f(x) = P IECEWISE([exp(-x), `if `*0 < x],[0, `if `*x <= 0]);" "6#/-%\"fG6#% \"xG-%*PIECEWISEG6$7$-%$expG6#,$F'!\"\"2*&%&if~~~G\"\"\"\"\"!F4F'7$F51 *&F3F4F'F4F5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "f := x -> piecewise(x>0,x*ex p(-x)):\nplot(f(x),x=-2..6,color=red,thickness=2,ytickmarks=4);" }} {PARA 13 "" 1 "" {GLPLOT2D 481 222 222 {PLOTDATA 2 "6(-%'CURVESG6#7bo7 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$e#=#oC\"F/$\"3!e1Mq[vNe$FB7$$\"3<+++X@4L8F/$\"3h%QM*y.%[^$FB7$$\"31++ +N;R(\\\"F/$\"3ibUh@'e)\\LFB7$$\"3wmmm;4#)o;F/$\"3y.X\"))o7_9$FB7$$\"3 jmmm6lCE=F/$\"3`#\\![pQcSHFB7$$\"3ELLL$G^g*>F/$\"3sIAqd'\\?r#FB7$$\"3o KLL=2Vs@F/$\"3[kM@>xSuCFB7$$\"3f*****\\`pfK#F/$\"3Qr:Tk!4AF#FB7$$\"3!H LLLm&z\"\\#F/$\"3olc8N0Ci?FB7$$\"3s******z-6jEF/$\"3=bhegM,d=FB7$$\"3< ******4#32$GF/$\"3s\"ph21*Hp;FB7$$\"3O*****\\#y'G*HF/$\"3`I>\\:ls+:FB7 $$\"3G******H%=H<$F/$\"3%>&*[61c)G8FB7$$\"35mmm1>qMLF/$\"3)[VleO%*z=\" FB7$$\"3%)*******HSu]$F/$\"3#\\T:r+/80\"FB7$$\"3'HLL$ep'Rm$F/$\"3'z2sB -!)4R*FR7$$\"3')******R>4NQF/$\"3/5TR\"G]NG)FR7$$\"3#emm;@2h*RF/$\"3SV 4F+InZtFR7$$\"3]*****\\c9W;%F/$\"3%*3BvNV+rkFR7$$\"3Lmmmmd'*GVF/$\"32) )*yPg%31dFR7$$\"3j*****\\iN7]%F/$\"3m^w%zEYU*\\FR7$$\"3aLLLt>:nYF/$\"3 K_frhhl'Q%FR7$$\"35LLL.a#o$[F/$\"3y&Ha+akm$QFR7$$\"3ammm^Q40]F/$\"3-*[ 5O&3FbLFR7$$\"3y******z]rf^F/$\"3uV$yil(RjHFR7$$\"3gmmmc%GpL&F/$\"3)Qw N`l&RnDFR7$$\"3/LLL8-V&\\&F/$\"3m9Cs&HVhD#FR7$$\"3=+++XhUkcF/$\"33&ovS 0MR'>FR7$$\"3=+++:o " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[f(x)] = 1/((1+i*omega)^2);" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&*&F&F&*$,&F&F&* &%\"iGF&%&omegaGF&F&\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ (1-i*omega)^2/((1+i*omega)^2*(1-i*omega)^2);" "6#/%!G*&,&\"\"\"F'*&%\" iGF'%&omegaGF'!\"\"\"\"#*&,&F'F'*&F)F'F*F'F'F,,&F'F'*&F)F'F*F'F+F,F+" }{XPPEDIT 18 0 "`` = (1-2*i*omega+i^2*omega^2)/((1-i^2*omega^2)^2);" " 6#/%!G*&,(\"\"\"F'*(\"\"#F'%\"iGF'%&omegaGF'!\"\"*&F*F)F+F)F'F'*$,&F'F '*&F*F)F+F)F,F)F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-omega^2)/((1+omega^2)^2)-2*i*omega/((1+omega^2 )^2);" "6#/%!G,&*&,&\"\"\"F(*$%&omegaG\"\"#!\"\"F(*$,&F(F(*$F*F+F(F+F, F(**F+F(%\"iGF(F*F(*$,&F(F(*$F*F+F(F+F,F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = A(omega) +B(omega)*i;" "6#/%!G,&-%\"AG6#%&omegaG\"\"\"*&-%\"BG6#F)F*%\"iGF*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A(omega)=(1-omega^2)/(1+omega^2)^2" "6#/-%\"AG6#%&omegaG*&,&\"\"\"F** $F'\"\"#!\"\"F**$,&F*F**$F'F,F*F,F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B(omega) = (-2*omega)/((1+omega^2)^2);" "6#/-%\"BG6#%&omegaG*&,$*& \"\"#\"\"\"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 0 201 "plo t([(1-w^2)/(1+w^2)^2,-2*w/(1+w^2)^2],w=-4..4,-1..1,\n ytickmarks=3,col or=[COLOR(RGB,.4,0,.9),COLOR(RGB,.6,.4,0)],\n title=`Fourier transfo rm F(w)=A(w) + i B(w) of f(x)`,\n legend=[`A(w)`,`B(w)`]);" }} {PARA 13 "" 1 "" {GLPLOT2D 633 328 328 {PLOTDATA 2 "6(-%'CURVESG6%7io7 $$!\"%\"\"!$!3%37&o=9J!>&!#>7$$!3ommmmFiDQ!#<$!3)RX$)[gNwd&F-7$$!35LLL o!)*Qn$F1$!37C\"Hq1bh%fF-7$$!3nmmmwxE.NF1$!3Iuo@]l)))R'F-7$$!3YmmmOk]J LF1$!3CL8+7G#*)*oF-7$$!3_LLL[9cgJF1$!3([?r')[FPW(F-7$$!3smmmhN2-IF1$!3 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f)['F[o7$Fa[l$!35i+8KxepkF[o7$$\"3ojm;H&fpT'F[o$!3YF@NH'e\"RkF[o7$Ff[l $!3'oLj#=)p#)R'F[o7$F[\\l$!3G-c1sq2bhF[o7$F`\\l$!3WZUw]mb;eF[o7$Fj\\l$ !3)p4kZ/fc.&F[o7$F_]l$!3G*QO??Ui:%F[o7$Fi]l$!3?8M'=1q/X$F[o7$Fc^l$!3r= TxzHU:GF[o7$F]_l$!3#HY.:)p*HM#F[o7$Fb_l$!3Tk(yN**>S#>F[o7$Fg_l$!3Hxfj& zoog\"F[o7$F\\`l$!35Q#p$eMaR8F[o7$Fa`l$!3h0M6r/oG6F[o7$Ff`l$!3X(>=V:$) 4]*F-7$F[al$!3\\>#=jE*)G5)F-7$F`al$!3nX7\\p=.JpF-7$Feal$!3Ah\"yG`'etfF -7$Fjal$!389P(RK(3Q_F-7$F_bl$!3b([h!e\"4?`%F-7$Fdbl$!3vNABwk?NF-7$F^cl$!3[hV9iIaGJF-7$Fccl$!3-)Rl**3m\"oFF--Ffcl6&Fhcl$ \"\"'FjclFiclF*-F^dl6#%%B(w)G-%&TITLEG6#%MFourier~transform~F(w)=A(w)~ +~i~B(w)~of~f(x)G-%*AXESTICKSG6$%(DEFAULTG\"\"$-%+AXESLABELSG6$Q\"w6\" Q!6\"-%%VIEWG6$;F(Fccl;$FjclF*$\"\"\"F*" 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = exp(-a*x^2);" "6#/-%\"fG6#%\"xG-%$expG6#,$*&%\"aG\"\"\"*$F' \"\"#F.!\"\"" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "g(x) = exp(-b*x^2 );" "6#/-%\"gG6#%\"xG-%$expG6#,$*&%\"bG\"\"\"*$F'\"\"#F.!\"\"" }{TEXT -1 9 ", where " }{XPPEDIT 18 0 "a > 0" "6#2\"\"!%\"aG" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "b > 0" "6#2\"\"!%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)*`*`*g(x) = Int(f(x-y)*g(y),y = -infinity .. infinity);" "6# /*(-%\"fG6#%\"xG\"\"\"%\"*GF)-%\"gG6#F(F)-%$IntG6$*&-F&6#,&F(F)%\"yG! \"\"F)-F,6#F5F)/F5;,$%)infinityGF6F<" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(exp(-a*(x-y)^2)*exp(-b*y^ 2),y = -infinity .. infinity);" "6#/%!G-%$IntG6$*&-%$expG6#,$*&%\"aG\" \"\"*$,&%\"xGF/%\"yG!\"\"\"\"#F/F4F/-F*6#,$*&%\"bGF/*$F3F5F/F4F//F3;,$ %)infinityGF4F?" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int(exp(-a*(x^2-2*x*y+y^2)-b*y^2),y = -infinity .. infinity);" "6#/%!G-%$IntG6$-%$expG6#,&*&%\"aG\"\"\",(*$%\"xG\"\"#F.* (F2F.F1F.%\"yGF.!\"\"*$F4F2F.F.F5*&%\"bGF.*$F4F2F.F5/F4;,$%)infinityGF 5F=" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-a*x^2)*Int(exp(2*a*x*y-(a+b)*y^2),y = -infinity .. infini ty);" "6#/%!G*&-%$expG6#,$*&%\"aG\"\"\"*$%\"xG\"\"#F,!\"\"F,-%$IntG6$- F'6#,&**F/F,F+F,F.F,%\"yGF,F,*&,&F+F,%\"bGF,F,*$F8F/F,F0/F8;,$%)infini tyGF0F@F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = y*sqrt(a+b)-a*x/sqrt(a+b );" "6#/%\"uG,&*&%\"yG\"\"\"-%%sqrtG6#,&%\"aGF(%\"bGF(F(F(*(F-F(%\"xGF (-F*6#,&F-F(F.F(!\"\"F4" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "u^2 = (a+b)*y^2-2*a*x*y-a^2*x^2/(a+b);" "6#/*$%\"uG\"\"#,(*&,&%\"aG\"\"\" %\"bGF+F+*$%\"yGF&F+F+**F&F+F*F+%\"xGF+F.F+!\"\"*(F*F&F0F&,&F*F+F,F+F1 F1" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "du/dy=sqrt(a+b)" "6#/*&%#duG\" \"\"%#dyG!\"\"-%%sqrtG6#,&%\"aGF&%\"bGF&" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(x)*`*`*g (x) = 1/sqrt(a+b);" "6#/*(-%\"fG6#%\"xG\"\"\"%\"*GF)-%\"gG6#F(F)*&F)F) -%%sqrtG6#,&%\"aGF)%\"bGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(- a*x^2)*exp(a^2*x^2/(a+b));" "6#*&-%$expG6#,$*&%\"aG\"\"\"*$%\"xG\"\"#F *!\"\"F*-F%6#*(F)F-F,F-,&F)F*%\"bGF*F.F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-u^2),u=-infinity..infinity)" "6#-%$IntG6$-%$expG6#,$*$% \"uG\"\"#!\"\"/F+;,$%)infinityGF-F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 37 "We can now use the well-known result " }{XPPEDIT 18 0 " Int(exp(-u^2),u = -infinity .. infinity) = sqrt(Pi)" "6#/-%$IntG6$-%$e xpG6#,$*$%\"uG\"\"#!\"\"/F,;,$%)infinityGF.F2-%%sqrtG6#%#PiG" }{TEXT -1 12 " to obtain " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)*`*`*g(x) = sqrt(Pi)/sqrt(a+b);" "6#/*(-%\"fG6#%\"xG\"\"\"%\"*G F)-%\"gG6#F(F)*&-%%sqrtG6#%#PiGF)-F06#,&%\"aGF)%\"bGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp((a^2/(a+b)-a)*x^2)" "6#-%$expG6#*&,&*&%\" aG\"\"#,&F)\"\"\"%\"bGF,!\"\"F,F)F.F,*$%\"xGF*F," }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = sqrt(Pi/(a+b));" "6#/%!G-%%sqrtG6#*&%#PiG\"\"\",&% \"aGF*%\"bGF*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-a*b*x^2/(a+b) );" "6#-%$expG6#,$**%\"aG\"\"\"%\"bGF)%\"xG\"\"#,&F(F)F*F)!\"\"F." } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "interface(s howassumed=0): a := 'a': b := 'b':\nassume(a>0,b>0):\nf := x -> exp(-a *x^2):\n'f(x)'=f(x);\ng := x -> exp(-b*x^2):\n'g(x)'=g(x);\nInt('f(x-y )*g(y)',y=-infinity..infinity)=Int(f(x-y)*g(y),y=-infinity..infinity); \n``=value(rhs(%));\n a := 'a': b := 'b':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$expG6#,$*&%#a|irG\"\"\")F'\"\"#F.!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%$expG6#,$*&%#b|irG \"\"\")F'\"\"#F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&- %\"fG6#,&%\"xG\"\"\"%\"yG!\"\"F--%\"gG6#F.F-/F.;,$%)infinityGF/F6-F%6$ *&-%$expG6#,$*&%#a|irGF-)F+\"\"#F-F/F--F;6#,$*&%#b|irGF-)F.FAF-F/F-F3 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(-%$expG6#**,&%#a|irG!\"\"%#b |irGF,F,F+\"\"\"%\"xG\"\"#F-F.F.,&F+F.F-F.#F,F0%#PiG#F.F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Fr*[f(x)]=F(omega " "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" }{TEXT -1 3 " = \+ " }{XPPEDIT 18 0 "sqrt(Pi/a)*exp(-omega^2/(4*a))" "6#*&-%%sqrtG6#*&%#P iG\"\"\"%\"aG!\"\"F)-%$expG6#,$*&%&omegaG\"\"#*&\"\"%F)F*F)F+F+F)" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "Fr*[g(x)]=G(omega)" "6#/*&%#FrG\" \"\"7#-%\"gG6#%\"xGF&-%\"GG6#%&omegaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(Pi/b)*exp(-omega^2/(4*b))" "6#*&-%%sqrtG6#*&%#PiG\"\"\"%\"bG! \"\"F)-%$expG6#,$*&%&omegaG\"\"#*&\"\"%F)F*F)F+F+F)" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 14 "Similarly, if " }{XPPEDIT 18 0 "h(x) = f (x)*`*`*g(x)" "6#/-%\"hG6#%\"xG*(-%\"fG6#F'\"\"\"%\"*GF,-%\"gG6#F'F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(Pi/(a+b))*exp(-c*x^2)" "6#*&-%% sqrtG6#*&%#PiG\"\"\",&%\"aGF)%\"bGF)!\"\"F)-%$expG6#,$*&%\"cGF)*$%\"xG \"\"#F)F-F)" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "c = a*b/(a+b)" "6# /%\"cG*(%\"aG\"\"\"%\"bGF',&F&F'F(F'!\"\"" }{TEXT -1 7 ", then " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[h(x)]=sqrt(Pi/(a+ b))" "6#/*&%#FrG\"\"\"7#-%\"hG6#%\"xGF&-%%sqrtG6#*&%#PiGF&,&%\"aGF&%\" bGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(Pi/c)*exp(-omega^2/(4* c))" "6#*&-%%sqrtG6#*&%#PiG\"\"\"%\"cG!\"\"F)-%$expG6#,$*&%&omegaG\"\" #*&\"\"%F)F*F)F+F+F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi/sqrt(a*b) ;" "6#/%!G*&%#PiG\"\"\"-%%sqrtG6#*&%\"aGF'%\"bGF'!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(-omega^2*``((a+b)/(4*a*b)));" "6#-%$expG6#,$*&%& omegaG\"\"#-%!G6#*&,&%\"aG\"\"\"%\"bGF0F0*(\"\"%F0F/F0F1F0!\"\"F0F4" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = Pi/sqrt(a*b);" "6#/%!G*&%#PiG\"\"\"-%%sqrtG6#*&%\"aGF'%\"bGF'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-omega^2*(1/(4*a)+1/(4*b))))" "6#- %$expG6#,$*&%&omegaG\"\"#,&*&\"\"\"F,*&\"\"%F,%\"aGF,!\"\"F,*&F,F,*&F. F,%\"bGF,F0F,F,F0" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = F(omega)*G(omega);" "6#/%!G*&-%\"FG6#%&omegaG\"\"\"-%\"GG6#F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 185 "interface(showassumed=0): a := 'a': b := 'b': \nassume(a>0,b>0):\nexp(-a*x^2*b/(a+b))*sqrt(Pi)/(sqrt(a+b));\n`Fourie r transform`=simplify(inttrans[fourier](%,x,omega));\n a := 'a': b := \+ 'b':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$expG6#,$**%#a|irG\"\"\"% \"xG\"\"#%#b|irGF*,&F)F*F-F*!\"\"F/F*%#PiG#F*F,F.#F/F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG**%#PiG\"\"\"-%$expG6#,$*,\" \"%!\"\"%&omegaG\"\"#%#a|irGF.%#b|irGF.,&F1F'F2F'F'F.F'F1#F.F0F2F4" }} }{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 5 " L et " }{XPPEDIT 18 0 "f(x)=PIECEWISE([1 , abs(x)a ])" "6 #/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"\"2-%$absG6#F'%\"aG7$\"\"!2F1-F/6# F'" }{TEXT -1 9 " , and " }{XPPEDIT 18 0 "g(x)=x*exp(-b*x^2)" "6#/-% \"gG6#%\"xG*&F'\"\"\"-%$expG6#,$*&%\"bGF)*$F'\"\"#F)!\"\"F)" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a>0" "6#2\"\"!%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b>0" "6#2\"\"!%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Note that " } {XPPEDIT 18 0 "f(x)=H(x+a)-H(x-a)" "6#/-%\"fG6#%\"xG,&-%\"HG6#,&F'\"\" \"%\"aGF-F--F*6#,&F'F-F.!\"\"F2" }{TEXT -1 46 ", where H is the Heavis ide function given by " }{XPPEDIT 18 0 "H(x)=PIECEWISE([0 , x<0],[1 , x>0])" "6#/-%\"HG6#%\"xG-%*PIECEWISEG6$7$\"\"!2F'F,7$\"\"\"2F,F'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) *`*`*g(x) = Int(f(x-y)*g(y),y = -infinity .. infinity);" "6#/*(-%\"fG6 #%\"xG\"\"\"%\"*GF)-%\"gG6#F(F)-%$IntG6$*&-F&6#,&F(F)%\"yG!\"\"F)-F,6# F5F)/F5;,$%)infinityGF6F<" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int((H(x-y+a)-H(x-y-a))*y*exp(-b*y^2),y \+ = -infinity .. infinity);" "6#/%!G-%$IntG6$*(,&-%\"HG6#,(%\"xG\"\"\"% \"yG!\"\"%\"aGF/F/-F+6#,(F.F/F0F1F2F1F1F/F0F/-%$expG6#,$*&%\"bGF/*$F0 \"\"#F/F1F//F0;,$%)infinityGF1FA" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 " Since " }{XPPEDIT 18 0 "H(-x)=-H(x)" "6#/-%\"HG6#,$%\"xG !\"\",$-F%6#F(F)" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "f(x)*`*`*g(x) = Int((H(y-x+a)-H(y-x-a))*y*exp(-b*y^2 ),y = -infinity .. infinity)" "6#/*(-%\"fG6#%\"xG\"\"\"%\"*GF)-%\"gG6# F(F)-%$IntG6$*(,&-%\"HG6#,(%\"yGF)F(!\"\"%\"aGF)F)-F46#,(F7F)F(F8F9F8F 8F)F7F)-%$expG6#,$*&%\"bGF)*$F7\"\"#F)F8F)/F7;,$%)infinityGF8FH" } {TEXT -1 3 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(y*exp(-b*y^2),y = x-a .. x+a);" "6#/%!G-%$IntG6$*&%\"yG\"\"\"-% $expG6#,$*&%\"bGF**$F)\"\"#F*!\"\"F*/F);,&%\"xGF*%\"aGF3,&F7F*F8F*" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-b*y^2)/(-2*b);" "6#/%!G*&-%$e xpG6#,$*&%\"bG\"\"\"*$%\"yG\"\"#F,!\"\"F,,$*&F/F,F+F,F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x+a, ``],[x-a, ``]);" "6#-%*PIECEWISE G6$7$,&%\"xG\"\"\"%\"aGF)%!G7$,&F(F)F*!\"\"F+" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(-2*b);" "6#/%!G*&\"\"\"F&,$*&\"\"#F&%\"bGF&!\" \"F+" }{XPPEDIT 18 0 " ``(exp(-b*(x+a)^2)-exp(-b*(x-a)^2))" "6#-%!G6#, &-%$expG6#,$*&%\"bG\"\"\"*$,&%\"xGF-%\"aGF-\"\"#F-!\"\"F--F(6#,$*&F,F- *$,&F0F-F1F3F2F-F3F3" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (exp(-b*(x-a)^2)-exp(-b*(x+a)^2))/(2*b);" "6# /%!G*&,&-%$expG6#,$*&%\"bG\"\"\"*$,&%\"xGF-%\"aG!\"\"\"\"#F-F2F--F(6#, $*&F,F-*$,&F0F-F1F-F3F-F2F2F-*&F3F-F,F-F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "ali as(H=Heaviside):\nf := x -> H(x+a)-H(x-a):\n'f(x)'=f(x);\ng := x -> x* exp(-b*x^2):\n'g(x)'=g(x);\nInt('f(x-y)*g(y)',y=-infinity..infinity)=I nt(f(x-y)*g(y),y=-infinity..infinity);\n``=simplify(value(rhs(%)));\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%\"HG6#,&F'\"\"\" %\"aGF-F--F*6#,&F'F-F.!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" gG6#%\"xG*&F'\"\"\"-%$expG6#,$*&%\"bGF))F'\"\"#F)!\"\"F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%\"fG6#,&%\"xG\"\"\"%\"yG!\"\"F--% \"gG6#F.F-/F.;,$%)infinityGF/F6-F%6$*(,&-%\"HG6#,(F,F-F.F/%\"aGF-F--F< 6#,(F,F-F.F/F?F/F/F-F.F--%$expG6#,$*&%\"bGF-)F.\"\"#F-F/F-F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"\"#F(*(,&F(!\"\"-%$expG6#,$ **\"\"%F(%\"bGF(%\"aGF(%\"xGF(F(F(F(F3F,-F.6#,$*&F3F(),&F5F(F4F(F)F(F, F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This is equivalent to the expression given above. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "(-1+exp (4*b*a*x))*exp(-b*(x+a)^2)/(2*b)=(exp(-b*(x-a)^2)-exp(-b*(x+a)^2))/(2* b);\ntesteq(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"\"#F' *(,&F'!\"\"-%$expG6#,$**\"\"%F'%\"bGF'%\"aGF'%\"xGF'F'F'F'F2F+-F-6#,$* &F2F'),&F4F'F3F'F(F'F+F'F'F',$*&F&F'*&,&-F-6#,$*&F2F'),&F4F'F3F+F(F'F+ F'F5F+F'F2F+F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "a = 2" "6#/%\"aG\"\"#" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "b = 1" "6#/%\"bG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(x)=H(x+1)-H(x-1)" "6#/-%\"fG6#%\"xG,&-%\"HG6# ,&F'\"\"\"F-F-F--F*6#,&F'F-F-!\"\"F1" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "g(x) = x*exp(-x^2" "6#/-%\"gG6#%\"xG*&F'\"\"\"-%$expG6#,$*$F'\"\"#!\" \"F)" }{TEXT -1 6 ") and " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "h(x) = f(x)*`*`*g(x)" "6#/-%\"hG6#%\"xG*(-%\"fG6#F'\"\" \"%\"*GF,-%\"gG6#F'F," }{XPPEDIT 18 0 "`` = (exp(-(x-2)^2)-exp(-(x+2)^ 2))/2;" "6#/%!G*&,&-%$expG6#,$*$,&%\"xG\"\"\"\"\"#!\"\"F/F0F.-F(6#,$*$ ,&F-F.F/F.F/F0F0F.F/F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The following picture shows the grap hs of " }{XPPEDIT 18 0 "f(x), g(x)" "6$-%\"fG6#%\"xG-%\"gG6#F&" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "alias(H=Heaviside):\nf := x -> H(x+2)-H(x-2):\n'f(x) '=f(x);\ng := x -> x*exp(-x^2):\n'g(x)'=g(x);\nh := x -> (exp(-(x-2)^2 )-exp(-(x+2)^2))/2:\n'h(x)'=h(x);\nplot([f(x),g(x),h(x)],x=-4..4,-1..1 ,ytickmarks=3,color=[red,brown,magenta]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%\"HG6#,&F'\"\"\"\"\"#F-F--F*6#,&F'F-F .!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&F'\"\"\"-% $expG6#,$*$)F'\"\"#F)!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"h G6#%\"xG,&*&#\"\"\"\"\"#F+-%$expG6#,$*$),&F'F+F,!\"\"F,F+F4F+F+*&#F+F, F+-F.6#,$*$),&F'F+F,F+F,F+F4F+F4" }}{PARA 13 "" 1 "" {GLPLOT2D 490 279 279 {PLOTDATA 2 "6(-%'CURVESG6$7co7$$!\"%\"\"!$F*F*7$$!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$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[h(x)]=Fr*[f(x)*`*`*g(x)]" "6#/*&%#FrG\"\"\"7 #-%\"hG6#%\"xGF&*&F%F&7#*(-%\"fG6#F+F&%\"*GF&-%\"gG6#F+F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = F(omega)*G(omega);" "6#/%!G*&-%\"FG6#%&omeg aG\"\"\"-%\"GG6#F)F*" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = i*sqrt(Pi)*sin*2*omega*exp(-omega^2/4);" "6#/ %!G*.%\"iG\"\"\"-%%sqrtG6#%#PiGF'%$sinGF'\"\"#F'%&omegaGF'-%$expG6#,$* &F.F-\"\"%!\"\"F5F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= i*sqrt(Pi) " "6 #/%!G*&%\"iG\"\"\"-%%sqrtG6#%#PiGF'" }{XPPEDIT 18 0 "``((exp(2*i*omega )-exp(-2*i*omega))/(2*i));" "6#-%!G6#*&,&-%$expG6#*(\"\"#\"\"\"%\"iGF- %&omegaGF-F--F)6#,$*(F,F-F.F-F/F-!\"\"F4F-*&F,F-F.F-F4" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(-omega^2/4)" "6#-%$expG6#,$*&%&omegaG\"\"#\"\"%! \"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sqrt(Pi)/2" "6#/%!G*&-%%sqrtG 6#%#PiG\"\"\"\"\"#!\"\"" }{XPPEDIT 18 0 "``(exp(-2*i*omega-omega^2/4)- exp(2*i*omega-omega^2/4))" "6#-%!G6#,&-%$expG6#,&*(\"\"#\"\"\"%\"iGF-% &omegaGF-!\"\"*&F/F,\"\"%F0F0F--F(6#,&*(F,F-F.F-F/F-F-*&F/F,F2F0F0F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "h := x -> ( exp(-(x-2)^2)-exp(-(x+2)^2))/2:\n'h(x)'=h(x);\n`Fourier transform`=int trans[fourier](h(x),x,omega);\n``=map(simplify,expand(convert(rhs(%),e xp)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG,&*&#\"\"\"\" \"#F+-%$expG6#,$*$),&F'F+F,!\"\"F,F+F4F+F+*&#F+F,F+-F.6#,$*$),&F'F+F,F +F,F+F4F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG** ^#!\"\"\"\"\"%#PiG#F(\"\"#-%$sinG6#,$*&F+F(%&omegaGF(F(F(-%$expG6#,$*& \"\"%F'F1F+F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\" #F(*&%#PiG#F(F)-%$expG6#,$*(\"\"%!\"\"%&omegaGF(,&F4F3^#\"\")F(F(F(F(F (F3*&F,F(*&F+F,-F.6#,$*(F2F3F4F(,&F4F(F6F(F(F3F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "OR " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "h := x -> ( exp(-(x-2)^2)-exp(-(x+2)^2))/2:\n'h(x)'=h(x);\n`Fourier transform`=map 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1*G64$=]9F17$$\"3y&******>am%**F[q$!3-IL$[`+YE\"F17$$\"3h****\\nQO?6F1 $!3ILt,Fa::5F17$$\"3k*****\\JigC\"F1$!39Cj()))yxqsF[q7$$\"3!)***\\i-#o n8F1$!3#3yu#Hlu(Q%F[q7$$\"3%*****\\Pl\"F[q7$$\"3*)* **\\7pRVi\"F1$\"3_v.Vsp9&z*FS7$$\"3#)*****\\kx$f-U\\F [q7$$\"32***\\(38WM?F1$\"3YH'))*['z#Q]F[q7$$\"3U****\\d(Ho1#F1$\"37Dl' QG2,5&F[q7$$\"3w***\\i?=#*4#F1$\"32p-A0d8H^F[q7$$\"3p*****\\l1;8#F1$\" 3Lb3H#F1$\"36.u:`,rP]F[q7$$\"3I++D,?xGAF1$\"3'oa(othLa\\F[q7$$\"3A+++]/;hA F1$\"3c\"3%RNq*z%[F[q7$$\"3#)***\\PWb&yBF1$\"32$plB_+TI%F[q7$$\"3Y**** \\P/&f\\#F1$\"3-qeY(>%=*e$F[q7$$\"3q+++5zj_FF1$\"3'pda#=b>r=F[q7$$\"3= ****\\<3;%*HF1$\"3OZO1g5uwaFS7$$\"3n****\\K8R?JF1$\"3c#\\HUL&y'e'F>7$$ \"3;++]Z=iYKF1$!3#Rv=\\]r-l#FS7$$\"3#)***\\([_.qLF1$!3GMJ1nY&FS7 $$\"3W****\\PM&=v$F1$!3rgK&o&yAK\\FS7$$\"3v+++gzs+SF1$!3VX)\\V@_k?$FS7 $$\"35+++0\"Q_D%F1$!3K>laEBa=:FS7$$\"3q++]x2k2XF1$!31mmKkgm!R%F>7$$\"3 d+++?EdRZF1$\"3dMO$*G$)\\0NF-7$$\"3M+++&o#R0]F1$\"3y2G#R)p%p'=F>7$$\"3 ++++?`9V_F1$\"3d]/!*)e?Gg\"F>7$$\"3G++]<#Rm\\&F1$\"3mtRIs#yYH*F-7$$\"3 F++]A_ERdF1$\"3!oL()=G0R;%F-7$$\"\"'F*$\"3]LMe[)*ot6F--%&TITLEG6#%*F(w )~G(w)G-%&COLORG6&%$RGBG$Fg[m!\"\"$\"\"%Fc\\mF*-%+AXESLABELSG6$Q\"w6\" Q!Fj\\m-%%VIEWG6$;F(Ff[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 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "f(x)=PIECEWISE ([1 , abs(x)a ])" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\" \"\"2-%$absG6#F'%\"aG7$\"\"!2F1-F/6#F'" }{TEXT -1 9 " , where " } {XPPEDIT 18 0 "a>0" "6#2\"\"!%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Note that " }{XPPEDIT 18 0 "f(x)=H(x+a)-H(x-a)" "6#/-%\"fG6#%\"xG,&-%\"HG6#,&F'\"\"\"%\"aGF- F--F*6#,&F'F-F.!\"\"F2" }{TEXT -1 46 ", where H is the Heaviside funct ion given by " }{XPPEDIT 18 0 "H(x)=PIECEWISE([0 , x<0],[1 , x>0])" " 6#/-%\"HG6#%\"xG-%*PIECEWISEG6$7$\"\"!2F'F,7$\"\"\"2F,F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)*`*`*f(x) = \+ Int(f(x-y)*f(y),y = -infinity .. infinity);" "6#/*(-%\"fG6#%\"xG\"\"\" %\"*GF)-F&6#F(F)-%$IntG6$*&-F&6#,&F(F)%\"yG!\"\"F)-F&6#F4F)/F4;,$%)inf inityGF5F;" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int((H(x-y+a)-H(x-y-a))*(H(y+a)-H(y-a)),y = -infin ity .. infinity);" "6#/%!G-%$IntG6$*&,&-%\"HG6#,(%\"xG\"\"\"%\"yG!\"\" %\"aGF/F/-F+6#,(F.F/F0F1F2F1F1F/,&-F+6#,&F0F/F2F/F/-F+6#,&F0F/F2F1F1F/ /F0;,$%)infinityGF1F@" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 " \+ Since " }{XPPEDIT 18 0 "H(-x)=-H(x)" "6#/-%\"HG6#,$%\"xG!\"\",$-F%6#F (F)" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)*`*`*f(x) = Int((H(y-x+a)-H(y-x-a))*(H(y+a)-H(y-a)),y = -inf inity .. infinity);" "6#/*(-%\"fG6#%\"xG\"\"\"%\"*GF)-F&6#F(F)-%$IntG6 $*&,&-%\"HG6#,(%\"yGF)F(!\"\"%\"aGF)F)-F36#,(F6F)F(F7F8F7F7F),&-F36#,& F6F)F8F)F)-F36#,&F6F)F8F7F7F)/F6;,$%)infinityGF7FF" }{TEXT -1 3 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(H(y+a)-H(y -a),y = x-a .. x+a);" "6#/%!G-%$IntG6$,&-%\"HG6#,&%\"yG\"\"\"%\"aGF.F. -F*6#,&F-F.F/!\"\"F3/F-;,&%\"xGF.F/F3,&F7F.F/F." }{TEXT -1 2 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = [PIECEWISE([0, y <= -a],[y+a, -a < y and y < a],[2*a, a <= y])];" "6#/%!G7#-%*PIECEWIS EG6%7$\"\"!1%\"yG,$%\"aG!\"\"7$,&F,\"\"\"F.F232,$F.F/F,2F,F.7$*&\"\"#F 2F.F21F.F," }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([x+a, ``],[``, \+ ``],[x-a, ``]);" "6#-%*PIECEWISEG6%7$,&%\"xG\"\"\"%\"aGF)%!G7$F+F+7$,& F(F)F*!\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = H(y+a)*(y+a)+H(y -a)*(a-y);" "6#/%!G,&*&-%\"HG6#,&%\"yG\"\"\"%\"aGF,F,,&F+F,F-F,F,F,*&- F(6#,&F+F,F-!\"\"F,,&F-F,F+F3F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIE CEWISE([x+a, ``],[x-a, ``])" "6#-%*PIECEWISEG6$7$,&%\"xG\"\"\"%\"aGF)% !G7$,&F(F)F*!\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = H(y+a)*(y+a)-H( y-a)*(y-a);" "6#/%!G,&*&-%\"HG6#,&%\"yG\"\"\"%\"aGF,F,,&F+F,F-F,F,F,*& -F(6#,&F+F,F-!\"\"F,,&F+F,F-F3F,F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PI ECEWISE([x+a, ``],[x-a, ``])" "6#-%*PIECEWISEG6$7$,&%\"xG\"\"\"%\"aGF) %!G7$,&F(F)F*!\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = H(x+2*a)*(x+2* a)-H(x)*x-[H(x)*x-H(x-2*a)*(x-2*a)];" "6#/%!G,(*&-%\"HG6#,&%\"xG\"\"\" *&\"\"#F,%\"aGF,F,F,,&F+F,*&F.F,F/F,F,F,F,*&-F(6#F+F,F+F,!\"\"7#,&*&-F (6#F+F,F+F,F,*&-F(6#,&F+F,*&F.F,F/F,F5F,,&F+F,*&F.F,F/F,F5F,F5F5" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = H(x+2*a)*(x+2*a)-2*H(x)*x+H(x-2*a) *(x-2*a);" "6#/%!G,(*&-%\"HG6#,&%\"xG\"\"\"*&\"\"#F,%\"aGF,F,F,,&F+F,* &F.F,F/F,F,F,F,*(F.F,-F(6#F+F,F+F,!\"\"*&-F(6#,&F+F,*&F.F,F/F,F5F,,&F+ F,*&F.F,F/F,F5F,F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([0, x < 2*a],[x+2*a, -2*a < x and x < 0],[2*a-x, 0 < x and x <= 2*a],[0, 2* a < x]);" "6#/%!G-%*PIECEWISEG6&7$\"\"!2%\"xG*&\"\"#\"\"\"%\"aGF.7$,&F +F.*&F-F.F/F.F.32,$*&F-F.F/F.!\"\"F+2F+F)7$,&*&F-F.F/F.F.F+F732F)F+1F+ *&F-F.F/F.7$F)2*&F-F.F/F.F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEW ISE([2*a-abs(x), abs(x) < 2*a],[0, 2*a < abs(x)]);" "6#/%!G-%*PIECEWIS EG6$7$,&*&\"\"#\"\"\"%\"aGF,F,-%$absG6#%\"xG!\"\"2-F/6#F1*&F+F,F-F,7$ \"\"!2*&F+F,F-F,-F/6#F1" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Fo r example, when " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "f(x)=H(x+1)-H(x-1)" "6#/-%\"fG6#%\"xG,&-%\"HG6#, &F'\"\"\"F-F-F--F*6#,&F'F-F-!\"\"F1" }{TEXT -1 6 " and " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "h(x) = f(x)*`*`*f(x);" "6#/-% \"hG6#%\"xG*(-%\"fG6#F'\"\"\"%\"*GF,-F*6#F'F," }{TEXT -1 3 " = " } {XPPEDIT 18 0 "H(x+2)*(x+2)-2*H(x)*x+H(x-2)*(x-2);" "6#,(*&-%\"HG6#,&% \"xG\"\"\"\"\"#F*F*,&F)F*F+F*F*F**(F+F*-F&6#F)F*F)F*!\"\"*&-F&6#,&F)F* F+F0F*,&F)F*F+F0F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([2-a bs(x), abs(x) < 2],[0, 2 < abs(x)]);" "6#/%!G-%*PIECEWISEG6$7$,&\"\"# \"\"\"-%$absG6#%\"xG!\"\"2-F-6#F/F*7$\"\"!2F*-F-6#F/" }{TEXT -1 3 " . \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "alias(H=Heaviside):\nf := x -> H(x+1)-H(x-1):\n'f(x)'=f(x);\n h := x -> H(x+2)*(x+2)-2*H(x)*x+H(x-2)*(x-2):\n'h(x)'=h(x);\nplot([f(x ),h(x)],x=-4..4,color=[red,magenta],thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%\"HG6#,&F'\"\"\"F-F-F--F*6#,&F'F-F-! \"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG,(*&-%\"HG6#,& F'\"\"\"\"\"#F.F.F-F.F.*(F/F.-F+F&F.F'F.!\"\"*&-F+6#,&F'F.F/F2F.F6F.F. " }}{PARA 13 "" 1 "" {GLPLOT2D 531 193 193 {PLOTDATA 2 "6'-%'CURVESG6$ 7co7$$!\"%\"\"!$F*F*7$$!3ommmmFiDQ!#1DBF/F +7$$!3kmmmw))yr@F/F+7$$!3;+++S(R#**>F/F+7$$!30++++@)f#=F/F+7$$!3-+++gi ,f;F/F+7$$!3qmmm\"G&R2:F/F+7$$!3XLLLtK5F8F/F+7$$!3_LLL$yP2D\"F/F+7$$!3 eLLL$HsV<\"F/F+7$$!39++v$oc*H6F/F+7$$!3!pmmT2Tb3\"F/F+7$$!3;+]PpKLj5F/ F+7$$!3mLLeka7T5F/F+7$$!3=+v=i:-I5F/F+7$$!3$pm\"zfw\"*=5F/F+7$$!3>]Pf3 dO85F/F+7$$!3oLeRdP\"y+\"F/F+7$$!3%p\"z>1=E-5F/F+7$$!3+-++]&)4n**!#=$ \"\"\"F*7$$!3cpmmT(******z-6j'F[qF\\q7$$\"3q\"** ****4#32$)F[qF\\q7$$\"3q#****\\qM8F/F+7$$\"3% )*******HSu]\"F/F+7$$\"3'HLL$ep'Rm\"F/F+7$$\"3')******R>4N=F/F+7$$\"3# emm;@2h*>F/F+7$$\"3]*****\\c9W;#F/F+7$$\"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$$\"3=+++:oF/$\"3p*)******z!*Q()!#>7$FR$\"3[*********y,u \"F[q7$FU$\"3!)*******RP)4MF[q7$FX$\"3.LLL$=Zg#\\F[q7$Fen$\"3clmmms'*G nF[q7$F[o$\"31kmmmqFc#)F[q7$Fip$\"3!)*****\\9!H.5F/7$Fbq$\"3Immm1:bg6F /7$Feq$\"3<+++X@4L8F/7$Fhq$\"31+++N;R(\\\"F/7$F[r$\"3wmmm;4#)o;F/7$F^r $\"3jmmm6lCE=F/7$$!3m0++]-6&)))Fcz$\"3%*****\\(*)[6\">F/7$Far$\"3ELLL$ G^g*>F/7$$\"3'oHLL3+TU)Fcz$\"3.nm;***ed\">F/7$Fer$\"3Knmm\"Gpv#=F/7$Fh r$\"3S+++l/.u;F/7$F[s$\"34nmmOV?3:F/7$F^s$\"3G+++?(*)oL\"F/7$Fas$\"3#3 +++z\"Hp6F/7$Fgs$\"3j+++v@825F/7$Feu$\"352+++d\"3F)F[q7$F[v$\"3'*QLLL4 )Hl'F[q7$F^v$\"3c,+++qfD\\F[q7$Fav$\"3Kqmm;/LgLF[q7$Fdv$\"3M,+++13\\;F [q7$$\"3&GLLed*f:>F/$\"3S:nm;C/S%)Fcz7$Fgv$\"3'R " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now \+ " }{XPPEDIT 18 0 "Fr*[f(x)]=F(omega)" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xG F&-%\"FG6#%&omegaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*sin(omega)/ome ga;" "6#*(\"\"#\"\"\"-%$sinG6#%&omegaGF%F)!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Fr*[h(x)] = Fr*[f(x)*`*`*f(x)];" "6#/*&%#FrG\"\"\"7# -%\"hG6#%\"xGF&*&F%F&7#*(-%\"fG6#F+F&%\"*GF&-F06#F+F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = F(omega)^2; " "6#/%!G*$-%\"FG6#%&omegaG\"\"#" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ 4*sin^2*omega/(omega^2);" "6#/%!G**\"\"%\"\"\"*$%$sinG\"\"#F'%&omegaGF '*$F+F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "alias(H=Heaviside):\nh := x -> H(x+2)*(x+2)-2*H(x)*x+H(x-2)*(x-2):\n'h(x)'=h(x);\n`Fourier transf orm`=inttrans[fourier](rhs(%),x,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG,(*&-%\"HG6#,&F'\"\"\"\"\"#F.F.F-F.F.*(F/F.-F+F&F. F'F.!\"\"*&-F+6#,&F'F.F/F2F.F6F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%2Fourier~transformG,$*(\"\"%\"\"\"%&omegaG!\"#-%$sinG6#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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 " L et " }{XPPEDIT 18 0 "f(x) = 1/(a^2+x^2);" "6#/-%\"fG6#%\"xG*&\"\"\"F), &*$%\"aG\"\"#F)*$F'F-F)!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a >0" "6#2\"\"!%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)*`*`*f(x) = Int( f(x-y)*f(y),y = -infinity .. infinity);" "6#/*(-%\"fG6#%\"xG\"\"\"%\"* GF)-F&6#F(F)-%$IntG6$*&-F&6#,&F(F)%\"yG!\"\"F)-F&6#F4F)/F4;,$%)infinit yGF5F;" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(``(1/(a^2+(x-y)^2))*``(1/(a^2-y^2)),y = -infinity .. in finity);" "6#/%!G-%$IntG6$*&-F$6#*&\"\"\"F,,&*$%\"aG\"\"#F,*$,&%\"xGF, %\"yG!\"\"F0F,F5F,-F$6#*&F,F,,&*$F/F0F,*$F4F0F5F5F,/F4;,$%)infinityGF5 F?" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 53 "1/((a^2+(x-y)^2)*(a^2+y^2));\n``=convert(%,p arfrac,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&,&*$)%\"aG\" \"#F$F$*$),&%\"xGF$%\"yG!\"\"F*F$F$F$,&F'F$*$)F/F*F$F$F$F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&**,&*&\"\"$\"\"\"%\"xGF*F**&\"\"#F*%\" yGF*!\"\"F*,**$)%\"aGF-F*F**$)F+F-F*F**(F-F*F+F*F.F*F/*$)F.F-F*F*F/F+F /,&F4F**&\"\"%F*F2F*F*F/F***,&F+F**&F-F*F.F*F*F*,&F1F*F7F*F/F+F/F9F/F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Using the identity" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "1/((a^2+(x-y)^2)*(a ^2+y^2)) = (x+2*y)/(x*(4*a^2+x^2)*(a^2+y^2))+(3*x-2*y)/(x*(x^2+4*a^2)* (a^2+(x-y)^2));" "6#/*&\"\"\"F%*&,&*$%\"aG\"\"#F%*$,&%\"xGF%%\"yG!\"\" F*F%F%,&*$F)F*F%*$F.F*F%F%F/,&*&,&F-F%*&F*F%F.F%F%F%*(F-F%,&*&\"\"%F%* $F)F*F%F%*$F-F*F%F%,&*$F)F*F%*$F.F*F%F%F/F%*&,&*&\"\"$F%F-F%F%*&F*F%F. F%F/F%*(F-F%,&*$F-F*F%*&F:F%*$F)F*F%F%F%,&*$F)F*F%*$,&F-F%F.F/F*F%F%F/ F%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x)*`*`*f(x) = 1/(x*(4*a^2+x^2)); " "6#/*(-%\"fG6#%\"xG\"\"\"%\"*GF)-F&6#F(F)*&F)F)*&F(F),&*&\"\"%F)*$% \"aG\"\"#F)F)*$F(F4F)F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(Int(( x+2*y)/(a^2+y^2),y = -infinity .. infinity)+Int((3*x-2*y)/(a^2+(x-y)^2 ),y = -infinity .. infinity));" "6#-%!G6#,&-%$IntG6$*&,&%\"xG\"\"\"*& \"\"#F-%\"yGF-F-F-,&*$%\"aGF/F-*$F0F/F-!\"\"/F0;,$%)infinityGF5F9F--F( 6$*&,&*&\"\"$F-F,F-F-*&F/F-F0F-F5F-,&*$F3F/F-*$,&F,F-F0F5F/F-F5/F0;,$F 9F5F9F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x+2*y)/(a^2+y^2) ,y = -R .. R) =x*Int(1/(a^2+y^2),y = -R .. R) + Int(2*y/(a^2+y^2),y = \+ -R .. R)" "6#/-%$IntG6$*&,&%\"xG\"\"\"*&\"\"#F*%\"yGF*F*F*,&*$%\"aGF,F **$F-F,F*!\"\"/F-;,$%\"RGF2F6,&*&F)F*-F%6$*&F*F*,&*$F0F,F**$F-F,F*F2/F -;,$F6F2F6F*F*-F%6$*(F,F*F-F*,&*$F0F,F**$F-F,F*F2/F-;,$F6F2F6F*" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*x/a;" "6#/%!G*(\"\"#\"\"\"%\"xGF '%\"aG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(R/a)" "6#-%'arctan G6#*&%\"RG\"\"\"%\"aG!\"\"" }{TEXT -1 5 " + 0," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Int((x+2*y)/(a^2+y ^2),y = -infinity .. infinity) =2*x/a" "6#/-%$IntG6$*&,&%\"xG\"\"\"*& \"\"#F*%\"yGF*F*F*,&*$%\"aGF,F**$F-F,F*!\"\"/F-;,$%)infinityGF2F6*(F,F *F)F*F0F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit( arctan(R/a),R=infini ty)" "6#-%&LimitG6$-%'arctanG6#*&%\"RG\"\"\"%\"aG!\"\"/F*%)infinityG" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*x/a;" "6#/%!G*(%#PiG\"\"\"%\"xGF'%\"aG!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "u = x - y" "6#/% \"uG,&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "du/dy =-1" "6#/*&%#duG\"\"\"%#dyG!\"\",$F&F(" }{TEXT -1 2 ", " }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int((3*x-2*y)/(a^2+(x-y)^2),y = -R .. R) = -Int((x+2*u)/(a^2+u^2),u = x+R .. x-R);" "6#/-%$IntG6$*&,& *&\"\"$\"\"\"%\"xGF+F+*&\"\"#F+%\"yGF+!\"\"F+,&*$%\"aGF.F+*$,&F,F+F/F0 F.F+F0/F/;,$%\"RGF0F9,$-F%6$*&,&F,F+*&F.F+%\"uGF+F+F+,&*$F3F.F+*$F@F.F +F0/F@;,&F,F+F9F+,&F,F+F9F0F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int((x+2*u)/(a^2+u^2),u = x-R .. x +R);" "6#/%!G-%$IntG6$*&,&%\"xG\"\"\"*&\"\"#F+%\"uGF+F+F+,&*$%\"aGF-F+ *$F.F-F+!\"\"/F.;,&F*F+%\"RGF3,&F*F+F7F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = ln((a^2+(x+R)^2)/(a^2+(x- R)^2))+x/a;" "6#/%!G,&-%#lnG6#*&,&*$%\"aG\"\"#\"\"\"*$,&%\"xGF.%\"RGF. F-F.F.,&*$F,F-F.*$,&F1F.F2!\"\"F-F.F7F.*&F1F.F,F7F." }{XPPEDIT 18 0 "` `(arctan((x+R)/a)+arctan((R-x)/a));" "6#-%!G6#,&-%'arctanG6#*&,&%\"xG \"\"\"%\"RGF-F-%\"aG!\"\"F--F(6#*&,&F.F-F,F0F-F/F0F-" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int((3*x-2*y)/(a^2+(x-y)^2),y = -infinity .. infinity) \+ = ``;" "6#/-%$IntG6$*&,&*&\"\"$\"\"\"%\"xGF+F+*&\"\"#F+%\"yGF+!\"\"F+, &*$%\"aGF.F+*$,&F,F+F/F0F.F+F0/F/;,$%)infinityGF0F9%!G" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``,R=infini ty)" "6#-%&LimitG6$%!G/%\"RG%)infinityG" }{XPPEDIT 18 0 "ln((a^2+(x+R) ^2)/(a^2+(x-R)^2))+x/a" "6#,&-%#lnG6#*&,&*$%\"aG\"\"#\"\"\"*$,&%\"xGF, %\"RGF,F+F,F,,&*$F*F+F,*$,&F/F,F0!\"\"F+F,F5F,*&F/F,F*F5F," }{XPPEDIT 18 0 "``(arctan((x+R)/a)+arctan((R-x)/a));" "6#-%!G6#,&-%'arctanG6#*&, &%\"xG\"\"\"%\"RGF-F-%\"aG!\"\"F--F(6#*&,&F.F-F,F0F-F/F0F-" }{TEXT -1 1 "\n" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=0+x/a" "6 #/%!G,&\"\"!\"\"\"*&%\"xGF'%\"aG!\"\"F'" }{XPPEDIT 18 0 "``(Pi/2+Pi/2) " "6#-%!G6#,&*&%#PiG\"\"\"\"\"#!\"\"F)*&F(F)F*F+F)" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*x/a;" "6#/%!G *(%#PiG\"\"\"%\"xGF'%\"aG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "It now follows that " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)*`*`*f(x) = 1/(x *(4*a^2+x^2));" "6#/*(-%\"fG6#%\"xG\"\"\"%\"*GF)-F&6#F(F)*&F)F)*&F(F), &*&\"\"%F)*$%\"aG\"\"#F)F)*$F(F4F)F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(2*Pi*x/a);" "6#-%!G6#**\"\"#\"\"\"%#PiGF(%\"xGF(%\"aG!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 6 " = " }{XPPEDIT 18 0 "2*Pi/(a*(4*a^2+x^2));" "6#*(\"\"# \"\"\"%#PiGF%*&%\"aGF%,&*&\"\"%F%*$F(F$F%F%*$%\"xGF$F%F%!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "f(x) = 1/(1+x^2);" "6#/-%\"fG6#%\" xG*&\"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "h(x) = f(x)*`*`*f(x);" "6#/-%\"hG6#%\"xG*(-%\"fG6#F'\"\"\"%\"*GF ,-F*6#F'F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*Pi/(4+x^2);" "6#*(\"\" #\"\"\"%#PiGF%,&\"\"%F%*$%\"xGF$F%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "alias(H=Heaviside):\nf := x -> 1/(1+x^2): \n'f(x)'=f(x);\nh := x -> 2*Pi/(4+x^2):\n'h(x)'=h(x);\nplot([f(x),h(x) ],x=-4..4,color=[red,magenta],thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&\"\"\"F),&*$)F'\"\"#F)F)F)F)!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG,$*(\"\"#\"\"\"%#PiGF+,& \"\"%F+*$)F'F*F+F+!\"\"F+" }}{PARA 13 "" 1 "" {GLPLOT2D 517 169 169 {PLOTDATA 2 "6'-%'CURVESG6$7ao7$$!\"%\"\"!$\"3_qk7$$!3ommmmF iDQ!#<$\"33)4LpEZdR'F-7$$!35LLLo!)*Qn$F1$\"3UdLrN\"Gx*oF-7$$!3nmmmwxE. 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" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[h(x)] = Fr*[f(x)* `*`*f(x)];" "6#/*&%#FrG\"\"\"7#-%\"hG6#%\"xGF&*&F%F&7#*(-%\"fG6#F+F&% \"*GF&-F06#F+F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = F(omega)^2;" "6#/%!G*$-%\"FG6#%&omegaG\"\"#" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi^2*exp(-2*abs(omega));" "6#/%!G* &%#PiG\"\"#-%$expG6#,$*&F'\"\"\"-%$absG6#%&omegaGF-!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "alias(H=Heaviside):\nh := x -> 2*Pi/(4+x^2):\n'h(x)' =h(x);\n`Fourier transform`=inttrans[fourier](h(x),x,omega);\n``=conve rt(rhs(%),piecewise);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"x G,$*(\"\"#\"\"\"%#PiGF+,&\"\"%F+*$)F'F*F+F+!\"\"F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%2Fourier~transformG*&)%#PiG\"\"#\"\"\",&*&-%$expG6# ,$*&F(F)%&omegaGF)!\"\"F)-%\"HG6#F1F)F)*&-F-6#,$*&F(F)F1F)F)F)-F46#,$F 1F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6%7$*&)% #PiG\"\"#\"\"\"-%$expG6#,$*&F,F-%&omegaGF-F-F-2F3\"\"!7$*&F*F-%*undefi nedGF-/F3F57$*&F*F--F/6#,$*&F,F-F3F-!\"\"F-2F5F3" }}}{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 5 " Let " }{XPPEDIT 18 0 "f(x)=PIECEWISE([1 , abs(x)a ])" "6#/-%\"fG6#%\"xG-%* PIECEWISEG6$7$\"\"\"2-%$absG6#F'%\"aG7$\"\"!2F1-F/6#F'" }{TEXT -1 9 " \+ , and " }{XPPEDIT 18 0 "g(x) = sin*b*x/x;" "6#/-%\"gG6#%\"xG**%$sinG \"\"\"%\"bGF*F'F*F'!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a>0" "6#2\"\"!%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b>0" "6#2\"\"!%\"b G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Note that " }{XPPEDIT 18 0 "f(x)=H(x+a)-H(x-a)" "6#/-%\" fG6#%\"xG,&-%\"HG6#,&F'\"\"\"%\"aGF-F--F*6#,&F'F-F.!\"\"F2" }{TEXT -1 46 ", where H is the Heaviside function given by " }{XPPEDIT 18 0 "H( x)=PIECEWISE([0 , x<0],[1 , x>0])" "6#/-%\"HG6#%\"xG-%*PIECEWISEG6$7$ \"\"!2F'F,7$\"\"\"2F,F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)*`*`*g(x) = Int(f(x-y)*g(y),y = -infinity .. \+ infinity);" "6#/*(-%\"fG6#%\"xG\"\"\"%\"*GF)-%\"gG6#F(F)-%$IntG6$*&-F& 6#,&F(F)%\"yG!\"\"F)-F,6#F5F)/F5;,$%)infinityGF6F<" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int((H(x-y+a)-H( x-y-a))*``(sin*b*y/y),y = -infinity .. infinity);" "6#/%!G-%$IntG6$*&, &-%\"HG6#,(%\"xG\"\"\"%\"yG!\"\"%\"aGF/F/-F+6#,(F.F/F0F1F2F1F1F/-F$6#* *%$sinGF/%\"bGF/F0F/F0F1F//F0;,$%)infinityGF1F>" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 8 " Since " }{XPPEDIT 18 0 "H(-x)=-H(x)" "6# /-%\"HG6#,$%\"xG!\"\",$-F%6#F(F)" }{TEXT -1 2 ", " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)*`*`*g(x) = Int((H(y-x+a)-H(y-x- a))*``(sin*b*y/y),y = -infinity .. infinity);" "6#/*(-%\"fG6#%\"xG\"\" \"%\"*GF)-%\"gG6#F(F)-%$IntG6$*&,&-%\"HG6#,(%\"yGF)F(!\"\"%\"aGF)F)-F4 6#,(F7F)F(F8F9F8F8F)-%!G6#**%$sinGF)%\"bGF)F7F)F7F8F)/F7;,$%)infinityG F8FF" }{TEXT -1 3 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(sin*b*y/y,y = x-a .. x+a);" "6#/%!G-%$IntG6$**%$sinG\" \"\"%\"bGF*%\"yGF*F,!\"\"/F,;,&%\"xGF*%\"aGF-,&F1F*F2F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "`` = Si(b*x);" "6#/%!G-%#SiG6#*&%\"bG\"\"\"%\"xGF*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x+a, ``],[``, ``],[x-a, ``] );" "6#-%*PIECEWISEG6%7$,&%\"xG\"\"\"%\"aGF)%!G7$F+F+7$,&F(F)F*!\"\"F+ " }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 41 "where Si is the sine \+ integral defined by " }{XPPEDIT 18 0 "Si(x) = Int(sin*t/t,t = 0 .. x); " "6#/-%#SiG6#%\"xG-%$IntG6$*(%$sinG\"\"\"%\"tGF-F.!\"\"/F.;\"\"!F'" } {TEXT -1 1 "." }{TEXT 264 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)*`*`*g(x) = Si( b*(x+a))-Si(b*(x-a))" "6#/*(-%\"fG6#%\"xG\"\"\"%\"*GF)-%\"gG6#F(F),&-% #SiG6#*&%\"bGF),&F(F)%\"aGF)F)F)-F06#*&F3F),&F(F)F5!\"\"F)F:" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "alias(H=Heaviside): \nf := x -> H(x+a)-H(x-a):\n'f(x)'=f(x);\ng := x -> sin(b*x)/x:\n'g(x) '=g(x);\nInt('f(x-y)*g(y)',y=-infinity..infinity)=Int(f(x-y)*g(y),y=-i nfinity..infinity);\n``=factor(value(rhs(%)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%\"HG6#,&F'\"\"\"%\"aGF-F--F*6#,&F'F-F .!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&-%$sinG6#* &%\"bG\"\"\"F'F.F.F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6 $*&-%\"fG6#,&%\"xG\"\"\"%\"yG!\"\"F--%\"gG6#F.F-/F.;,$%)infinityGF/F6- F%6$*(,&-%\"HG6#,(F,F-F.F/%\"aGF-F--F<6#,(F,F-F.F/F?F/F/F--%$sinG6#*&% \"bGF-F.F-F-F.F/F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%#SiG6#*& %\"bG\"\"\",&%\"xGF+%\"aGF+F+F+-F'6#*&F*F+,&F-!\"\"F.F+F+F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "a = 2" "6#/%\"aG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b = 3 " "6#/%\"bG\"\"$" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "f(x)=H(x+1)-H( x-1)" "6#/-%\"fG6#%\"xG,&-%\"HG6#,&F'\"\"\"F-F-F--F*6#,&F'F-F-!\"\"F1 " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "g(x) = sin*3*x/x;" "6#/-%\"gG6#%\"x G**%$sinG\"\"\"\"\"$F*F'F*F'!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "h(x) = f(x)*`*`*g(x)" "6#/-%\"hG6#%\"xG*(-%\"fG6#F'\"\"\"%\"*GF,-% \"gG6#F'F," }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Si(3*(x+2))-Si(3*(x-2)) ;" "6#,&-%#SiG6#*&\"\"$\"\"\",&%\"xGF)\"\"#F)F)F)-F%6#*&F(F),&F+F)F,! \"\"F)F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "alias(H=Heaviside):\nf := x -> H(x +2)-H(x-2): 'f(x)'=f(x);\ng := x -> sin(3*x)/x: 'g(x)'=g(x);\nh := x - > Si(3*(x+2))-Si(3*(x-2)): 'h(x)'=h(x);\nplot([f(x),g(x),h(x)],x=-8..8 ,ytickmarks=3,color=[red,brown,magenta]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%\"HG6#,&F'\"\"\"\"\"#F-F--F*6#,&F'F-F .!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&-%$sinG6#, $*&\"\"$\"\"\"F'F/F/F/F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" hG6#%\"xG,&-%#SiG6#,&*&\"\"$\"\"\"F'F/F/\"\"'F/F/-F*6#,&*&F.F/F'F/F/F0 !\"\"F5" }}{PARA 13 "" 1 "" {GLPLOT2D 592 250 250 {PLOTDATA 2 "6(-%'CU RVESG6$7co7$$!\")\"\"!$F*F*7$$!3OLLLLbC^w!#l$F/F+7$$!3/ +++?D.=LF/F+7$$!3SLLLj0z9IF/F+7$$!3!pmmma1Ul#F/F+7$$!3/nmmmbZ,DF/F+7$$ !3=nmm'eW([BF/F+7$$!3G++]nL\"*fAF/F+7$$!3yLLL[@3r@F/F+7$$!3M++vQlmE@F/ F+7$$!3Knm;H4D#3#F/F+7$$!3O+]PCJ/g?F/F+7$$!3'QL$e>`$y.#F/F+7$$!3Q+v=<9 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" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[h(x)]=Fr*[f(x)*`*`*g(x)]" "6#/ *&%#FrG\"\"\"7#-%\"hG6#%\"xGF&*&F%F&7#*(-%\"fG6#F+F&%\"*GF&-%\"gG6#F+F &F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "`` = F(omega)*G(omega);" "6#/%!G*&-%\"FG6#%&omegaG\"\"\"-%\"GG6# F)F*" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*sin*2*omega/omega;" "6#/%!G*.\"\"#\"\"\"%#PiGF'%$sinGF 'F&F'%&omegaGF'F*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[H(omega+3)-H( omega-3)];" "6#7#,&-%\"HG6#,&%&omegaG\"\"\"\"\"$F*F*-F&6#,&F)F*F+!\"\" F/" }{TEXT -1 2 ". 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