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{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
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0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
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0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 31 "EXAMPLES OF FOURIER TRAN
SFORMS:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Maple has a package inttrans that \+
performs Fourier transformations and integrals.\nTo load it type" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 75 "We will here be concerned only with the f
unctions fourier,  and invfourier " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "fourier  evaluates \n" }{XPPEDIT 
18 0 "F( omega):=int(f(t)*e^(-i omega t),t=- infinity..infinity)" "6#>
-%\"FG6#%&omegaG-%$intG6$*&-%\"fG6#%\"tG\"\"\")%\"eG,$*(%\"iGF0F'F0F/F
0!\"\"F0/F/;,$%)infinityGF6F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "
while invfourier evaluates\n" }{XPPEDIT 18 0 "f(t):=int( F( omega) /(2
* pi)*e^(i *omega *t), omega=- infinity..infinity)" "6#>-%\"fG6#%\"tG-
%$intG6$*(-%\"FG6#%&omegaG\"\"\"*&\"\"#F0%#piGF0!\"\")%\"eG*(%\"iGF0F/
F0F'F0F0/F/;,$%)infinityGF4F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "E
XAMPLE:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=t->1/(t^2+1);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"tG6\"6$%)operatorG%&ar
rowGF(*&\"\"\"F-,&*$)9$\"\"#F-F-F-F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "F:=fourier(f(t),t,w);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"FG,&*(-%$expG6#%\"wG\"\"\"%#PiGF+-%*HeavisideG6#,$F
*!\"\"F+F+*(-F(F/F+F,F+-F.F)F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "unapply(F,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%
\"xG6\"6$%)operatorG%&arrowGF&,&*(-%$expG6#%\"wG\"\"\"%#PiGF0-%*Heavis
ideG6#,$F/!\"\"F0F0*(-F-F4F0F1F0-F3F.F0F0F&F&F&" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 48 "The Heaviside function is just the step function" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Heaviside(x)=  1 if x>0, " }}
{PARA 0 "" 0 "" {TEXT -1 31 "                       0 if x<0" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "so F(w)  is just " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "F1:=omega->exp(-abs(omega))*pi;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#F1GR6#%&omegaG6\"6$%)operatorG%&arr
owGF(*&-%$expG6#,$-%$absG6#9$!\"\"\"\"\"%#piGF6F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "h:=invfourier(F1(omega),omega,t);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG*&%#piG\"\"\"*&,&*$)%\"tG\"\"#
F'F'F'F'F'%#PiGF'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "For som
e reason Maple doesn't recognize that we have the same " }{XPPEDIT 18 
0 "pi" "6#%#piG" }{TEXT -1 33 " in the numerator and denominator" }}}
{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "A
N IMPORTANT POINT:" }}{PARA 0 "" 0 "" {TEXT -1 102 "You must use the f
ourier and invfourier commands rather the definition of the transform \+
as an integral" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Let us first ch
eck that I haven't inadvertently changed the definition of f(t):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&\"\"\"F$,&*$)%\"tG\"\"#F$F$F$F$!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "F:=int(f(t)*exp(-I*w*t),t=-infinity
..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Which is WRONG! " }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 16 "ANOTHER EXAMPLE " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "f:=x->exp(-x^2/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*$)9$\"\"#\"\"\"#!
\"\"F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g:=fourier(
f(t),t,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*(-%%sqrtG6#\"
\"#\"\"\"-F'6#%#PiGF*-%$expG6#,$*$)%&omegaGF)F*#!\"\"F)F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "F:=unapply(g,omega);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"FGR6#%&omegaG6\"6$%)operatorG%&arrowGF(*(-%%
sqrtG6#\"\"#\"\"\"-F.6#%#PiGF1-%$expG6#,$*$)9$F0F1#!\"\"F0F1F(F(F(" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "invfourier(F(omega),omega,t
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$*$)%\"tG\"\"#\"\"\"#!
\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 22 "So this time it works!" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}}{MARK "20 0 0" 12 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 
}{PAGENUMBERS 0 1 2 33 1 1 }