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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "Approximations for Fourier integr
als" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Can
ada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:  27.3.2007" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 17 "Fourier integrals" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 17 
"Fourier transform" }{TEXT -1 30 " of a (non-periodic) function " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(omega)=Int(f(x)*exp(-i*omega*
x),x=-infinity..infinity)" "6#/-%\"FG6#%&omegaG-%$IntG6$*&-%\"fG6#%\"x
G\"\"\"-%$expG6#,$*(%\"iGF0F'F0F/F0!\"\"F0/F/;,$%)infinityGF7F;" }
{TEXT -1 16 "  -------  (i). " }}{PARA 0 "" 0 "" {TEXT -1 29 "Under su
itable conditions on " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 
-1 23 ", we expect to recover " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 26 " from the Fourier integral" }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "f(x)=1/(2*Pi)" "6#/-%\"fG6#%\"xG*&\"\"\"F)*&
\"\"#F)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(F(omega)*exp(
i*omega*x),omega=-infinity..infinity)" "6#-%$IntG6$*&-%\"FG6#%&omegaG
\"\"\"-%$expG6#*(%\"iGF+F*F+%\"xGF+F+/F*;,$%)infinityG!\"\"F5" }{TEXT 
-1 17 "  -------  (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 127 "We can consider (i) to be the analogue of the det
ermination of the coefficients in a the Fourier series of a periodic f
unction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 121 ", and
 (ii) to be the analogue of the Fourier series which, under suitable c
onditions, converges to the periodic function " }{XPPEDIT 18 0 "f(x)" 
"6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x
G" }{TEXT -1 21 " is a (non-periodic) " }{TEXT 261 11 "real valued" }
{TEXT -1 47 " function, then (ii) can be written in the form" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=1/(2*Pi)" "6#/-%\"fG6#
%\"xG*&\"\"\"F)*&\"\"#F)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
Int((A(omega)+i*B(omega))*(cos*omega*x+i*sin*omega*x),omega = -infinit
y .. infinity);" "6#-%$IntG6$*&,&-%\"AG6#%&omegaG\"\"\"*&%\"iGF,-%\"BG
6#F+F,F,F,,&*(%$cosGF,F+F,%\"xGF,F,**F.F,%$sinGF,F+F,F5F,F,F,/F+;,$%)i
nfinityG!\"\"F;" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 6 "where \+
" }{XPPEDIT 18 0 "A(omega)" "6#-%\"AG6#%&omegaG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "B(omega)" "6#-%\"BG6#%&omegaG" }{TEXT -1 37 " are the r
eal and imaginary parts of " }{XPPEDIT 18 0 "F(omega)" "6#-%\"FG6#%&om
egaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 
256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "f(x) = 1/(2*Pi);" "6#/-%
\"fG6#%\"xG*&\"\"\"F)*&\"\"#F)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "Int(A(omega)*cos(omega*x)-B(omega)*sin*omega*x+i*(A(omega)*sin*o
mega*x+B(omega)*cos*omega*x),omega = -infinity .. infinity);" "6#-%$In
tG6$,(*&-%\"AG6#%&omegaG\"\"\"-%$cosG6#*&F+F,%\"xGF,F,F,**-%\"BG6#F+F,
%$sinGF,F+F,F1F,!\"\"*&%\"iGF,,&**-F)6#F+F,F6F,F+F,F1F,F,**-F46#F+F,F.
F,F+F,F1F,F,F,F,/F+;,$%)infinityGF7FD" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "
f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 96 " is a real valued function, then
 the imaginary part of (iii) must be identically 0, which gives " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=1/(2*Pi)" "6#/-%
\"fG6#%\"xG*&\"\"\"F)*&\"\"#F)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "Int(A(omega)*cos*omega*x-B(omega)*sin*omega*x,omega = -infinity \+
.. infinity);" "6#-%$IntG6$,&**-%\"AG6#%&omegaG\"\"\"%$cosGF,F+F,%\"xG
F,F,**-%\"BG6#F+F,%$sinGF,F+F,F.F,!\"\"/F+;,$%)infinityGF4F8" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "But " }{XPPEDIT 18 0 "A(omega)*
cos(omega*x)-B(omega)*sin(omega*x)" "6#,&*&-%\"AG6#%&omegaG\"\"\"-%$co
sG6#*&F(F)%\"xGF)F)F)*&-%\"BG6#F(F)-%$sinG6#*&F(F)F.F)F)!\"\"" }{TEXT 
-1 24 " is an even function of " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }
{TEXT -1 4 ", so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f
(x)=1/Pi" "6#/-%\"fG6#%\"xG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(A(omega)*cos*omega*x-B(omega)*sin*omega*x,omega = 0
 .. infinity);" "6#-%$IntG6$,&**-%\"AG6#%&omegaG\"\"\"%$cosGF,F+F,%\"x
GF,F,**-%\"BG6#F+F,%$sinGF,F+F,F.F,!\"\"/F+;\"\"!%)infinityG" }{TEXT 
-1 17 "  -------  (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 38 "If, in addition to being real valued, " }{XPPEDIT 
18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " is an " }{TEXT 261 4 "even
" }{TEXT -1 16 " function, then " }{XPPEDIT 18 0 "B(omega)" "6#-%\"BG6
#%&omegaG" }{TEXT -1 27 " is identically 0, so that " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=1/Pi" "6#/-%\"fG6#%\"xG*&\"\"
\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(A(omega)*cos*omega
*x,omega = 0 .. infinity);" "6#-%$IntG6$**-%\"AG6#%&omegaG\"\"\"%$cosG
F+F*F+%\"xGF+/F*;\"\"!%)infinityG" }{TEXT -1 16 "  -------  (iv)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "If, in ad
dition to being real valued, " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 7 " is an " }{TEXT 261 3 "odd" }{TEXT -1 16 " function, th
en " }{XPPEDIT 18 0 "A(omega);" "6#-%\"AG6#%&omegaG" }{TEXT -1 27 " is
 identically 0, so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "f(x) = -1/Pi;" "6#/-%\"fG6#%\"xG,$*&\"\"\"F*%#PiG!\"\"F
," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(B(omega)*sin*omega*x,omega = 0 \+
.. infinity);" "6#-%$IntG6$**-%\"BG6#%&omegaG\"\"\"%$sinGF+F*F+%\"xGF+
/F*;\"\"!%)infinityG" }{TEXT -1 15 "  -------  (v)." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "By analogy with Fourier
 series, we can obtain approximations for the function " }{XPPEDIT 18 
0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 31 " by integrating from 0 to som
e " }{TEXT 261 18 "finite upper limit" }{TEXT -1 1 " " }{TEXT 265 1 "L
" }{TEXT -1 55 ", instead of infinity, in formulas (iii), (iv) and (v)
." }}{PARA 0 "" 0 "" {TEXT -1 38 "Thus we would expect that the functi
on" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)=1/Pi" "6#/
-%\"gG6#%\"xG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int
(A(omega)*cos(omega*x)-B(omega)*sin(omega*x),omega=0..L)" "6#-%$IntG6$
,&*&-%\"AG6#%&omegaG\"\"\"-%$cosG6#*&F+F,%\"xGF,F,F,*&-%\"BG6#F+F,-%$s
inG6#*&F+F,F1F,F,!\"\"/F+;\"\"!%\"LG" }{TEXT -1 2 "  " }}{PARA 256 "" 
0 "" {TEXT -1 2 "  " }{TEXT 262 27 "___________________________" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 19 "should approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"
xG" }{TEXT -1 7 ", when " }{TEXT 266 1 "L" }{TEXT -1 87 " is a reasona
bly large positive real number, or, in the case of even and odd functi
ons " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "," }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)=1/Pi" "6#/-%\"gG6#%\"x
G*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(A(omega)*co
s(omega*x),omega=0..L)" "6#-%$IntG6$*&-%\"AG6#%&omegaG\"\"\"-%$cosG6#*
&F*F+%\"xGF+F+/F*;\"\"!%\"LG" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{TEXT 263 17 "_________________" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "g(x) = -1/Pi;" "6#/-%\"gG6#%\"xG,$*&\"\"\"F*%#PiG!\"\"F
," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(B(omega)*sin(omega*x),omega = 0
 .. L);" "6#-%$IntG6$*&-%\"BG6#%&omegaG\"\"\"-%$sinG6#*&F*F+%\"xGF+F+/
F*;\"\"!%\"LG" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{TEXT 264 18 "__________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "should approximate " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 15 ", respectively." 
}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 31 "Consider the pulse of width 1, " }
{XPPEDIT 18 0 "f(x)=PIECEWISE([1 , abs(x)<1],[ 0, abs(x)>=1])" "6#/-%
\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"\"2-%$absG6#F'F,7$\"\"!1F,-F/6#F'" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 25 "The Fourier transform of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f
G6#%\"xG" }{TEXT -1 5 " is  " }{XPPEDIT 18 0 "F(omega) = 2*sin*omega/o
mega;" "6#/-%\"FG6#%&omegaG**\"\"#\"\"\"%$sinGF*F'F*F'!\"\"" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x)" 
"6#-%\"fG6#%\"xG" }{TEXT -1 56 " is an even real valued function, func
tions of the form " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 
0 "g(x) = 1/Pi" "6#/-%\"gG6#%\"xG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(2*sin*omega*cos*omega*x/omega,omega = 0 .. L);" 
"6#-%$IntG6$*0\"\"#\"\"\"%$sinGF(%&omegaGF(%$cosGF(F*F(%\"xGF(F*!\"\"/
F*;\"\"!%\"LG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "should a
pproximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " 
}}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "Int(2*sin*omega*cos*omega*x/omega,omega = 0 .. L
) = Int(sin((x+1)*omega)/omega-sin((x-1)*omega)/omega,omega = 0 .. L);
" "6#/-%$IntG6$*0\"\"#\"\"\"%$sinGF)%&omegaGF)%$cosGF)F+F)%\"xGF)F+!\"
\"/F+;\"\"!%\"LG-F%6$,&*&-F*6#*&,&F-F)F)F)F)F+F)F)F+F.F)*&-F*6#*&,&F-F
)F)F.F)F+F)F)F+F.F./F+;F1F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 5 "Now  " }{XPPEDIT 18 0 "Int(sin*a*x/x,x) = Si(a*x)+c;" "6#/-%$Int
G6$**%$sinG\"\"\"%\"aGF)%\"xGF)F+!\"\"F+,&-%#SiG6#*&F*F)F+F)F)%\"cGF)
" }{TEXT -1 18 ", where Si is the " }{TEXT 261 13 "sine integral" }
{TEXT -1 12 " defined by " }{XPPEDIT 18 0 "Si(x)=Int(sin(t)/t,t=0..x)
" "6#/-%#SiG6#%\"xG-%$IntG6$*&-%$sinG6#%\"tG\"\"\"F/!\"\"/F/;\"\"!F'" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "This follows since " }
{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "[Si(a*x)] = a*sin*a*x/(a*x);" "6#/7#-%#SiG6#*&%\"aG\"\"
\"%\"xGF**,F)F*%$sinGF*F)F*F+F**&F)F*F+F*!\"\"" }{XPPEDIT 18 0 "`` = s
in*a*x/x;" "6#/%!G**%$sinG\"\"\"%\"aGF'%\"xGF'F)!\"\"" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "g(x) = (Si((x+1)*omega)-Si((x-1)*omega))/Pi;" "6#/
-%\"gG6#%\"xG*&,&-%#SiG6#*&,&F'\"\"\"F/F/F/%&omegaGF/F/-F+6#*&,&F'F/F/
!\"\"F/F0F/F5F/%#PiGF5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([L, \+
``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%\"LG%!G7$F(F(7$\"\"!F(" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (Si((x+1)*L)-Si((x-1)*L))/Pi;" "6#
/%!G*&,&-%#SiG6#*&,&%\"xG\"\"\"F-F-F-%\"LGF-F--F(6#*&,&F,F-F-!\"\"F-F.
F-F3F-%#PiGF3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "1/Pi*Int(2*sin(omega)*cos(om
ega*x)/omega,omega =0..L);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#*&-%$IntG6$,$*&*&-%$sinG6#%&omegaG\"\"\"-%$cosG6#*&F-F.%\"xGF.F.F.F
-!\"\"\"\"#/F-;\"\"!%\"LGF.%#PiGF4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
*&,&-%#SiG6#*&,&!\"\"\"\"\"%\"xGF+F+%\"LGF+F*-F&6#*&,&F+F+F,F+F+F-F+F+
F+%#PiGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 225 "alias(H=Heaviside):\nf := x -> H(x+1)-H(x-1);\nF :
= (x,L) -> (Si((x+1)*L)-Si((x-1)*L))/Pi;\nplot([f(x),F(x,2),F(x,4),F(x
,8),F(x,16)],x=-5..5,linestyle=[3,1$4],\n                 color=[black
,red,green,blue,magenta],numpoints=60);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%\"HG6#,&9$\"\"\"F2F
2F2-F.6#,&F1F2F2!\"\"F6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"F
Gf*6$%\"xG%\"LG6\"6$%)operatorG%&arrowGF)*&,&-%#SiG6#*&,&9$\"\"\"F5F5F
59%F5F5-F06#*&,&F4F5F5!\"\"F5F6F5F;F5%#PiGF;F)F)F)" }}{PARA 13 "" 1 "
" {GLPLOT2D 671 226 226 {PLOTDATA 2 "6)-%'CURVESG6%7^p7$$!\"&\"\"!$F*F
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" }}}}{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 32 "Consider the \"wedge function\",  " }{XPPEDIT 18 0 "f(x)=
PIECEWISE([1-abs(x), abs(x) < 1],[0, 1 <= abs(x)])" "6#/-%\"fG6#%\"xG-
%*PIECEWISEG6$7$,&\"\"\"F--%$absG6#F'!\"\"2-F/6#F'F-7$\"\"!1F--F/6#F'
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 25 "The Fourier transform of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f
G6#%\"xG" }{TEXT -1 5 " is  " }{XPPEDIT 18 0 "F(omega) = 2*(1-cos*omeg
a)/(omega^2);" "6#/-%\"FG6#%&omegaG*(\"\"#\"\"\",&F*F**&%$cosGF*F'F*!
\"\"F**$F'F)F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since \+
" }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 56 " is an even re
al valued function, functions of the form " }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "g(x) = 1/Pi" "6#/-%\"gG6#%\"xG*&\"\"\"F)%#Pi
G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*(1-cos*omega)*cos*omega*
x/(omega^2),omega = 0 .. L);" "6#-%$IntG6$*.\"\"#\"\"\",&F(F(*&%$cosGF
(%&omegaGF(!\"\"F(F+F(F,F(%\"xGF(*$F,F'F-/F,;\"\"!%\"LG" }{TEXT -1 1 "
 " }}{PARA 0 "" 0 "" {TEXT -1 19 "should approximate " }{XPPEDIT 18 0 
"f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
64 "We can find an analytical formula for this integral using Maple." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 89 "1/Pi*Int(2*(1-cos(omega))*cos(omega*x)/(omega^2),omega=0..L);\nF
 := unapply(value(%),x,L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$Int
G6$,$**\"\"#\"\"\",&F*F*-%$cosG6#%&omegaG!\"\"F*-F-6#*&%\"xGF*F/F*F*F/
!\"#F*/F/;\"\"!%\"LG*$%#PiGF0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"F
Gf*6$%\"xG%\"LG6\"6$%)operatorG%&arrowGF),$*(%#PiG!\"\",2*&\"\"#\"\"\"
-%$cosG6#*&9%F49$F4F4F4**F3F4-%#SiGF7F4F:F4F9F4F4-F66#*&,&F:F4F4F4F4F9
F4F0*(-F=F?F4F9F4F:F4F0*&FCF4F9F4F0-F66#*&,&F:F4F4F0F4F9F4F0*(-F=FFF4F
9F4F:F4F0*&FJF4F9F4F4F4F9F0F0F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "alias(H=Heaviside):\nf :=
 x -> H(x+1)*(1+x)-2*H(x)*(x)-H(x-1)*(1-x);\nplot([f(x),F(x,2),F(x,3),
F(x,4),F(x,5)],x=-4..4,linestyle=[3,1$4],\n                 color=[bla
ck,red,green,blue,magenta]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG
f*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&-%\"HG6#,&9$\"\"\"F3F3F3F1F3F3*
(\"\"#F3-F/6#F2F3F2F3!\"\"*&-F/6#,&F2F3F3F8F3,&F3F3F2F8F3F8F(F(F(" }}
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" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{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 24 "Consider the functi
on,  " }{XPPEDIT 18 0 "f(x) = PIECEWISE([exp(-x), 0 < x],[0, x <= 0]);
" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$-%$expG6#,$F'!\"\"2\"\"!F'7$F21F'F
2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 25 "The Fourier transform of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f
G6#%\"xG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "F(omega)" "6#-%\"FG6#%&om
egaG" }{TEXT -1 4 " =  " }{XPPEDIT 18 0 "1/(1+i*omega) = 1/(1+omega^2)
-i*omega/(1+omega^2);" "6#/*&\"\"\"F%,&F%F%*&%\"iGF%%&omegaGF%F%!\"\",
&*&F%F%,&F%F%*$F)\"\"#F%F*F%*(F(F%F)F%,&F%F%*$F)F/F%F*F*" }{TEXT -1 2 
". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x)" "6#-%
\"fG6#%\"xG" }{TEXT -1 50 " is a real valued function, functions of th
e form " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "g(x) = 1/
Pi" "6#/-%\"gG6#%\"xG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "Int((cos*omega*x+omega*sin*omega*x)/(1+omega^2),omega = 0 .. L);
" "6#-%$IntG6$*&,&*(%$cosG\"\"\"%&omegaGF*%\"xGF*F***F+F*%$sinGF*F+F*F
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0 "" 0 "" {TEXT -1 19 "should approximate " }{XPPEDIT 18 0 "f(x)" "6#-
%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 151 "The anal
ytical expression given by Maple for this integral is rather complicat
ed, so it is probably better to use numerical integration to evaluate \+
it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "F := (x,L) -> 1/Pi*Int((cos(omega*x)+omega*sin(omega*
x))/(1+omega^2),omega=0..L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG
f*6$%\"xG%\"LG6\"6$%)operatorG%&arrowGF),$-%$IntG6$*&,&-%$cosG6#*&%&om
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"> " 0 "" {MPLTEXT 1 0 38 "value(F(2,5));\nevalf(Re(evalf(%,14)));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#*&%#PiG!\"\",6*&#\"\"\"\"\"#F)*&-%#SiG
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{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "f := x -> p
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estyle=[3,1$3],\n                               color=[black,red,green
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