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{SECT 0 {EXCHG {PARA 264 "" 0 "" {TEXT -1 19 "Energy Computations" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The energ
y  " }{XPPEDIT 18 0 "E" "6#%\"EG" }{TEXT -1 26 "  of a periodic functi
on  " }{XPPEDIT 18 0 " f " "6#%\"fG" }{TEXT -1 15 "  with period  " }
{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 28 "   is defi
ned by the formula" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {XPPEDIT 18 0 "E = int(f(x)^2,x=-Pi..Pi)/Pi" "6#/%\"EG*&-%$intG6$*$-
%\"fG6#%\"xG\"\"#/F-;,$%#PiG!\"\"F2\"\"\"F2F3" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 51 "Recall that the Fourier coefficients for  f    are \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "a
[0] = int(f(x),x = -Pi .. Pi)/(2*Pi);" "6#/&%\"aG6#\"\"!*&-%$intG6$-%
\"fG6#%\"xG/F/;,$%#PiG!\"\"F3\"\"\"*&\"\"#F5F3F5F4" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "a[k] \+
= int(f(x)*cos(k*x),x = -Pi .. Pi)/Pi;" "6#/&%\"aG6#%\"kG*&-%$intG6$*&
-%\"fG6#%\"xG\"\"\"-%$cosG6#*&F'F1F0F1F1/F0;,$%#PiG!\"\"F9F1F9F:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "b[k] \+
= int(f(x)*sin(k*x),x = -Pi .. Pi)/Pi;" "6#/&%\"bG6#%\"kG*&-%$intG6$*&
-%\"fG6#%\"xG\"\"\"-%$sinG6#*&F'F1F0F1F1/F0;,$%#PiG!\"\"F9F1F9F:" }
{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 31 "and that by the Energy Theorem," }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "E = Sum(A[k]^2,k=1..infinit
y)" "6#/%\"EG-%$SumG6$*$&%\"AG6#%\"kG\"\"#/F,;\"\"\"%)infinityG" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "where" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "A[0] \+
= sqrt(2)*a[0] " "6#/&%\"AG6#\"\"!*&-%%sqrtG6#\"\"#\"\"\"&%\"aG6#F'F-
" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "A[k]^2 = a[k]^2+b[k]^2" "6#/*$&
%\"AG6#%\"kG\"\"#,&*$&%\"aG6#F(F)\"\"\"*$&%\"bG6#F(F)F/" }{TEXT -1 2 "
, " }{XPPEDIT 18 0 "k >=1" "6#1\"\"\"%\"kG" }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 12 "Square Wave " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Consider the Square Wave function \+
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "
SW = piecewise(x<0,0,1)" "6#/%#SWG-%*piecewiseG6%2%\"xG\"\"!F*\"\"\"" 
}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "It Fourier approximations and energy are computed as foll
ows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "SW:=piecewise(x<0,0,1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#SWG-%*PIECEWISEG6$7$\"\"!2%\"xGF)7$\"\"\"%*otherwiseG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "FourierDegree := 50:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a[0]:=1/(2*Pi)*int(SW,x=-Pi.
.Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for k to FourierDe
gree do a[k]:=1/Pi*int(SW*cos(k*x),x=-Pi..Pi): od:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 68 "for k to FourierDegree do b[k]:=1/Pi*int(SW
*sin(k*x),x=-Pi..Pi): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
109 "for k from 1 to FourierDegree do Fourier[k]:=a[0] + sum(a[n]*cos(
n*x),n=1..k) + sum(b[n]*sin(n*x),n=1..k):od:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Now the \+
Fourier approximations have been formed.  Let us plot them:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "plot([SW,Fourier[2],Fourier[
10],Fourier[50]],x=-Pi..Pi,color=[black,red,blue,green]);" }}{PARA 13 
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