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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 22 "Fourier approximations" }
}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Conside
r the following wave function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Wave:=piecewise(x>=-Pi and x
< -Pi/2, 1, x>=-Pi/2 and x<0, 0, x>=0 and x<Pi/2,1,0);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%WaveG-%*PIECEWISEG6&7$\"\"\"31,&%#PiG!\"\"%\"xG
F.\"\"!2,&F/F)*&#F)\"\"#F)F-F)F)F07$F031,&F-#F.F5F/F.F02F/F07$F)31,$F/
F.F02,&F/F)*&#F)F5F)F-F)F.F07$F0%*otherwiseG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Its grap
h looks like" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(Wave, \+
x=-Pi..Pi,thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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LTG" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 91 "Next we compute the Fourier coefficients for the abo
ve function using the defining formulae" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a[0] = int(f(x),x = -Pi .. Pi)/(
2*Pi);" "6#/&%\"aG6#\"\"!*&-%$intG6$-%\"fG6#%\"xG/F/;,$%#PiG!\"\"F3\"
\"\"*&\"\"#F5F3F5F4" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 257 "" 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 258 "" 0 "" {XPPEDIT 18 0 "b[k] = int(f(x)*sin(k*x),x = -Pi .. P
i)/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 52 "The computation  for the \+
above wave function follows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "FourierDegree := 50:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ConstantTerm := int(Wave,x=-
Pi..Pi)/2/Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-ConstantTermG#\"\"
\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a:= unapply(1/Pi
*int(Wave*cos(m*x),x=-Pi..Pi),m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"aGf*6#%\"mG6\"6$%)operatorG%&arrowGF(*(%#PiG!\"\"-%$sinG6#*&F-\"\"
\"9$F3F3F4F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "b := \+
unapply(1/Pi*int(Wave*sin(m*x),x=-Pi..Pi),m);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"bGf*6#%\"mG6\"6$%)operatorG%&arrowGF(*(%#PiG!\"\",(
-%$cosG6#,$*&F-\"\"\"9$F5#F5\"\"#!\"#-F16#F4F5F5F5F5F6F.F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Now we are ready to form the Fouri
er approximations as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 118 "for k from 5 to FourierDegree do Fourier[k]:= ConstantTerm + su
m(a(n)*cos(n*x),n=1..k) + sum(b(n)*sin(n*x),n=1..k):od:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Fourier[5];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&#\"\"\"\"\"#F%*(F&F%%#PiG!\"\"-%$sinG6#,$%\"xGF&F%F%
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "To get an idea about how goo
d the approximations are, we next plot some approximations together wi
th the graph of the original wave function:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 99 "plot([Wave,Fourier[5],Fourier[10],Fourier[50]],x=-
Pi..Pi,color=[black,green,red,blue],thickness=3);" }}{PARA 13 "" 1 "" 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "By increasing the value of the var
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