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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 11 "Math 2270-1" }}{PARA 
257 "" 0 "" {TEXT -1 20 "Wednesday October 25" }}{PARA 258 "" 0 "" 
{TEXT -1 22 "Fourier Series example" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "with(linalg):with(plots):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "Dot:=(f,g)->(1/Pi) *int(f(x)*g(x),x=-Pi..Pi);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DotGR6$%\"fG%\"gG6\"6$%)operatorG%&
arrowGF)*&-%$intG6$*&-9$6#%\"xG\"\"\"-9%F4F6/F5;,$%#PiG!\"\"F<F6F<F=F)
F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "Mag:=f->sqrt(Dot(f
,f)); #computes the magnitude of\n     #a vector\nDist:=(f,g)->Mag(f-g
);  #computes the ``distance'' between\n     #two vectors.  \nangle:=(
f,g)->arccos(Dot(f,g)/(Mag(f)*Mag(g)));\n     #computes the ``angle'' \+
between two vectors" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MagGR6#%\"fG
6\"6$%)operatorG%&arrowGF(-%%sqrtG6#-%$DotG6$9$F2F(F(F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%DistGR6$%\"fG%\"gG6\"6$%)operatorG%&arrowGF)-
%$MagG6#,&9$\"\"\"9%!\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
angleGR6$%\"fG%\"gG6\"6$%)operatorG%&arrowGF)-%'arccosG6#*&-%$DotG6$9$
9%\"\"\"*&-%$MagG6#F4F6-F96#F5F6!\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 29 "Dot(x->sin(3*x),x->sin(4*x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "f:
=x->abs(x);  #we'll do Fourier for the absolute value function\nn:=4; \+
  #order 10 expansion\na:=vector(n); #sin coefficients\nb:=vector(n); \+
 #cos coefficients\nc:=Dot(f,1/sqrt(2));  #order zero Fourier coeffici
ent" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$absG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"nG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG
-%&arrayG6$;\"\"\"\"\"%7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%
&arrayG6$;\"\"\"\"\"%7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG,$*&
%#PiG\"\"\"-%%sqrtG6#\"\"#F(#F(F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 115 "for i from 1 to n do  #compute the projection coeffi
cients\na[i]:=Dot(f,x->sin(i*x));\nb[i]:=Dot(f,x->cos(i*x));\nod:\n" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "evalm(a);#why will an even functio
n have Fourier sine\n  #coefficients all equal to zero?" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'vectorG6#7&\"\"!F'F'F'" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "evalm(b); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'vectorG6#7&,$*&\"\"\"F)%#PiG!\"\"!\"%\"\"!,$F(#F,\"\"*F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "approx:=x->sum(a[k]*sin(k*x)
,k=1..n)\n           +sum(b[m]*cos(m*x),m=1..n)\n           +c/sqrt(2)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'approxGR6#%\"xG6\"6$%)operator
G%&arrowGF(,(-%$sumG6$*&&%\"aG6#%\"kG\"\"\"-%$sinG6#*&F4F59$F5F5/F4;F5
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qrtG6#\"\"#!\"\"F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 
"approx(x); #order 4 Fourier expansion for abs(x):" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,(*&-%$cosG6#%\"xG\"\"\"%#PiG!\"\"!\"%*&#\"\"%\"\"*F)
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0 "" {MPLTEXT 1 0 118 "almost:=plot(approx(x),x=-Pi..Pi,color=`black`)
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