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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "15. FOURIER SERIES WITH MA
PLE" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plots):" }}
{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
edefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "We will give a few \+
example of evaluation and summation of Fourier series using Maple:" }}
{PARA 0 "" 0 "" {TEXT -1 10 "EXAMPLE 1:" }}{PARA 0 "" 0 "" {TEXT -1 
20 "Trapezoidal function" }}{PARA 0 "" 0 "" {TEXT -1 33 "f(x)=x, for 0
<x <1;=1, for 1<x<2;" }}{PARA 0 "" 0 "" {TEXT -1 20 "      =2-x for 2<
x<3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Th
e first thing to do is to define f(x).  One way to define a function i
s   " }}{PARA 0 "" 0 "" {TEXT -1 21 "f:=x->expression(x)  " }}{PARA 0 
"" 0 "" {TEXT -1 53 "This definition  can be extended to several varia
bles" }}{PARA 0 "" 0 "" {TEXT -1 31 "f:=(x,y...)->expression(x,y...)" 
}}{PARA 0 "" 0 "" {TEXT -1 68 "Our function is defined piecewise and t
he maple command for that is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "f:=x->piecewise(x<1,x,x<2,1\n,x<3,3-x):f(x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%*PIECEWISEG6%7$%\"xG2F'\"\"\"7$F)2F'\"\"#7$,&\"\"$F
)F'!\"\"2F'F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 40 "The period in the periodic extension is " }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 5 "a:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"a
G\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Plot the function to ch
eck that we have got it right " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "plot(f(x),x=0..a);" }}{PARA 13 "" 1 "" {GLPLOT2D 267 162 162 
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Fin$\"37)******4!=)*\\F17$$\"3-++DE&4Qc#Fin$\"3u****\\PZ!>O%F17$$\"3=+
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$$\"33++Dr\"[8v#Fin$\"39****\\(G=l[#F17$$\"3++++Ijy5GFin$\"3+++++n8#*=
F17$$\"31+]P/)fT(GFin$\"3G***\\i&>Se7F17$$\"31+]i0j\"[$HFin$\"3V$***\\
P%p$=lF-7$$\"\"$F)F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q
\"x6\"Q!6\"-%%VIEWG6$;F(Few%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 38 "The Maple command for integration is  " }{TEXT 256 5 "int
, " }{TEXT 263 11 "the command" }{TEXT 264 1 " " }{TEXT -1 73 "attempt
s a symbolic integration, but will do it numerically if that fails" }}
{PARA 0 "" 0 "" {TEXT -1 56 "and there are no undefined constants in t
he expression. " }}{PARA 0 "" 0 "" {TEXT -1 23 "SINE AND COSINE SERIES
." }}{PARA 0 "" 0 "" {TEXT -1 39 "We want to write the series on the f
orm" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "f(x)=alpha(0)+sum(alpha(n)*cos(2
*n *Pi*x/a)+beta(n)*sin(2*n*Pi*x/a),n=1..infinity) " "6#/-%\"fG6#%\"xG
,&-%&alphaG6#\"\"!\"\"\"-%$sumG6$,&*&-F*6#%\"nGF--%$cosG6#*,\"\"#F-F5F
-%#PiGF-F'F-%\"aG!\"\"F-F-*&-%%betaG6#F5F--%$sinG6#*,F:F-F5F-F;F-F'F-F
<F=F-F-/F5;F-%)infinityGF-" }}{PARA 0 "" 0 "" {TEXT -1 41 "Using the f
ormulas of lecture 13 we find:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "alpha0:=int(f(x)/a,x=0..a);
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'alpha0G#\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "an:=(int(2/a*f(x)*cos(2*n*Pi*x/a),x=0..a));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#anG,$*&,(*$)-%$cosG6#*&%\"nG\"\"\"%#PiGF/\"\"#F/F1-F
+6#,$F-#\"\"%\"\"$!\"\"-F+6#,$F-#F1F7F8F/*&)F.F1F/)F0F1F/F8#!\"$F1" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The command unapply assigns a fun
ction " }{XPPEDIT 18 0 "alpha(n)" "6#-%&alphaG6#%\"nG" }{TEXT -1 14 " \+
to the result" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "alpha:=una
pply(an,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaGR6#%\"nG6\"6$%
)operatorG%&arrowGF(,$*&,(*$)-%$cosG6#*&9$\"\"\"%#PiGF6\"\"#F6F8-F26#,
$F4#\"\"%\"\"$!\"\"-F26#,$F4#F8F>F?F6*&)F5F8F6)F7F8F6F?#!\"$F8F(F(F(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "bn:=(int(2/a*f(x)*sin(2*n
*Pi*x/a),x=0..a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bnG,$*&,(*&-%
$sinG6#*&%\"nG\"\"\"%#PiGF.F.-%$cosGF+F.!\"#-F*6#,$F,#\"\"%\"\"$F.-F*6
#,$F,#\"\"#F8F.F.*&)F-F=F.)F/F=F.!\"\"#F8F=" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 56 "bn is actually zero as can be seen by using the identit
y" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "2*cos(A)*sin(B) = sin(A+B)+sin(B-A
);" "6#/*(\"\"#\"\"\"-%$cosG6#%\"AGF&-%$sinG6#%\"BGF&,&-F,6#,&F*F&F.F&
F&-F,6#,&F.F&F*!\"\"F&" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "B=n *Pi" 
"6#/%\"BG*&%\"nG\"\"\"%#PiGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "A= \+
-n*Pi/3" "6#/%\"AG,$*(%\"nG\"\"\"%#PiGF(\"\"$!\"\"F+" }{TEXT -1 9 ", a
nd the" }}{PARA 0 "" 0 "" {TEXT -1 10 "fact that " }{XPPEDIT 18 0 "sin
(n*Pi)=0" "6#/-%$sinG6#*&%\"nG\"\"\"%#PiGF)\"\"!" }{TEXT -1 75 " when \+
n is an integer. But this is perhaps something which it is easier for
" }}{PARA 0 "" 0 "" {TEXT -1 30 "a human than for Maple to see." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 51 "g:=x->alpha0+sum(alpha(n)*cos(2*n*Pi*x/a),n=1..20);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&%'alp
ha0G\"\"\"-%$sumG6$*&-%&alphaG6#%\"nGF.-%$cosG6#,$*&*(F6F.%#PiGF.9$F.F
.%\"aG!\"\"\"\"#F./F6;F.\"#?F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "plot(g(x),x=-a..3*a,numpoints=500);" }}{PARA 13 "" 1 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "We see that if we extend the domai
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"" 0 "" {TEXT -1 96 "which also happens to be the even periodic extens
ion. This is actually an easier way to see that" }}{PARA 0 "" 0 "" 
{TEXT -1 97 "the sine terms in the general fourier series must be zero
! If we include more terms in the sum we" }}{PARA 0 "" 0 "" {TEXT -1 
18 "get more accuracy." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 22 "COMPLEX FOURIER SERIES" }}{PARA 0 "" 0 "
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{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
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" {XPPMATH 20 "6#-%*PIECEWISEG6%7$%\"xG2F'\"\"\"7$F)2F'\"\"#7$,&\"\"$F
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "We wish to express this in a serie
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)infinityGF9F=" }}{PARA 0 "" 0 "" {TEXT -1 51 "to compute the coeffici
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0 43 "g:=1/a*int(exp(-2*Pi*I*n*x/a)*f(x),x=0..a);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gG,$*&,*-%$expG6#*(^##!\"%\"\"$\"\"\"%#PiGF0%\"nGF0
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1 0 16 "c:=unapply(g,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGR6#%
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case" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "c0:=1/a*int(f(x),x=
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(Ref(x),x=-6..6);" }}
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Which looks very much like the odd
 periodic extension of f(x)!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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