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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 277 "This wor
ksheet shows how the partial Fourier sine series converges to a functi
on g(t) on the interval (0, Pi).  Although the domain of g is defined \+
only on this interval, the Fourier series is defined for all t- so we'
ll plot the Fourier sine series on a larger domain for fun." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "g
:=t->signum(sin(t));  #This is a square wave" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "plot(g(t),t=-10..10);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 210 "Now we'll compute the \+
first 40 Fourier sine coefficients.  We'll do this by constructing an \+
array of 40 numbers, corresponding to the 40 coefficients.  You can do
 more or less by changing the value of NN below:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 7 "NN:=40;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "a:=array(1..NN,[]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
72 "for k from 1 to NN do a[k]:=evalf((2/Pi)*int(g(t)*sin(k*t),t=0..Pi
)) od;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 151 "We'll construct a function of  two variables, n and t.  \+
The n will define how many terms of the series to use, and t will be o
ur usual domain variable:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ts:=(n,t)->sum(a[j]*sin(j*t),j=1..n
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 111 "Just so we can see what this function is, let's show the parti
al sum with 7 terms (some coefficients are zero):" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ts(7,t);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "animate(ts(n,t),t=-10..10,n=1..NN,f
rames=40, numpoints=500);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 98 "Notice what is happening at the corners- \+
This kind of \"bunching up\" is called the Gibbs phenomena." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "Let's see what
 kind of convergence we get by looking at the difference between g and
 the partial sine series:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "animate(g(t)-ts(n,t),t=-10..10,n=1.
.NN,frames=40, numpoints=500);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
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