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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 52 "Animation of the Fourie
r Analysis of a Triangle Wave" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 145 "   The coefficients for the cosine compo
nents of the Triangle wave are :  8/(nPi)^2, for odd n values  (here i
t is done with (2*n-1)  for \"all n\"" }}{PARA 0 "" 0 "" {TEXT -1 85 "
     m=number of terms to keep - then we create a \"sequence\" of the \+
sums (myseqsum) :" }}{PARA 0 "" 0 "" {TEXT -1 68 "           first ter
m of the sequence is just the first fourier term" }}{PARA 0 "" 0 "" 
{TEXT -1 119 "           second term of the sequence is the sum of the
 first two terms, third in sequence is sum of first three terms" }}
{PARA 0 "" 0 "" {TEXT -1 71 "   Thus, we can watch the final waveform \+
evolve into the Triangle Wave." }}{PARA 0 "" 0 "" {TEXT -1 135 "   Als
o, we could create a sequence of just the fourier terms (myseq2) - we \+
can see the amplitude of each successive term gets smaller." }}{PARA 
0 "" 0 "" {TEXT -1 139 "                              [ Once the \"gra
phs\" are created, click once on the graph to highlight it, and you wi
ll see \"player controls\" " }}{PARA 0 "" 0 "" {TEXT -1 91 "          \+
                       appear in a toolbar above - to let you run the \+
animation.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 21 "restart; with(plots):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "m:=10: # Number of terms to keep" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 53 "mycoeffs := seq( (8/((2*n-1)*(2*n-1)*Pi*P
i)),n=1..m);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "myseq := se
q( (8/((2*n-1)*(2*n-1)*Pi*Pi))*cos((2*n-1)*Pi*x/2),n=1..m):  # notice \+
only odd terms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "myseqsum \+
:= seq(sum(myseq[nn],nn=1..n),n=1..m):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 152 "display(seq(plot(myseqsum[i], x=-1..3,color=blue, th
ickness=2),i=1..m), insequence=true,          title=`Evolution of Four
ier series to Triangle Wave`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 164 "#  Note:  Watch the top corner .. the series will very quickl
y approach the triangle function - there is no \"ringing\" of the Gibb
's Phenomena as in the Square Wave." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "myseq2 := seq(myseq[n],n=1..m): # these are the indiv
idual terms of the sequence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
174 "display(seq(plot(myseq2[i], x=-1..3,color=blue, thickness=2),i=1.
.m), insequence=true,          title=`Individual terms of the Fourier \+
series solution to the Triangle Wave`);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "11" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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