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{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
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0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 
0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 
-1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 29 "Analyzing a Periodic Fun
ction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "
The cell below illustrates the basic step for finding a component of t
he form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 257 "" 0 "" {TEXT 256 41 "a cos 2 pi t/lambda + b sin 2 pi \+
t/lambda" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "in a function" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT 257 32 "f(t) on the interval  [0, lambda" }{TEXT -1 2 "]." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 258 15 "Evaluate it now" }{TEXT -1 317 ".  Warning: Ma
ny of the functions that we work with in this module have discontinuti
es or places where they are not differentiable.  As a result the usual
 integration and graphing algorithms in Maple have difficulty with the
se functions.  Notice that the integrations in this module use some ad
ditional parameters -- " }{TEXT 261 1 "4" }{TEXT -1 6 "  and " }{TEXT 
262 5 "_Dexp" }{TEXT -1 289 ". These parameters relax the usual Maple \+
precision requirements and enable us to obtain results even though the
 results are less precise than usual.  Notice also that we need to sur
round some of the functions with  single quotes to enable the Maple pl
ot and integration routines to work.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 375 "f := proc(t)\n        \+
if (t < 1) then -1 else 1 fi\n     end:\n\nlambda := 2:\n\na := (2/lam
bda) * evalf(Int('f(t) * cos(2 * Pi * t/lambda)', \n           t = 0..
lambda, 4, _Dexp));\nb := (2/lambda) * evalf(Int('f(t) * sin(2 * Pi * \+
t/lambda)', \n           t = 0..lambda, 4, _Dexp));\n\ng := t -> a * c
os(2 * Pi * t/2) + b * sin(2 * Pi * t/lambda):\n\nplot(\{'f(t)', g(t)
\}, t = 0..lambda);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 39 "The cell below computes and graphs the " }
{TEXT 259 1 "n" }{TEXT -1 38 "-th Fourier polynomial of a function  " 
}{TEXT 260 4 "f(t)" }{TEXT -1 18 "  on the interval " }}{PARA 259 "" 
0 "" {TEXT -1 30 "[0, lambda].  Evaluate it now." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 656 "f := proc(
t)\n        if (t < 1) then -1 else 1 fi\n     end:\n\nlambda := 2:\n
\nn := 3:\n\na0 := (1/lambda) * evalf(Int('f(t)', t = 0..lambda, 4, _D
exp)):\n\naa := k -> (2/lambda) * evalf(Int('f(t) * cos(2 * Pi * k * t
/lambda)', \n            t = 0..lambda, 4, _Dexp)):\n\nbb := k -> (2/l
ambda) * evalf(Int('f(t) * sin(2 * Pi * k * t/lambda)', \n            \+
t = 0..lambda, 4, _Dexp)):\n\na := [seq(i, i=1..n)]:\nb := [seq(i, i=1
..n)]:\n\nfor j from 1 to n do\n    a[j] := aa(j):\n    b[j] := bb(j):
\nod:\n\nFourier := t -> a0 + sum(a[k] * cos(2 * Pi * k * t/lambda) +
\n                         b[k] * sin(2 * Pi * k * t/lambda), k=1..n):
\n\nplot(\{'f(t)', Fourier(t)\}, t=0..lambda);\n" }}}}{MARK "0 0 0" 
29 }{VIEWOPTS 1 1 0 1 1 1803 }