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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 37 "Calculating Fourier Series with
 Maple" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
125 "Here are some Maple procedures to calculate the Fourier coefficie
nts and the partial sum of the Fourier series of a function " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 1 "\000" }{TEXT -1 53 ".  I ass
ume the function is defined on the interval [" }{XPPEDIT 18 0 "-L,L;" 
"6$,$%\"LG!\"\"F$" }{TEXT -1 30 "].  I need to tell Maple that " }
{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT -1 22 " is positive and that " }
{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 78 " (the index for the Fourier
 coefficients) is an integer.   I do this with the " }{TEXT 256 6 "ass
ume" }{TEXT -1 121 " command.  When you have assumed something about a
 variable, Maple indicates that by following the variable with a tilde
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "assume(L,positive);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "assume(k,integer);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 52 "The following procedure calculates the coefficients " }
{XPPEDIT 18 0 "a[k];" "6#&%\"aG6#%\"kG" }{TEXT -1 18 " for the functio
n " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 31 ".  I have called the p
rocedure " }{TEXT 257 1 "a" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT 
-1 18 "and it depends on " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 5 "
 and " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a := proc(
f,k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "   integrate(f(x)*cos(k*Pi*
x/L),x=-L..L)/L;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Similarly, the \+
following procedure calculates the coefficients " }{XPPEDIT 18 0 "b[k]
;" "6#&%\"bG6#%\"kG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 
"f;" "6#%\"fG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "b := proc(f,k)" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 43 "   integrate(f(x)*sin(k*Pi*x/L),x=-L..L)/L;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The following procedure calculates
 the " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 54 "th partial sum of t
he Fourier series for the function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "L := 'L';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "fourier_partial_sum := " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
   proc(f,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "      a(f,0)/2 +   \+
      sum(a(f,k)*cos(k*Pi*x/L)+b(f,k)*sin(k*Pi*x/L),k=1..n);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "   end;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 50 "Here is an example of how to use it.  I
 will take " }{XPPEDIT 18 0 "f(x) = abs(x);" "6#/-%\"fG6#%\"xG-%$absG6
#F'" }{TEXT -1 0 "" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x -> abs(x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a(f,k);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 7 "b(f,k);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 53 "Here is the second partial sum of the Fou
rier series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 25 "fourier_partial_sum(f,2);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Here is the fifth partial
 sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "fourier_partial_sum(f,5);" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "By taking " }{XPPEDIT 18 0 "n;
" "6#%\"nG" }{TEXT -1 7 " to be " }{XPPEDIT 18 0 "infinity;" "6#%)infi
nityG" }{TEXT -1 40 ", I can even write down the full series." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "fourier_partial_sum(f,infinity);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 93 "I can compare the graph of the origin
al function.  To do this, I need to specify a value for " }{XPPEDIT 
18 0 "L;" "6#%\"LG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "L := 1;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 46 "plot(\{f(x),fourier_partial_sum(f,2)\},x=-1..
1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(\{f(x),fourier_
partial_sum(f,5)\},x=-1..1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(\{f(x),fourier_partial_
sum(f,10)\},x=-1..1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 22 "Now I'll do this with " }{XPPEDIT 18 0 "f(x) = x;" "
6#/-%\"fG6#%\"xGF'" }{TEXT -1 17 ".  I;ll think of " }{XPPEDIT 18 0 "f
;" "6#%\"fG" }{TEXT -1 90 " as defined on (-1,1) and extend it to be p
iecewise continuous and periodic with period 2." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f := x -> pi
ecewise(x>=-3 and x < -1,x+2,x>-1 and x<1, x, x>1 and x<=3, x-2);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "What does
 the plot of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 11 " look like?
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 48 "plot(f,-3..3,discont=true,scaling=constrained);\n" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Now I'll plot " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 232 " with the second partial s
um of its Fourier series.  This time I didn't tell Maple that it is pl
otting a discontinuous function, so Maple adds straight vertical lines
 connecting the values from the two sides at a jump discontinuity." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
66 "plot(\{f(x),fourier_partial_sum(f,2)\},x=-3..3,scaling=constrained
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(\{f(x),fourier_p
artial_sum(f,5)\},x=-3..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "plot(\{f(x),fourier_partial_sum(f,10)\},x=-1..1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(\{f(x),fourier_partial_sum(f,2
0)\},x=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(\{f(
x),fourier_partial_sum(f,50)\},x=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "plot(\{f(x),fourier_partial_sum(f,50)\},x=-1..1,numpo
ints=1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "35 0" 
5 }{VIEWOPTS 1 1 0 1 1 1803 }