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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 43 "Section 10.4 Fourier sine and c
osine series" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"fG6#%\"xGF'" }
{TEXT -1 73 " on the interval [0,1].   First I calculate the Fourier c
osine series of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "assume(k,integer);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f
 := x -> x;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 12 "a := proc(k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 
"  2*integrate(f(x)*cos(k*Pi*x),x=0..1);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "fou
rier_cosine_partial_sum := proc(n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "  a(0)/2 + sum(a(k)*cos(k*Pi*x),k=1..n);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "fou
rier_cosine_partial_sum(infinity);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "plot(\{f(x),fourier_cosine_partial_sum(2)\},x=0..1);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(\{f(x),fourier_cos
ine_partial_sum(5)\},x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 51 "plot(\{f(x),fourier_cosine_partial_sum(10)\},x=0..1);" }}}{PARA 
0 "" 0 "" {TEXT -1 88 "Recall that the cosine series is the Fourier se
ries of the even 2-periodic extension of " }{XPPEDIT 18 0 "f;" "6#%\"f
G" }{TEXT -1 10 ".  (Here, " }{XPPEDIT 18 0 "L = 1;" "6#/%\"LG\"\"\"" 
}{TEXT -1 41 ".)  The even extension to [-1,1] is just " }{XPPEDIT 18 
0 "abs(x);" "6#-%$absG6#%\"xG" }{TEXT -1 3 ".  " }{TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot(\{abs(x),fourier_cosine
_partial_sum(2)\},x=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "plot(\{abs(x),fourier_cosine_partial_sum(5)\},x=-1..1);" }}}{PARA 
0 "" 0 "" {TEXT -1 46 "Now I'll calculate the Fourier sine series of \+
" }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 1 "." }{TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "b := proc(k)" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "   2*integrate(f(x)*sin(k*Pi*x),x=0..1);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "fourier_sin_partial_sum := proc(n)" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 32 "   sum(b(k)*sin(k*Pi*x),k=1..n);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "fourier_sin_partial_sum(infinity);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "plot(\{f(x),fourier_sin_partial_sum(2)\},x=0..1);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(\{f(x),fourier_sin_par
tial_sum(5)\},x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "p
lot(\{f(x),fourier_sin_partial_sum(10)\},x=0..1);" }}}{PARA 0 "" 0 "" 
{TEXT -1 21 "The odd extension of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }
{TEXT -1 19 " to [-1,1] is just " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"f
G6#%\"xGF'" }{TEXT -1 3 ".  " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "plot(\{x,fourier_sin_partial_sum(10)\},x=-1..1);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "26" 0 }
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