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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "I     Topics in Differenti
al Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 33 "                i) Fourier Series" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "                        A)Orthogon
ality Properties of sin(x) and cos(x)" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 77 "The functions sin(ax) and cos(ax) are
 eigenfunctions (or basic solutions) in " }}{PARA 0 "" 0 "" {TEXT -1 
80 "various boundary value problems in partial differential equations.
  We begin by " }}{PARA 0 "" 0 "" {TEXT -1 81 "showing that the set of
 function\{sin(n*Pi*x/c),1,cos(n*Pi*x/c) | 0<n<Infty\} is an" }}{PARA 
0 "" 0 "" {TEXT -1 85 "orthogonal set of functions on the interval (-c
,c) under the inner product defined by" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 44 "  < f(x),g(x) > = int(f(x)*g(x),x=  -
c..c )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 76 "We define the  first n terms of the \+
Fourier Series  for the function f(x) by" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "  f(x) = (1/2) a(0) + sum(a(k)*cos
(k*Pi*x/c) + b(k) *sin(k*Pi*x /c), k=1..n)" }}{PARA 0 "" 0 "" {TEXT 
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0 "> " 0 "" {MPLTEXT 1 0 11 "f := x ->x;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&arrowGF(9$F(F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "L := 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
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20 "6#>%\"aG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "b := u
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