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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "Fourier series for periodic funct
ions with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }
{TEXT 278 44 " using various intervals for the integration" }}{PARA 0 
"" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 
"" 0 "" {TEXT -1 19 "Version:  26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 52 "load Fourier series and Fourier transform procedures" }}{PARA 
0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 280 9 "fourier.m" }
{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 13 "Fouri
erSeries" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" 
{TEXT -1 121 "It can be read into a Maple session by a command similar
 to the one that follows, where the file path gives its location." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/procdrs/f
ourier.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 55 "Integrating a periodic function over a complete period \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 36 "If we integrate a periodic function " }{XPPEDIT 18 0 "f(x
)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " with period " }{TEXT 264 1 "p" }
{TEXT -1 48 " over a complete period, the value obtained is  " }{TEXT 
260 34 "independent of the starting point " }{XPPEDIT 18 0 "x = alpha;
" "6#/%\"xG%&alphaG" }{TEXT 260 19 " of the integration" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 70 "This is intuitively clear from the d
escription of a definite integral " }{XPPEDIT 18 0 "Int(f(x),x = alpha
 .. alpha+p);" "6#-%$IntG6$-%\"fG6#%\"xG/F);%&alphaG,&F,\"\"\"%\"pGF.
" }{TEXT -1 36 " as the limit of a sum of the form  " }{XPPEDIT 18 0 "
Sum(f(x)*delta);" "6#-%$SumG6#*&-%\"fG6#%\"xG\"\"\"%&deltaGF+" }{TEXT 
276 1 "x" }{TEXT -1 58 ", where the summation extends over a finite co
llection of " }{TEXT 275 1 "x" }{TEXT -1 16 " values between " }
{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 
"alpha+p;" "6#,&%&alphaG\"\"\"%\"pGF%" }{TEXT -1 30 ". Changing the st
arting point " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 82 " onl
y changes the order of the terms in any finite sum approximating the i
ntegral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
92 "An alternative explanation is based on the fact that, for an (inte
grable) periodic function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 13 " with period " }{TEXT 263 1 "p" }{TEXT -1 70 ", translati
ng the interval of integration of any definite integral of " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 15 " by the period " 
}{TEXT 262 1 "p" }{TEXT -1 36 ", doesn't change its value, that is," }
}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(f(x),x = a .. b
) = Int(f(x),x = a+p .. b+p);" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;%\"aG%\"b
G-F%6$-F(6#F*/F*;,&F-\"\"\"%\"pGF6,&F.F6F7F6" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 93 "This is essentially because, the graph fu
nction \"does the same thing\" over the interval from " }{XPPEDIT 18 
0 "a+p" "6#,&%\"aG\"\"\"%\"pGF%" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "b+
p" "6#,&%\"bG\"\"\"%\"pGF%" }{TEXT -1 35 " as it does over the interva
l from " }{TEXT 268 1 "a" }{TEXT -1 4 " to " }{TEXT 269 1 "b" }{TEXT 
-1 43 ". In particular, if the graph is above the " }{TEXT 270 1 "x" }
{TEXT -1 148 " axis throughout each of these intervals, the regions un
der the graph corresponding to each of these integrals have exactly th
e same size and shape." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 
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(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;F]ioF\\ewF_g\\l" 1 2 0 1 10 
0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv
e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv
e 10" "Curve 11" "Curve 12" "Curve 13" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 27 "For example, the fact that " }
{XPPEDIT 18 0 "Int(sin(x),x = Pi/4 .. 2*Pi/3) = Int(sin(x),x = 9*Pi/4 \+
.. 8*Pi/3);" "6#/-%$IntG6$-%$sinG6#%\"xG/F*;*&%#PiG\"\"\"\"\"%!\"\"*(
\"\"#F/F.F/\"\"$F1-F%6$-F(6#F*/F*;*(\"\"*F/F.F/F0F1*(\"\")F/F.F/F4F1" 
}{TEXT -1 223 "  is illustrated by the following picture in which the \+
vertical boundary lines of the region for the first integral are drawn
 in blue and the vertical boundary lines of the region for the second \+
integral are drawn in green." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "with(plots):\na1 := Pi/4: b
1 := 2*Pi/3:\nleft1 := listplot([[a1,0],[a1,sin(a1)]],color=blue):\nri
ght1 := listplot([[b1,0],[b1,sin(b1)]],color=blue):\na2 := a1+2*Pi: b2
 := b1 + 2*Pi:\nleft2 := listplot([[a2,0],[a2,sin(a2)]],color=green):
\nright2 := listplot([[b2,0],[b2,sin(b2)]],color=green):\ncurve := plo
t(sin(x),x=0..3*Pi):\ndisplay(\{curve,left1,right1,left2,right2\});" }
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0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv
e 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(sin(x),x=Pi/4..2*Pi/3);\n``=val
ue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$sinG6#%\"xG/F);,
$*&\"\"%!\"\"%#PiG\"\"\"F1,$*(\"\"#F1\"\"$F/F0F1F1" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%!G,&*&\"\"#!\"\"F'#\"\"\"F'F*F)F*" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int(sin(x
),x=Pi/4+2*Pi..2*Pi/3+2*Pi);\n``=value(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$-%$sinG6#%\"xG/F);,$*(\"\"*\"\"\"\"\"%!\"\"%#P
iGF/F/,$*(\"\")F/\"\"$F1F2F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G
,&*&\"\"#!\"\"F'#\"\"\"F'F*F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
24 "For a formal proof that " }{XPPEDIT 18 0 "Int(f(x),x = a .. b) = I
nt(f(x),x = a+p .. b+p);" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;%\"aG%\"bG-F%6
$-F(6#F*/F*;,&F-\"\"\"%\"pGF6,&F.F6F7F6" }{TEXT -1 7 "  when " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " has period " }
{TEXT 266 1 "p" }{TEXT -1 13 ", substitute " }{XPPEDIT 18 0 "t = x - p
" "6#/%\"tG,&%\"xG\"\"\"%\"pG!\"\"" }{TEXT -1 44 " in the second integ
ral, which then becomes " }{XPPEDIT 18 0 "Int(f(t+p),t = a .. b);" "6#
-%$IntG6$-%\"fG6#,&%\"tG\"\"\"%\"pGF+/F*;%\"aG%\"bG" }{TEXT -1 57 ". T
he result follows by using the periodicity to replace " }{XPPEDIT 18 
0 "f(t+p)" "6#-%\"fG6#,&%\"tG\"\"\"%\"pGF(" }{TEXT -1 4 " by " }
{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 
"" {TEXT -1 62 "We now return to the result at the start of this secti
on that " }{XPPEDIT 18 0 "Int(f(x),x = alpha .. alpha+p);" "6#-%$IntG6
$-%\"fG6#%\"xG/F);%&alphaG,&F,\"\"\"%\"pGF." }{TEXT -1 21 "  does not \+
depend on " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 7 ", when \+
" }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " has period " 
}{TEXT 267 1 "p" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
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F`er-Figr6#%(DEFAULTG-%*AXESTICKSG6$F)F)-%%VIEWG6$;F\\blFbdoFhhr" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2
" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9
" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" }}
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 18 "In the case where " }{XPPEDIT 18 0 "0 < alpha;" "6#2\"\"!
%&alphaG" }{XPPEDIT 18 0 "`` < p;" "6#2%!G%\"pG" }{TEXT -1 9 "  we hav
e" }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "Int(f(x),x = a
lpha .. p+alpha) = Int(f(x),x = alpha .. p)+Int(f(x),x = p .. p+alpha)
;" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;%&alphaG,&%\"pG\"\"\"F-F0,&-F%6$-F(6#
F*/F*;F-F/F0-F%6$-F(6#F*/F*;F/,&F/F0F-F0F0" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 3 "=  " }{XPPEDIT 18 0 "Int(f(x),x = alpha .. p)+
Int(f(x),x = 0 .. alpha);" "6#,&-%$IntG6$-%\"fG6#%\"xG/F*;%&alphaG%\"p
G\"\"\"-F%6$-F(6#F*/F*;\"\"!F-F/" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "
" {TEXT -1 2 "= " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. alpha)+Int(f(x),x \+
= alpha .. p);" "6#,&-%$IntG6$-%\"fG6#%\"xG/F*;\"\"!%&alphaG\"\"\"-F%6
$-F(6#F*/F*;F.%\"pGF/" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 
3 " = " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. p);" "6#-%$IntG6$-%\"fG6#%\"
xG/F);\"\"!%\"pG" }{TEXT -1 25 "                         " }}{PARA 0 "
" 0 "" {TEXT -1 38 "Since this integral is independent of " }{XPPEDIT 
18 0 "alpha;" "6#%&alphaG" }{TEXT -1 71 ", this proves the result.  Ot
her cases can be handled in a similar way." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We consider some illustrative e
xamples involving the function " }{XPPEDIT 18 0 "f(x) = exp(sin*x);" "
6#/-%\"fG6#%\"xG-%$expG6#*&%$sinG\"\"\"F'F-" }{TEXT -1 31 " which is p
eriodic with period " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" 
}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 99 "The following integrals
 are all approximated numerically, since Maple cannot obtain \"exact\"
 values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "plot(exp(sin(x)),x=-8..8,y=0..3,ytickmarks=4);" }}
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m#F17$$\"1++DE%4ep(F1$\"1I+s#)*fXo#F17$$\"1++]A_ERxF1$\"1sS`CRZ+FF17$$
\"1++v=5s#y(F1$\"1.C#fn\"R6FF17$$\"1+++:o<EyF1$\"1(=KlLJsr#F17$$\"1++D
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1$\"12#)o4\\./FF17$$\"\")F*$\"1%y4w\"z]*o#F1-%'COLOURG6&%$RGBG$\"#5!\"
\"F*F*-%+AXESLABELSG6$Q\"x6\"Q\"yFiem-%*AXESTICKSG6$%(DEFAULTG\"\"%-%%
VIEWG6$;F(Fjdm;F*$\"\"$F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int(exp(sin(x)),x=-Pi..Pi);\nevalf(
%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#-%$sinG6#%\"x
G/F,;,$%#PiG!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@l#\\&z!\"*
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "Int(exp(sin(x)),x=0..2*Pi);\nevalf(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#-%$sinG6#%\"xG/F,;\"\"!,$%#PiG
\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@l#\\&z!\"*" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Int(e
xp(sin(x)),x=sqrt(2)..sqrt(2)+2*Pi);\nevalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$-%$expG6#-%$sinG6#%\"xG/F,;*$-%%sqrtG6#\"\"#\"
\"\",&F/F4%#PiGF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@l#\\&z!\"*" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 53 "Int(exp(sin(x)),x=-sqrt(2)..-sqrt(2)+2*Pi);\nevalf(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#-%$sinG6#%\"xG/F,;,$
*$-%%sqrtG6#\"\"#\"\"\"!\"\",&F0F6%#PiGF4" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+@l#\\&z!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 50 "Fourier series for periodic functions with period " }
{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 279 44 " using va
rious intervals for the integration" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 
0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 50 " be an periodic (integrable) \+
function with period " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 14 "F
ourier series" }{TEXT -1 17 " of the function " }{XPPEDIT 18 0 "f(x)" 
"6#-%\"fG6#%\"xG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "F(x) = c+``;" "6#/-%\"FG6#%\"xG,&%\"cG\"\"\"%!GF*" }
{XPPEDIT 18 0 "Sum(a[k]*cos*k*x+b[k]*sin*k*x,k = 1 .. infinity);" "6#-
%$SumG6$,&**&%\"aG6#%\"kG\"\"\"%$cosGF,F+F,%\"xGF,F,**&%\"bG6#F+F,%$si
nGF,F+F,F.F,F,/F+;F,%)infinityG" }{TEXT -1 12 " ------- (i)" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = c+a[1]*cos*x+b[1]*sin*
x+a[2]*cos*2*x+b[2]*sin*2*x+a[3]*cos*3*x+b[3]*sin*3*x+` . . . `;" "6#/
%!G,2%\"cG\"\"\"*(&%\"aG6#F'F'%$cosGF'%\"xGF'F'*(&%\"bG6#F'F'%$sinGF'F
-F'F'**&F*6#\"\"#F'F,F'F6F'F-F'F'**&F06#F6F'F2F'F6F'F-F'F'**&F*6#\"\"$
F'F,F'F=F'F-F'F'**&F06#F=F'F2F'F=F'F-F'F'%(~.~.~.~GF'" }{TEXT -1 2 ", \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "  wh
ere  " }{XPPEDIT 18 0 "c = 1/(2*Pi)" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%#PiG
F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = alpha .. alpha+2
*Pi);" "6#-%$IntG6$-%\"fG6#%\"xG/F);%&alphaG,&F,\"\"\"*&\"\"#F.%#PiGF.
F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k] = 1/Pi;" "6#/&%\"aG6#%\"kG*&
\"\"\"F)%#PiG!\"\"" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(f(x)*cos*k*x,
x = alpha .. alpha+2*Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF+%
\"kGF+F*F+/F*;%&alphaG,&F0F+*&\"\"#F+%#PiGF+F+" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 "and  " }
{XPPEDIT 18 0 "b[k] = 1/Pi;" "6#/&%\"bG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin*k*x,x = alpha .. alpha+2*P
i);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$sinGF+%\"kGF+F*F+/F*;%&alphaG,
&F0F+*&\"\"#F+%#PiGF+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 13 "The constant " }{XPPEDIT 18 0 "alpha
;" "6#%&alphaG" }{TEXT -1 100 " can be any real number so that the int
egrals are computed over any complete period of the function " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 
"" {TEXT -1 31 "However, the usual choices are:" }}{PARA 15 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "alpha = -Pi;" "6#/%&alphaG,$%#PiG!\"\"
" }{TEXT -1 58 ", for which the integrals are computed between the lim
its " }{XPPEDIT 18 0 "-Pi;" "6#,$%#PiG!\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "alpha = 0;" "6#/%&alphaG\"\"!" }{TEXT 
-1 64 ", for which the integrals are computed between the limits 0 and
 " }{XPPEDIT 18 0 "2*pi;" "6#*&\"\"#\"\"\"%#piGF%" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 5 "Notes" }
{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 95 "We are assuming here t
hat the integrals involved in the definition of the Fourier coefficien
ts " }{XPPEDIT 18 0 "c, a[k]" "6$%\"cG&%\"aG6#%\"kG" }{TEXT -1 5 " and
 " }{XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" }{TEXT -1 47 " all exist, a
nd give finite real number values." }}{PARA 15 "" 0 "" {TEXT -1 25 "In
 general, the function " }{XPPEDIT 18 0 "F(x);" "6#-%\"FG6#%\"xG" }
{TEXT -1 65 " given by the Fourier series may not be the same as the f
unction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 ".\nIf \+
" }{XPPEDIT 18 0 "F(x) = f(x)" "6#/-%\"FG6#%\"xG-%\"fG6#F'" }{TEXT -1 
9 " for all " }{TEXT 265 1 "x" }{TEXT -1 39 ", the Fourier series (i) \+
is called the " }{TEXT 260 24 "Fourier series expansion" }{TEXT -1 4 "
 of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examp
le 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 18 "Define a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%
\"xG" }{TEXT -1 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "f(x) = (Pi-x)/2;" "6#/-%\"fG6#%\"xG*&,&%#PiG\"\"\"F'!\"
\"F+\"\"#F," }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"
xG" }{XPPEDIT 18 0 "`` < 2*Pi;" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT 
-1 3 ",  " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 4 "and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 
25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%
#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 30 "In order to draw the graph of " }{XPPEDIT 18 0 "f(x)
" "6#-%\"fG6#%\"xG" }{TEXT -1 23 ", we start by defining " }{XPPEDIT 
18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }
{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` < 2*Pi;" "6
#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 36 ", and then extend the definit
ion of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 37 " to all
 real numbers by periodicity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "f := x -> (Pi-x)/2:\n'f(x)'
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(x/(2*Pi)))';\n``=value(rhs(%));\nplot(f_(x),x=-7..14,color=COLOR(RGB,
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/-%\"fG6#%\"xG,&*&\"\"#!\"\"%#PiG\"\"\"F-*&F*F+F'F-F+" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%
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-$!3)yLqDYwNc\"F-7$$\"3_u=#\\N:NG'F-$\"3U_P#HIJ1d\"F-7$$\"3b(o/t!\\08j
F-$\"3!fMKn_heb\"F-7$$\"3o*\\(ofWfUjF-$\"3$)R4a]<4T:F-7$$\"39\\(=#pEvg
kF-$\"37:`xXE,#[\"F-7$$\"3Y***\\(y3\"*ylF-$\"3%**o45aLHU\"F-7$$\"3+**
\\(=p^7z'F-$\"3=!>ZW8jnJ\"F-7$$\"3_)****\\]#f.qF-$\"3T!p%)ys#f57F-7$$
\"3#3++vyIqX(F-$\"3o#*oMmetQ)*Fhp7$$\"3/**\\ilK\"z'yF-$\"3_,>svMK%y(Fh
p7$$\"3K++]Uj6<$)F-$\"3<&*oM\"43$QbFhp7$$\"3o(*\\iI9yR()F-$\"3L3>s]E)
\\U$Fhp7$$\"3g,]7L#)e\"=*F-$\"3q))=AQ'[f@\"Fhp7$$\"3x(***\\P^`8'*F-$!3
G@4`O)eyV*F]r7$$\"3\")*\\i:Nul+\"!#;$!3&R4yUxF[?$Fhp7$$\"3++++$RF,0\"F
abl$!39.J:Y)zCQ&Fhp7$$\"3%***\\Pomm%4\"Fabl$!3t*4.\\hV%4wFhp7$$\"37+D1
Or$)Q6Fabl$!3M4\"y#**p'z\")*Fhp7$$\"3-++]3_Uz6Fabl$!3Q5`6iqt%=\"F-7$$
\"3***\\(=kWo-7Fabl$!3G5GbSL.,8F-7$$\"3(***\\()>P%fA\"Fabl$!3=5.**='Ht
T\"F-7$$\"3+]il(RXjB\"Fabl$!3Mgl*y+Q$p9F-7$$\"3/+vVvquY7Fabl$!3Z5G!oRY
8_\"F-7$$\"3k7G)[\\Z$\\7Fabl$!3Yt$HS\\[V`\"F-7$$\"31D\"GV\"z%>D\"Fabl$
!3aNfD\"f]ta\"F-7$$\"3M\"y]S7[KD\"Fabl$!3/<#p)R;&Qb\"F-7$$\"3YPMxL$[XD
\"Fabl$!3k(\\#[)o_.c\"F-7$$\"3w$4'\\V&[eD\"Fabl$!39zd4PP&oc\"F-7$$\"31
](=Kv[rD\"Fabl$\"3]K2)y'yBo:F-7$$\"33v$4@f\\BE\"Fabl$\"3U2wUtOBU:F-7$$
\"35+++J/bn7Fabl$\"3N#[u*y%Hi^\"F-7$$\"3/]7.Z6t*G\"Fabl$\"3nK#=))*eK09
F-7$$\"3'**\\iI'=\">J\"Fabl$\"3)H)>m=BU%H\"F-7$$\"3=+v$RTrVN\"Fabl$\"3
!>)pGkX7#3\"F-7$$\"#9F*$\"3!z#[uRj\")R&)Fhp-%*THICKNESSG6#\"\"#-%+AXES
LABELSG6$Q\"x6\"Q!F`il-%&COLORG6&%$RGBG$\"\"%!\"\"$F*F*$\"\"\"F*-%*AXE
STICKSG6$%(DEFAULTG\"\"&-%%VIEWG6$;F(FdhlF_jl" 1 2 0 1 10 2 2 6 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 37 "It is convenient to use the interval " }{XPPEDIT 18 0 "
[0, 2*Pi];" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 69 " for the int
egrals involved in calculating the Fourier coefficients. " }}{PARA 0 "
" 0 "" {TEXT -1 50 "The constant coefficient in the Fourier series of \+
" }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 
256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c = 1/(2*Pi)" "6#/%\"cG*&
\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((Pi
-x)/2,x = 0 .. 2*Pi);" "6#-%$IntG6$*&,&%#PiG\"\"\"%\"xG!\"\"F)\"\"#F+/
F*;\"\"!*&F,F)F(F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%
!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "``
(-(Pi-x)^2/4);" "6#-%!G6#,$*&,&%#PiG\"\"\"%\"xG!\"\"\"\"#\"\"%F,F," }
{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2*Pi, ``],[``, ``],[0, ``]);
" "6#-%*PIECEWISEG6%7$*&\"\"#\"\"\"%#PiGF)%!G7$F+F+7$\"\"!F+" }{TEXT 
-1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "``=1/(2*Pi)" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF
&!\"\"" }{XPPEDIT 18 0 "``(-Pi^2/4+Pi^2/4);" "6#-%!G6#,&*&%#PiG\"\"#\"
\"%!\"\"F+*&F(F)F*F+\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0;" "6
#/%!G\"\"!" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 
";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "The coefficients of the co
sine terms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 
257 "" 0 "" {TEXT -1 51 "The coefficients of the cosine terms are give
n by: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "a[k] = 1/P
i;" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "Int(f(x)*cos*k*x,x = 0 .. 2*Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"
\"\"%$cosGF+%\"kGF+F*F+/F*;\"\"!*&\"\"#F+%#PiGF+" }{TEXT -1 2 "  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(``((Pi-x)/2)*cos*k*x,x = 0 .. 2*Pi);" "6#-%$IntG
6$**-%!G6#*&,&%#PiG\"\"\"%\"xG!\"\"F-\"\"#F/F-%$cosGF-%\"kGF-F.F-/F.;
\"\"!*&F0F-F,F-" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now  \+
" }{XPPEDIT 18 0 "Int(``((Pi-x)/2)*cos*k*x,x) = Int(u*``(dv/dx),x);" "
6#/-%$IntG6$**-%!G6#*&,&%#PiG\"\"\"%\"xG!\"\"F.\"\"#F0F.%$cosGF.%\"kGF
.F/F.F/-F%6$*&%\"uGF.-F)6#*&%#dvGF.%#dxGF0F.F/" }{TEXT -1 8 ", where \+
" }{XPPEDIT 18 0 "u = (Pi-x)/2;" "6#/%\"uG*&,&%#PiG\"\"\"%\"xG!\"\"F(
\"\"#F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = sin*k*x/k;" "6#/%\"vG
**%$sinG\"\"\"%\"kGF'%\"xGF'F(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 47 "Hence, using the integration by parts formula: " }
{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$In
tG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6
$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 11 ", we have: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k]=1/Pi" "6#/&%\"aG6#%\"k
G*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``((Pi-x)/2
)*cos*k*x,x = 0 .. 2*Pi)" "6#-%$IntG6$**-%!G6#*&,&%#PiG\"\"\"%\"xG!\"
\"F-\"\"#F/F-%$cosGF-%\"kGF-F.F-/F.;\"\"!*&F0F-F,F-" }{TEXT -1 9 "   .
..   " }{XPPEDIT 18 0 "PIECEWISE([ u = (Pi-x)/2,v = sin*k*x/k],[du/dx=
-1  ,dv/dx=cos*k*x  ])" "6#-%*PIECEWISEG6$7$/%\"uG*&,&%#PiG\"\"\"%\"xG
!\"\"F,\"\"#F./%\"vG**%$sinGF,%\"kGF,F-F,F4F.7$/*&%#duGF,%#dxGF.,$F,F.
/*&%#dvGF,F9F.*(%$cosGF,F4F,F-F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``((Pi-x)/2)*``(sin*k*x/k);" "6#*&-%
!G6#*&,&%#PiG\"\"\"%\"xG!\"\"F*\"\"#F,F*-F%6#**%$sinGF*%\"kGF*F+F*F2F,
F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2*Pi, ``],[``, ``],[0, \+
``]);" "6#-%*PIECEWISEG6%7$*&\"\"#\"\"\"%#PiGF)%!G7$F+F+7$\"\"!F+" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "-1/Pi;" "6#,$*&\"\"\"F%%#PiG!\"\"F'" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(-sin*k*x/k,x = 0 .. 2*Pi);" "6#-%$I
ntG6$,$**%$sinG\"\"\"%\"kGF)%\"xGF)F*!\"\"F,/F+;\"\"!*&\"\"#F)%#PiGF)
" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 
"" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0-cos*k*x/(Pi*k^2);" "6#/%!G,&
\"\"!\"\"\"**%$cosGF'%\"kGF'%\"xGF'*&%#PiGF'*$F*\"\"#F'!\"\"F0" }
{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([2*Pi, ``],[``, ``],[0, ``])
;" "6#-%*PIECEWISEG6%7$*&\"\"#\"\"\"%#PiGF)%!G7$F+F+7$\"\"!F+" }
{XPPEDIT 18 0 " ``=0" "6#/%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "interface
(showassumed=0): k := 'k': assume(k,integer):\na[k]=1/Pi*Int((Pi-x)/2*
cos(k*x),x=0..2*Pi);\n``=value(rhs(%)); k := 'k':\n" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/&%\"aG6#%#k|irG,$-%$IntG6$,$*&#\"\"\"\"\"#F/*&,&%#Pi
GF/%\"xG!\"\"F/-%$cosG6#*&F'F/F4F/F/F/F//F4;\"\"!,$*&F0F/F3F/F/*$F3F5
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"!" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 35 "The coefficients of the sine terms " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" 
{TEXT -1 49 "The coefficients of the sine terms are given by: " }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "b[k] = 1/Pi;" "6#/&%
\"bG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f
(x)*sin*k*x,x = 0 .. 2*Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$sinGF+
%\"kGF+F*F+/F*;\"\"!*&\"\"#F+%#PiGF+" }{TEXT -1 2 "  " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``
 = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"Int(``((Pi-x)/2)*sin*k*x,x = 0 .. 2*Pi);" "6#-%$IntG6$**-%!G6#*&,&%#P
iG\"\"\"%\"xG!\"\"F-\"\"#F/F-%$sinGF-%\"kGF-F.F-/F.;\"\"!*&F0F-F,F-" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now  " }{XPPEDIT 18 0 "I
nt(``((Pi-x)/2)*sin*k*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$**-%!G6
#*&,&%#PiG\"\"\"%\"xG!\"\"F.\"\"#F0F.%$sinGF.%\"kGF.F/F.F/-F%6$*&%\"uG
F.-F)6#*&%#dvGF.%#dxGF0F.F/" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u \+
= (Pi-x)/2;" "6#/%\"uG*&,&%#PiG\"\"\"%\"xG!\"\"F(\"\"#F*" }{TEXT -1 5 
" and " }{XPPEDIT 18 0 "v = -cos*k*x/k;" "6#/%\"vG,$**%$cosG\"\"\"%\"k
GF(%\"xGF(F)!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "He
nce, using the integration by parts formula: " }{XPPEDIT 18 0 "Int(u*`
`(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6
#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F
/F0F)F1F0" }{TEXT -1 11 ", we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "b[k] = 1/Pi;" "6#/&%\"bG6#%\"kG*&\"\"\"F)%#PiG!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``((Pi-x)/2)*sin*k*x,x = 0 .. 2*
Pi);" "6#-%$IntG6$**-%!G6#*&,&%#PiG\"\"\"%\"xG!\"\"F-\"\"#F/F-%$sinGF-
%\"kGF-F.F-/F.;\"\"!*&F0F-F,F-" }{TEXT -1 9 "   ...   " }{XPPEDIT 18 
0 "PIECEWISE([u = (Pi-x)/2, v = -cos*k*x/k],[du/dx = -1, dv/dx = sin*k
*x]);" "6#-%*PIECEWISEG6$7$/%\"uG*&,&%#PiG\"\"\"%\"xG!\"\"F,\"\"#F./%
\"vG,$**%$cosGF,%\"kGF,F-F,F5F.F.7$/*&%#duGF,%#dxGF.,$F,F./*&%#dvGF,F:
F.*(%$sinGF,F5F,F-F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G
*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``((Pi-x)/2)*``(
-cos*k*x/k);" "6#*&-%!G6#*&,&%#PiG\"\"\"%\"xG!\"\"F*\"\"#F,F*-F%6#,$**
%$cosGF*%\"kGF*F+F*F3F,F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE
([2*Pi, ``],[``,``],[0, ``])" "6#-%*PIECEWISEG6%7$*&\"\"#\"\"\"%#PiGF)
%!G7$F+F+7$\"\"!F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-1/Pi;" "6#,$*&\"
\"\"F%%#PiG!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*k*x/k,x = \+
0 .. 2*Pi);" "6#-%$IntG6$**%$cosG\"\"\"%\"kGF(%\"xGF(F)!\"\"/F*;\"\"!*
&\"\"#F(%#PiGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(x/(2*Pi)-1/2)
*``(cos*k*x/k);" "6#/%!G*&-F$6#,&*&%\"xG\"\"\"*&\"\"#F+%#PiGF+!\"\"F+*
&F+F+F-F/F/F+-F$6#**%$cosGF+%\"kGF+F*F+F5F/F+" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([2*Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWIS
EG6%7$*&\"\"#\"\"\"%#PiGF)%!G7$F+F+7$\"\"!F+" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "- sin*k*x/(Pi*k^2)" "6#,$**%$sinG\"\"\"%\"kGF&%\"xGF&*&
%#PiGF&*$F'\"\"#F&!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([
2*Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$*&\"\"#\"\"\"%#PiGF)
%!G7$F+F+7$\"\"!F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(2*k)+1/(2*k
)-0" "6#/%!G,(*&\"\"\"F'*&\"\"#F'%\"kGF'!\"\"F'*&F'F'*&F)F'F*F'F+F'\"
\"!F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/k" "6#/%!G*&\"\"\"F&%\"kG!
\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6,
 7, 8, 9, 10], [b[k], `|`, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, \+
1/10]])" "6#-%'matrixG6#7$7.%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"
'\"\"(\"\")\"\"*\"#57.&%\"bG6#F(F)F**&F*F*F+!\"\"*&F*F*F,F9*&F*F*F-F9*
&F*F*F.F9*&F*F*F/F9*&F*F*F0F9*&F*F*F1F9*&F*F*F2F9*&F*F*F3F9" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 218 "interface(showassumed=0): k := 'k': assume(k,integer
):\nb[k]=1/Pi*Int((Pi-x)/2*sin(k*x),x=0..2*Pi);\n``=value(rhs(%));\nbb
 := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..10)],['b'[
k],`|`,seq(bb(k),k=1..10)]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%
\"bG6#%#k|irG,$-%$IntG6$,$*&#\"\"\"\"\"#F/*&,&%#PiGF/%\"xG!\"\"F/-%$si
nG6#*&F'F/F4F/F/F/F//F4;\"\"!,$*&F0F/F3F/F/*$F3F5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%!G*&\"\"\"F&%#k|irG!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#K%'matrixG6#7$7.%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&
\"\"'\"\"(\"\")\"\"*\"#57.&%\"bG6#F(F)F*#F*F+#F*F,#F*F-#F*F.#F*F/#F*F0
#F*F1#F*F2#F*F3Q)pprint426\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x)=Sum(sin*k*x/k,k=1..inf
inity)" "6#/-%\"FG6#%\"xG-%$SumG6$**%$sinG\"\"\"%\"kGF-F'F-F.!\"\"/F.;
F-%)infinityG" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = sin*x+sin*2*x/2+sin*3*x/3+sin*4*x/4+sin*5*x/5+sin*
6*x/6+sin*7*x/7+sin*8*x/8+` . . . `;" "6#/%!G,4*&%$sinG\"\"\"%\"xGF(F(
**F'F(\"\"#F(F)F(F+!\"\"F(**F'F(\"\"$F(F)F(F.F,F(**F'F(\"\"%F(F)F(F0F,
F(**F'F(\"\"&F(F)F(F2F,F(**F'F(\"\"'F(F)F(F4F,F(**F'F(\"\"(F(F)F(F6F,F
(**F'F(\"\")F(F)F(F8F,F(%(~.~.~.~GF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture
 compares the graphs of some truncated Fourier series with the graph o
f the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 225 "f := x -> (Pi-x)/2;\nf_ := x -> f(x-2*Pi*floor(x/(2*
Pi))):\nFS := (x,n) -> sum(sin(k*x)/k,k=1..n);\nplot([f_(x),FS(x,1),FS
(x,2),FS(x,3),FS(x,4),FS(x,5)],x=-4..10,\n   color=[black,red,blue,gre
en,magenta,coral],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"\"#!\"\"%#PiG\"\"\"F
1*&#F1F.F19$F1F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%
\"xG%\"nG6\"6$%)operatorG%&arrowGF)-%$sumG6$*&-%$sinG6#*&%\"kG\"\"\"9$
F6F6F5!\"\"/F5;F69%F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 607 293 293 
{PLOTDATA 2 "6*-%'CURVESG6%7co7$$!\"%\"\"!$\"3UM50Kn.#H%!#=7$$!3cmmmT)
R[p$!#<$\"3[pVQSfBmFF-7$$!3PLLe>;KHMF1$\"3\\.x'*H[kQ9F-7$$!3ymm;4'=28$
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The conve
rgence of the Fourier series is rather slow near the discontinuities o
f " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "f :
= x -> (Pi-x)/2:\n'f(x)'=f(x);\nf_ := x -> f(x-2*Pi*floor(x/(2*Pi))):
\nFS := (x,n) -> sum(sin(k*x)/k,k=1..n):\nplot([f_(x),FS(x,50)],x=-4..
10,numpoints=80,\n        color=[black,COLOR(RGB,.8,0,1)],linestyle=[3
,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&\"\"#!\"\"%
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/.MJ$yF\"F17$$\"3a=/E0t(R<'F1$!3+*HG.E8<,\"F17$$\"3*>++]*3T6iF1$!3X0Gp
_u;WpF-7$$\"3RV&Q.%>-HiF1$!3\"em4w@c`J&F-7$$\"3![3xc)HjYiF1$!3_e?duF.C
OF-7$$\"3AEc,JSCkiF1$!3'R\\H4Ds(*)=F-7$$\"3inTNw]&=G'F1$!3FY\"e:s:-L\"
Ffw7$$\"3-4Fp@hY*H'F1$\"3noB*)pZJD;F-7$$\"3W]7.nr2<jF1$\"31\"\\%4****H
kLF-7$$\"3t#zpB@)oMjF1$\"3'=Xrr6\"Rj]F-7$$\"39M$3xD*H_jF1$\"3=--$\\XbG
q'F-7$$\"3%yT&Q[8_(Q'F1$\"3x,RGqwPI(*F-7$$\"3n+D1RMuAkF1$\"36&3zV;]>B
\"F17$$\"3[$eR(Hb'zX'F1$\"3*)3f.2#[xV\"F17$$\"3ImmT?w=$\\'F1$\"3[XUg#Q
j_e\"F17$$\"3IL$eR&[+IlF1$\"3KQ()yx$Rjn\"F17$$\"3I++](3Aoc'F1$\"3>hy&p
m)*oq\"F17$$\"3Hn;/@$ROg'F1$\"3g.Vl)>G`o\"F17$$\"3RLLealXSmF1$\"30uf-)
pLLi\"F17$$\"3\\mmm@549nF1$\"3,)4v>43LV\"F17$$\"3[++v)[Dxy'F1$\"3bZMx&
zwRC\"F17$$\"3z$e9T1@P#oF1$\"3'y'3c!fQj<\"F17$$\"35n\"z%RmrfoF1$\"3oX:
^XH$=8\"F17$$\"3?f9;FWrxoF1$\"33'HA)Q^i=6F17$$\"3S]P%[@7d*oF1$\"3yOuIc
,C66F17$$\"3iTg_-+r8pF1$\"3pTuVk\\H46F17$$\"3sL$3-z2<$pF1$\"3l81bn\"\\
A6\"F17$$\"3M+v$4%*)p.qF1$\"3+KE_7Q@e6F17$$\"3'pmm;4!pvqF1$\"3a&4K$*3A
r@\"F17$$\"3+v=U2@`%4(F1$\"3rZ#4rzV!G7F17$$\"31$3xJ7uL6(F1$\"3]@'Gm=^c
B\"F17$$\"37\"HK*Qh@KrF1$\"3\\AK0.fVR7F17$$\"31+voa\"e5:(F1$\"3YE^gPD0
R7F17$$\"3/<z>'=U()=(F1$\"3*4%=l4QCD7F17$$\"3;L$3x@EkA(F1$\"37:E!fb#z%
>\"F17$$\"3Dm\"H2G%z,tF1$\"3/7]\"yT.v4\"F17$$\"3M***\\PMirP(F1$\"3UBFF
2z[s)*F-7$$\"3+Le9Y2v\\uF1$\"3)G*='yi&=#3*F-7$$\"3_n;a[\"RB_(F1$\"3i,%
RBjdpv)F-7$$\"3A&eR(\\LjevF1$\"3c9xuR-Kd()F-7$$\"3/-v$4bF\\f(F1$\"31=2
^R0AI))F-7$$\"3()=a8_<AJwF1$\"3Ag\")Q+KoO*)F-7$$\"3oNLL`f^nwF1$\"3u!*4
w`<*[.*F-7$$\"3/Ne9;?j/xF1$\"3!RYo,@&Q'3*F-7$$\"3GN$e*y![<u(F1$\"39gg,
\"Q$3c!*F-7$$\"3_N3xTT')yxF1$\"3zGbhE$=Y#*)F-7$$\"3([L$e/-)f\"yF1$\"3M
]NoZa4)o)F-7$$\"3WM$3-L7-*yF1$\"3k\"3'3%Q\\\"fzF-7$$\"31ML$eXWW'zF1$\"
3%e'H'\\y^S4(F-7$$\"3momTSU\"*e#)F1$\"3#pC.8$f*Q*fF-7$$\"3&=+++R,&H&)F
1$\"3S%4e1McaZ%F-7$$\"3lnm;*zC'R))F1$\"3yUVfY*[v)HF-7$$\"3#RLLL(G+<\"*
F1$\"375jX<:h&e\"F-7$$\"3L++v`du7%*F1$\"3OPnr9)Gbf\"F>7$$\"3=,+DE%4ep*
F1$!3N$HmE:b$=7F-7$$\"#5F*$!3AY\"eg#ou))HF--%*THICKNESSG6#\"\"#-%+AXES
LABELSG6$Q\"x6\"Q!Fh_o-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fa`o-%%VIEW
G6$;F(F\\_o%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }
}}{PARA 0 "" 0 "" {TEXT -1 18 "Define a function " }{XPPEDIT 18 0 "f(x
)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 5 ",  \+
  " }{XPPEDIT 18 0 "-Pi/2 <= x;" "6#1,$*&%#PiG\"\"\"\"\"#!\"\"F)%\"xG
" }{XPPEDIT 18 0 "`` < 3*Pi/2;" "6#2%!G*(\"\"$\"\"\"%#PiGF'\"\"#!\"\"
" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 
-1 25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"
\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%
\"xG" }{TEXT -1 25 " can be drawn as follows." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "f := x -> x
:\n'f(x)'=f(x);\nf_ := x -> f(x-2*Pi*floor(x/(2*Pi)+1/4)):\n'f_(x)'='f
(x-2*Pi*floor(x/(2*Pi)+1/4))';\nplot(f_(x),x=-2*Pi-0.1..6*Pi,\n       \+
   color=COLOR(RGB,.4,0,1),ytickmarks=5,thickness=2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*(\"\"#F,%#PiGF,-%&floorG6#,&*(F.!\"
\"F'F,F/F5F,#F,\"\"%F,F,F5" }}{PARA 13 "" 1 "" {GLPLOT2D 627 187 187 
{PLOTDATA 2 "6(-%'CURVESG6#7_q7$$!3g***>XH&=$Q'!#<$!2wLTSt)******!#=7$
$!3!3Q1]M$=LeF*$\"3Ga?t@'>+]%F-7$$!3!R-z&*eHYN&F*$\"3CBc+wrb&G*F-7$$!3
34hATeW;[F*$\"3;xM&fYRnY\"F*7$$!3?%GT;m%puUF*$\"3.-$Qbk!\\3?F*7$$!3d1f
%3L=bt$F*$\"3mzOLwpmZDF*7$$!3WhS#G&GjNKF*$\"3![_bVX_v/$F*7$$!3;Z$H[RI!
=FF*$\"31R-N7\\:lNF*7$$!3!eCLB>DF=#F*$\"3VSj%[6g/5%F*7$$!30u=Pc4$f\">F
*$\"3i7x!3NasO%F*7$$!3I-0T?n8\\;F*$\"3$R3pne[Sj%F*7$$!340AI()G$[h\"F*$
\"3O\"Qx)>CNoYF*7$$!3)y!R>a!H0e\"F*$\"3\"zn&)HDcEq%F*7$$!3Qf(Rw8xLc\"F
*Feo7$$!3!4h&3@_AY:F*Fho7$$!3<i9`/L2H:F*F[p7$$!3o8t(zQ@>^\"F*F^p7$$!3[
>2w@PJV9F*Fap7$$!31DTabgqu8F*Fdp7$$!3WO46B2\\P7F*Fgp7$$!3#yux1Rv-5\"F*
Fjp7$$!3)==xo6nbe)F-F]q7$$!3a&)o(pK!QohF-F`q7$$!3I>g*Q%p?gs!#>Fcq7$$\"
3:?-u*y&oQZF-Fgq7$$\"3#*3(osI%\\+5F*Fjq7$$\"3Y(pPPh2'R7F*F]r7$$\"3+'o1
-#4sy9F*F`r7$$\"3CqC8p(\\Iw\"F*Fcr7$$\"3]a#e!='yt/#F*Ffr7$$\"3K2\\d[CC
)G#F*Fir7$$\"3;g:4zi5HDF*F\\s7$$\"3[f0ap+Z*3$F*F_s7$$\"3Sr\"yS(p[&e$F*
Fbs7$$\"3R>i!G8'pHTF*Fes7$$\"3i%\\)[RS!))Q%F*Fhs7$$\"3')p2<Y>\"zk%F*F[
t7$$\"3AK>2T*3[m%F*F^t7$$\"3q$4tf$fq\"o%F*Fat7$$\"31cU(3$Hg)p%F*Fdt7$$
\"3V=axD**\\:ZF*$!3!y;/9Q&on:F*7$$\"3GUxd:RH\\ZF*$!3'R%=g\"R\"*Q`\"F*7
$$\"37m+Q0z3$y%F*$!36?&*z,u4+:F*7$$\"3!Qr%)\\)en][F*$!3Vs[>A%4DV\"F*7$
$\"3\\h$*ekQE=\\F*$!3uC-fU9#\\O\"F*7$$\"3'ol)zB)RM0&F*$!3PH4Q$[X(H7F*7
$$\"37`z+$y:')=&F*$!35L;<C&pX4\"F*7$$\"3v88&)QK)oV&F*$!3%[s#G$o?IY)F-7
$$\"3QuYp%p]^o&F*$!3a=\"\\[7Y.)fF-7$$\"3O(Q&[w1t?iF*$!3i())>%pIYXiFeq7
$$\"35cUbbx/xnF*$\"3g)pYP[C'Q\\F-7$$\"3I:\"y#3MKhsF*$\"3c!H&)4,\"Q\"y*
F-7$$\"3)[#\\Q!QaVy(F*$\"3mQ`?t!p6]\"F*7$$\"3R,()=`lpC$)F*$\"3::\"4gC6
:/#F*7$$\"3!>#[!>&fJ`))F*$\"3mN_sW18qDF*7$$\"3-v_ew\"\\!4\"*F*$\"3z)o0
%pQ'e#GF*7$$\"39GdE,Cyk$*F*$\"3!>9'3%4(f\"3$F*7$$\"3Sg@&faI(['*F*$\"3;
uDxQ_alLF*7$$\"3l#fQ1pyE$**F*$\"3U1!fMQ$\\\\OF*7$$\"3bFVxs=y=5!#;$\"3B
*oj0ULY!RF*7$$\"3$ez%[weHW5Fhz$\"3/s$owXt(fTF*7$$\"3ir2I@v`r5Fhz$\"3**
H\"Ge!**=KWF*7$$\"3UZn6m\"z()4\"Fhz$\"3'z)y)RN1Yq%F*7$$\"3'\\)4Jx>K+6F
hz$!3!HK\\7%3:j:F*7$$\"3\\A_])yk=5\"Fhz$!3aZpIHFsZ:F*7$$\"3&)f%*p*f2M5
\"Fhz$!3&Rdkth%HK:F*7$$\"3R(p$*3T]\\5\"Fhz$!3f)>Aa]mo^\"F*7$$\"3Gs@GLg
.36Fhz$!3k\\u`\"G5g[\"F*7$$\"3OZ1nb;766Fhz$!3!*)p_w0a^X\"F*7$$\"3L(fZ/
!HH<6Fhz$!3?*>$))4;W$R\"F*7$$\"3IZXAXTYB6Fhz$!3^*p8@;H<L\"F*7$$\"3CZ%y
Zj1e8\"Fhz$!39+ZdmUI37F*7$$\"3=ZBLC\"\\\"[6Fhz$!3w+d.r$z[3\"F*7$$\"3qx
XlEj8v6Fhz$!3aaR8yM2]\")F-7$$\"3B3o(*GN7-7Fhz$!3a,4\"fC`8X&F-7$$\"3eKN
UP#4HD\"Fhz$!3(Gn%Q7S#ys$Feq7$$\"3:%4()f$\\*fI\"Fhz$\"3d!p<^X(yN\\F-7$
$\"3Ec_^4f*yN\"Fhz$\"3.!R$z![)e75F*7$$\"3Rdt')=9B79Fhz$\"3W,WJuN%fb\"F
*7$$\"3Q->IsHck9Fhz$\"3I^)f'3\"f#z?F*7$$\"3vhC5W&z!=:Fhz$\"32XamE[U9EF
*7$$\"3]y&o;(H:r:Fhz$\"3c7mK-\"f^9$F*7$$\"3A$))3'R>#*>;Fhz$\"3vf'H<y[G
j$F*7$$\"3_\"oV/o;en\"Fhz$\"3eUw2!>'z\">%F*7$$\"3lC!3<983q\"Fhz$\"3)R2
@F!3wTWF*7$$\"3znB(Hg4es\"Fhz$\"3O0XO:as\"p%F*7$$\"3OO?0=`ZF<Fhz$\"36
\">hhc#Q3ZF*7$$\"3e/<8L59H<Fhz$!3'Hr@-fX\"e:F*7$$\"3!GP6#[n!3t\"Fhz$!3
yI]UR%)[T:F*7$$\"3,T5HjCZK<Fhz$!3g[$G')GJ[_\"F*7$$\"3!yP]M*Q!et\"Fhz$!
3q!)\\.()p^\"\\\"F*7$$\"3f9(4OKN\"R<Fhz$!3!GhTao-#e9F*7$$\"3#yQGR=)zX<
Fhz$!3a!)[D#3u:R\"F*7$$\"3/hqCW5Y_<Fhz$!3I[\"o!za%\\K\"F*7$$\"3'yS%)[w
'yl<Fhz$!3C!o%ps#)o\">\"F*7$$\"3ma<_&[7\"z<Fhz$!3?77Km5Ve5F*7$$\"3L<*o
aJ@Y!=Fhz$!3)Q&e\\owUL!)F-7$$\"3-!3;a9I,$=Fhz$!3w&ezPnWD[&F-7$$\"3!***
**Q)eb\\)=Fhz$!3U)*fXmf(Qw$!#D-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x
6\"Q!Fggl-%&COLORG6&%$RGBG$\"\"%!\"\"$\"\"!Fahl$\"\"\"Fahl-%*AXESTICKS
G6$%(DEFAULTG\"\"&-%%VIEWG6$;$!+3`=$Q'!\"*$\"+#fb\\)=!\")Fghl" 1 2 0 
1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "It is convenient to use the int
erval " }{XPPEDIT 18 0 "[-Pi/2,3*Pi/2]" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"
F)*(\"\"$F'F&F'F(F)" }{TEXT -1 69 " for the integrals involved in calc
ulating the Fourier coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 50 "The
 constant coefficient in the Fourier series of " }{XPPEDIT 18 0 "f(x)
" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 
2 "  " }{XPPEDIT 18 0 "c = 1/(2*Pi)" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%#PiG
F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x,x = -Pi/2 .. 3*Pi/2);" 
"6#-%$IntG6$%\"xG/F&;,$*&%#PiG\"\"\"\"\"#!\"\"F.*(\"\"$F,F+F,F-F." }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"
#F&%#PiGF&!\"\"" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "``(x^2/2);" "6#-%!G6
#*&%\"xG\"\"#F(!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3*Pi/
2, ``],[-Pi/2, ``]);" "6#-%*PIECEWISEG6$7$*(\"\"$\"\"\"%#PiGF)\"\"#!\"
\"%!G7$,$*&F*F)F+F,F,F-" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(2*Pi)" "
6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{XPPEDIT 18 0 " ``(9*Pi^2/8-Pi
^2/8)" "6#-%!G6#,&*(\"\"*\"\"\"*$%#PiG\"\"#F)\"\")!\"\"F)*&F+F,F-F.F.
" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`
` = Pi/2;" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "The coefficients of the cosine te
rms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 
"" {TEXT -1 51 "The coefficients of the cosine terms are given by: " }
}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "a[k] = 1/Pi;" "6#/&
%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(
f(x)*cos*k*x,x = -Pi/2 .. 3*Pi/2);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%
$cosGF+%\"kGF+F*F+/F*;,$*&%#PiGF+\"\"#!\"\"F4*(\"\"$F+F2F+F3F4" }
{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+
= 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
Int(x*cos*k*x,x = -Pi/2 .. 3*Pi/2);" "6#-%$IntG6$**%\"xG\"\"\"%$cosGF(
%\"kGF(F'F(/F';,$*&%#PiGF(\"\"#!\"\"F1*(\"\"$F(F/F(F0F1" }{TEXT -1 3 "
.  " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now  " }{XPPEDIT 18 0 "Int(x*cos*k*
x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$**%\"xG\"\"\"%$cosGF)%\"kGF)F
(F)F(-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxG!\"\"F)F(" }{TEXT -1 8 ", where
 " }{XPPEDIT 18 0 "u = x" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "v = sin*k*x/k;" "6#/%\"vG**%$sinG\"\"\"%\"kGF'%\"xGF'F(
!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the
 integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = \+
u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dx
G!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }
{TEXT -1 10 ", we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "a[k] = 1/Pi;" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "``(x*sin*k*x/k);" "6#-%!G6#*,%\"xG\"\"\"%$sinGF(
%\"kGF(F'F(F*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3*Pi/2,
 ``],[-Pi/2, ``]);" "6#-%*PIECEWISEG6$7$*(\"\"$\"\"\"%#PiGF)\"\"#!\"\"
%!G7$,$*&F*F)F+F,F,F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-1/Pi;" "6#,$*&
\"\"\"F%%#PiG!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*k*x/k,x \+
= -Pi/2 .. 3*Pi/2);" "6#-%$IntG6$**%$sinG\"\"\"%\"kGF(%\"xGF(F)!\"\"/F
*;,$*&%#PiGF(\"\"#F+F+*(\"\"$F(F0F(F1F+" }{TEXT -1 1 " " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "`` = ``(x*sin*k*x/(Pi*k)+cos*k*x/(Pi*k^2));" "6#/%!G-F$6#,&*,%\"xG
\"\"\"%$sinGF*%\"kGF*F)F**&%#PiGF*F,F*!\"\"F***%$cosGF*F,F*F)F**&F.F**
$F,\"\"#F*F/F*" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([3*Pi/2, ``
],[-Pi/2, ``]);" "6#-%*PIECEWISEG6$7$*(\"\"$\"\"\"%#PiGF)\"\"#!\"\"%!G
7$,$*&F*F)F+F,F,F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 3/(2*k);" "6#/%!
G*&\"\"$\"\"\"*&\"\"#F'%\"kGF'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "s
in(3*k*Pi/2)+1/(k^2);" "6#,&-%$sinG6#**\"\"$\"\"\"%\"kGF)%#PiGF)\"\"#!
\"\"F)*&F)F)*$F*F,F-F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(3*k*Pi/2)+
1/(2*k);" "6#,&-%$cosG6#**\"\"$\"\"\"%\"kGF)%#PiGF)\"\"#!\"\"F)*&F)F)*
&F,F)F*F)F-F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(-k*Pi/2)-1/(k^2);" 
"6#,&-%$sinG6#,$*(%\"kG\"\"\"%#PiGF*\"\"#!\"\"F-F**&F*F**$F)F,F-F-" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(-k*Pi/2);" "6#-%$cosG6#,$*(%\"kG\"
\"\"%#PiGF)\"\"#!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*k);" "
6#/%!G*&\"\"\"F&*&\"\"#F&%\"kGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"``(3*sin(3*k*Pi/2)-sin(k*Pi/2))+1/(k^2);" "6#,&-%!G6#,&*&\"\"$\"\"\"-
%$sinG6#**F)F*%\"kGF*%#PiGF*\"\"#!\"\"F*F*-F,6#*(F/F*F0F*F1F2F2F**&F*F
**$F/F1F2F*" }{XPPEDIT 18 0 "``(cos(3*k*Pi/2)-cos(k*Pi/2));" "6#-%!G6#
,&-%$cosG6#**\"\"$\"\"\"%\"kGF,%#PiGF,\"\"#!\"\"F,-F(6#*(F-F,F.F,F/F0F
0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "
" {TEXT -1 12 "The formula " }{XPPEDIT 18 0 "cos*alpha-cos*beta=-2*sin
((alpha+beta)/2)*sin((alpha-beta)/2)" "6#/,&*&%$cosG\"\"\"%&alphaGF'F'
*&F&F'%%betaGF'!\"\",$*(\"\"#F'-%$sinG6#*&,&F(F'F*F'F'F.F+F'-F06#*&,&F
(F'F*F+F'F.F+F'F+" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "cos(3*k*Pi/2)-cos(k*Pi/2) = -2*sin*k*Pi*sin(k
*Pi/2);" "6#/,&-%$cosG6#**\"\"$\"\"\"%\"kGF*%#PiGF*\"\"#!\"\"F*-F&6#*(
F+F*F,F*F-F.F.,$*,F-F*%$sinGF*F+F*F,F*-F46#*(F+F*F,F*F-F.F*F." }
{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 4 ",   " }{XPPEDIT 18 
0 "k = 1, 2, 3, ` . . . `" "6&/%\"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k
] = 1/(2*k);" "6#/&%\"aG6#%\"kG*&\"\"\"F)*&\"\"#F)F'F)!\"\"" }
{XPPEDIT 18 0 "``(3*sin(3*k*Pi/2)-sin(k*Pi/2));" "6#-%!G6#,&*&\"\"$\"
\"\"-%$sinG6#**F(F)%\"kGF)%#PiGF)\"\"#!\"\"F)F)-F+6#*(F.F)F/F)F0F1F1" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 12 "The formula " }{XPPEDIT 18 0 "sin*alpha+sin*beta = 2*sin(
(alpha+beta)/2)*cos((alpha-beta)/2);" "6#/,&*&%$sinG\"\"\"%&alphaGF'F'
*&F&F'%%betaGF'F'*(\"\"#F'-F&6#*&,&F(F'F*F'F'F,!\"\"F'-%$cosG6#*&,&F(F
'F*F1F'F,F1F'" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "sin(3*k*Pi/2)+sin(k*Pi/2) = 2*sin*k*Pi*cos(k*Pi/2);
" "6#/,&-%$sinG6#**\"\"$\"\"\"%\"kGF*%#PiGF*\"\"#!\"\"F*-F&6#*(F+F*F,F
*F-F.F**,F-F*F&F*F+F*F,F*-%$cosG6#*(F+F*F,F*F-F.F*" }{XPPEDIT 18 0 "``
 = 0;" "6#/%!G\"\"!" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "k = 1, 2, 3, `
 . . . `" "6&/%\"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 1 "." }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence  \+
" }{XPPEDIT 18 0 "sin(3*k*Pi/2)=-sin(k*Pi/2)" "6#/-%$sinG6#**\"\"$\"\"
\"%\"kGF)%#PiGF)\"\"#!\"\",$-F%6#*(F*F)F+F)F,F-F-" }{TEXT -1 4 ",   " 
}{XPPEDIT 18 0 "k = 1, 2, 3, ` . . . `" "6&/%\"kG\"\"\"\"\"#\"\"$%(~.~
.~.~G" }{TEXT -1 7 ", and  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "a[k] = -2/k;" "6#/&%\"aG6#%\"kG,$*&\"\"#\"\"\"F'!\"\"F,
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(k*Pi/2);" "6#-%$sinG6#*(%\"kG\"
\"\"%#PiGF(\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 277 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 
0 "sin(k*Pi/2)" "6#-%$sinG6#*(%\"kG\"\"\"%#PiGF(\"\"#!\"\"" }{TEXT -1 
31 " takes the sequence of values: " }{XPPEDIT 18 0 "1,0,-1,0,1,0,-1,0
,1,0,` . . . `;" "6-\"\"\"\"\"!,$F#!\"\"F$F#F$,$F#F&F$F#F$%(~.~.~.~G" 
}{TEXT -1 7 ",  for " }{XPPEDIT 18 0 "k = 1, 2, 3, ` . . . `" "6&/%\"k
G\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k,
 `|`, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [a[k], `|`, -2, 0, 2/3, 0, -2/5,
 0, 2/7, 0, -2/9, 0]])" "6#-%'matrixG6#7$7.%\"kG%\"|grG\"\"\"\"\"#\"\"
$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"aG6#F(F),$F+!\"\"\"\"!*&F+F*F
,F9F:,$*&F+F*F.F9F9F:*&F+F*F0F9F:,$*&F+F*F2F9F9F:" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
217 "interface(showassumed=0): k := 'k': assume(k,integer):\na[k]=1/Pi
*Int(x*cos(k*x),x=-Pi/2..3*Pi/2);\n``=value(rhs(%));\naa := unapply(rh
s(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..10)],['a'[k],`|`,seq(aa(
k),k=1..10)]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%#k|irG,
$-%$IntG6$*&%\"xG\"\"\"-%$cosG6#*&F'F.F-F.F./F-;,$*&\"\"#!\"\"%#PiGF.F
8,$*(\"\"$F.F7F8F9F.F.*$F9F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$
*&#\"\"\"\"\"#F(*(%#PiG!\"\",**&F)F(-%$cosG6#,$*(F)F,F+F(%#k|irGF(F(F(
F,*(F4F(-%$sinGF1F(F+F(F,*&F)F(-F06#,$**\"\"$F(F)F,F+F(F4F(F(F(F(**F=F
(F4F(-F7F:F(F+F(F(F(F4!\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m
atrixG6#7$7.%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*
\"#57.&%\"aG6#F(F)!\"#\"\"!#F+F,F9#F8F.F9#F+F0F9#F8F2F9Q)pprint436\"" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "The coefficients of
 the sine terms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 257 "" 0 "" {TEXT -1 49 "The coefficients of the sine terms are \+
given by: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "b[k] =
 1/Pi" "6#/&%\"bG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(f(x)*sin*k*x,x = -Pi/2 .. 3*Pi/2);" "6#-%$IntG6$**-
%\"fG6#%\"xG\"\"\"%$sinGF+%\"kGF+F*F+/F*;,$*&%#PiGF+\"\"#!\"\"F4*(\"\"
$F+F2F+F3F4" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(x*sin*k*x,x = -Pi/2 .. 3*Pi/2);" "6#-%$IntG6$**%
\"xG\"\"\"%$sinGF(%\"kGF(F'F(/F';,$*&%#PiGF(\"\"#!\"\"F1*(\"\"$F(F/F(F
0F1" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now  " }{XPPEDIT 
18 0 "Int(x*sin*k*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$**%\"xG\"\"
\"%$sinGF)%\"kGF)F(F)F(-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxG!\"\"F)F(" }
{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x" "6#/%\"uG%\"xG" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "v = -cos*k*x/k;" "6#/%\"vG,$**%$cosG\"\"
\"%\"kGF(%\"xGF(F)!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
47 "Hence, using the integration by parts formula: " }{XPPEDIT 18 0 "I
nt(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"
\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%
#duGF)F/F0F)F1F0" }{TEXT -1 10 ", we have:" }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "b[k] = 1/Pi;" "6#/&%\"bG6#%\"kG*&\"\"\"F)%#Pi
G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-x*cos*k*x/k);" "6#-%!G6#,$
*,%\"xG\"\"\"%$cosGF)%\"kGF)F(F)F+!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 
18 0 "PIECEWISE([3*Pi/2, ``],[-Pi/2, ``]);" "6#-%*PIECEWISEG6$7$*(\"\"
$\"\"\"%#PiGF)\"\"#!\"\"%!G7$,$*&F*F)F+F,F,F-" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "-1/Pi;" "6#,$*&\"\"\"F%%#PiG!\"\"F'" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(-cos*k*x/k,x = -Pi/2 .. 3*Pi/2);" "6#-%$IntG6$,$**%
$cosG\"\"\"%\"kGF)%\"xGF)F*!\"\"F,/F+;,$*&%#PiGF)\"\"#F,F,*(\"\"$F)F1F
)F2F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(-x*cos*k*x/(Pi*k)+sin*k
*x/(Pi*k^2));" "6#/%!G-F$6#,&*,%\"xG\"\"\"%$cosGF*%\"kGF*F)F**&%#PiGF*
F,F*!\"\"F/**%$sinGF*F,F*F)F**&F.F**$F,\"\"#F*F/F*" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "PIECEWISE([3*Pi/2, ``],[-Pi/2, ``]);" "6#-%*PIECEWISEG6
$7$*(\"\"$\"\"\"%#PiGF)\"\"#!\"\"%!G7$,$*&F*F)F+F,F,F-" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = -3/(2*k);" "6#/%!G,$*&\"\"$\"\"\"*&\"\"#F(%\"kGF(!
\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(3*k*Pi/2)+1/(k^2);" "6#,&-
%$cosG6#**\"\"$\"\"\"%\"kGF)%#PiGF)\"\"#!\"\"F)*&F)F)*$F*F,F-F)" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(3*k*Pi/2)-1/(2*k);" "6#,&-%$sinG6#*
*\"\"$\"\"\"%\"kGF)%#PiGF)\"\"#!\"\"F)*&F)F)*&F,F)F*F)F-F-" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "cos(-k*Pi/2)-1/(k^2);" "6#,&-%$cosG6#,$*(%\"kG\"
\"\"%#PiGF*\"\"#!\"\"F-F**&F*F**$F)F,F-F-" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "sin(-k*Pi/2)" "6#-%$sinG6#,$*(%\"kG\"\"\"%#PiGF)\"\"#!\"\"F," }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/(2*k);" "6#/%!G,$*&\"\"\"F'*&\"
\"#F'%\"kGF'!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(3*cos(3*k*Pi/2
)+cos(k*Pi/2)) + 1/(k^2)" "6#,&-%!G6#,&*&\"\"$\"\"\"-%$cosG6#**F)F*%\"
kGF*%#PiGF*\"\"#!\"\"F*F*-F,6#*(F/F*F0F*F1F2F*F**&F*F**$F/F1F2F*" }
{XPPEDIT 18 0 "``(sin(3*k*Pi/2)+sin(k*Pi/2))" "6#-%!G6#,&-%$sinG6#**\"
\"$\"\"\"%\"kGF,%#PiGF,\"\"#!\"\"F,-F(6#*(F-F,F.F,F/F0F," }{TEXT -1 2 
". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "
The formula " }{XPPEDIT 18 0 "sin*alpha+sin*beta = 2*sin((alpha+beta)/
2)*cos((alpha-beta)/2);" "6#/,&*&%$sinG\"\"\"%&alphaGF'F'*&F&F'%%betaG
F'F'*(\"\"#F'-F&6#*&,&F(F'F*F'F'F,!\"\"F'-%$cosG6#*&,&F(F'F*F1F'F,F1F'
" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "sin(3*k*Pi/2)+sin(k*Pi/2) = 2*sin*k*Pi*cos(k*Pi/2);" "6#/,&-%$si
nG6#**\"\"$\"\"\"%\"kGF*%#PiGF*\"\"#!\"\"F*-F&6#*(F+F*F,F*F-F.F**,F-F*
F&F*F+F*F,F*-%$cosG6#*(F+F*F,F*F-F.F*" }{XPPEDIT 18 0 "`` = 0;" "6#/%!
G\"\"!" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "k = 1, 2, 3, ` . . . `" "6&
/%\"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"b[k] = -1/(2*k);" "6#/&%\"bG6#%\"kG,$*&\"\"\"F**&\"\"#F*F'F*!\"\"F-" 
}{XPPEDIT 18 0 "``(3*cos(3*k*Pi/2)+cos(k*Pi/2)))" "6#-%!G6#,&*&\"\"$\"
\"\"-%$cosG6#**F(F)%\"kGF)%#PiGF)\"\"#!\"\"F)F)-F+6#*(F.F)F/F)F0F1F)" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 12 "The formula " }{XPPEDIT 18 0 "cos*alpha-cos*beta=-2*sin((
alpha+beta)/2)*sin((alpha-beta)/2)" "6#/,&*&%$cosG\"\"\"%&alphaGF'F'*&
F&F'%%betaGF'!\"\",$*(\"\"#F'-%$sinG6#*&,&F(F'F*F'F'F.F+F'-F06#*&,&F(F
'F*F+F'F.F+F'F+" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "cos(3*k*Pi/2)-cos(k*Pi/2) = -2*sin*k*Pi*sin(k*Pi
/2);" "6#/,&-%$cosG6#**\"\"$\"\"\"%\"kGF*%#PiGF*\"\"#!\"\"F*-F&6#*(F+F
*F,F*F-F.F.,$*,F-F*%$sinGF*F+F*F,F*-F46#*(F+F*F,F*F-F.F*F." }{XPPEDIT 
18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "k = 1,
 2, 3, ` . . . `" "6&/%\"kG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence \+
 " }{XPPEDIT 18 0 "cos(3*k*Pi/2) = cos(k*Pi/2);" "6#/-%$cosG6#**\"\"$
\"\"\"%\"kGF)%#PiGF)\"\"#!\"\"-F%6#*(F*F)F+F)F,F-" }{TEXT -1 4 ",   " 
}{XPPEDIT 18 0 "k = 1, 2, 3, ` . . . `" "6&/%\"kG\"\"\"\"\"#\"\"$%(~.~
.~.~G" }{TEXT -1 7 ", and  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "b[k] = -2/k;" "6#/&%\"bG6#%\"kG,$*&\"\"#\"\"\"F'!\"\"F,
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(k*Pi/2);" "6#-%$cosG6#*(%\"kG\"
\"\"%#PiGF(\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 277 4 
"Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "cos(k*Pi/2);" "6#-%$cosG6#*(%
\"kG\"\"\"%#PiGF(\"\"#!\"\"" }{TEXT -1 31 " takes the sequence of valu
es: " }{XPPEDIT 18 0 "0,1,0,-1,0,1,0,-1,0,1,` . . . `;" "6-\"\"!\"\"\"
F#,$F$!\"\"F#F$F#,$F$F&F#F$%(~.~.~.~G" }{TEXT -1 7 ",  for " }
{XPPEDIT 18 0 "k = 1, 2, 3, ` . . . `" "6&/%\"kG\"\"\"\"\"#\"\"$%(~.~.
~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k,`|`, 1, 2, 3, 4, 5, 6, \+
7, 8, 9, 10],[b[k],`|`, 0, 1, 0, -1/2, 0, 1/3, 0, -1/4, 0, 1/5]])" "6#
-%'matrixG6#7$7.%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")
\"\"*\"#57.&%\"bG6#F(F)\"\"!F*F8,$*&F*F*F+!\"\"F;F8*&F*F*F,F;F8,$*&F*F
*F-F;F;F8*&F*F*F.F;" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "interface(showassumed=0): \+
k := 'k': assume(k,integer):\nb[k]=1/Pi*Int(x*sin(k*x),x=-Pi/2..3*Pi/2
);\n``=value(rhs(%));\nbb := unapply(rhs(%),k): k := 'k':\nmatrix([[k,
`|`,seq(k,k=1..10)],['b'[k],`|`,seq(bb(k),k=1..10)]]);\n" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$-%$IntG6$*&%\"xG\"\"\"-%$sinG6
#*&F'F.F-F.F./F-;,$*&\"\"#!\"\"%#PiGF.F8,$*(\"\"$F.F7F8F9F.F.*$F9F8" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"\"#F(*(%#PiG!\"\",**
&F)F(-%$sinG6#,$*(F)F,F+F(%#k|irGF(F(F(F(*(F4F(-%$cosGF1F(F+F(F,*&F)F(
-F06#,$**\"\"$F(F)F,F+F(F4F(F(F(F(**F=F(F4F(-F7F:F(F+F(F,F(F4!\"#F(F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7.%\"kG%\"|grG\"\"\"
\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57.&%\"bG6#F(F)\"\"!F*F8#!
\"\"F+F8#F*F,F8#F:F-F8#F*F.Q)pprint446\"" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 22 "The Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG
6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "F(x) = Pi/2-``;" "6#/-%\"FG6#%\"xG,&*&%#PiG\"\"\"\"\"#!
\"\"F+%!GF-" }{XPPEDIT 18 0 "Sum(``(2/k)*(sin(k*Pi/2)*cos*k*x+cos(k*Pi
/2)*sin*k*x),k = 1 .. infinity);" "6#-%$SumG6$*&-%!G6#*&\"\"#\"\"\"%\"
kG!\"\"F,,&**-%$sinG6#*(F-F,%#PiGF,F+F.F,%$cosGF,F-F,%\"xGF,F,**-F66#*
(F-F,F5F,F+F.F,F2F,F-F,F7F,F,F,/F-;F,%)infinityG" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=Pi/2-2*cos*x+sin*2*x+2/3*cos*3*x-sin*4*x/2-2/5*cos*5
*x+sin*6*x/3+2/7*cos*7*x-sin*8*x/4-2/9*cos*9*x+` . . . `" "6#/%!G,8*&%
#PiG\"\"\"\"\"#!\"\"F(*(F)F(%$cosGF(%\"xGF(F**(%$sinGF(F)F(F-F(F(*,F)F
(\"\"$F*F,F(F1F(F-F(F(**F/F(\"\"%F(F-F(F)F*F**,F)F(\"\"&F*F,F(F5F(F-F(
F***F/F(\"\"'F(F-F(F1F*F(*,F)F(\"\"(F*F,F(F9F(F-F(F(**F/F(\"\")F(F-F(F
3F*F**,F)F(\"\"*F*F,F(F=F(F-F(F*%(~.~.~.~GF(" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 4 "Note" }
{TEXT -1 15 ": The formula  " }{XPPEDIT 18 0 "sin(alpha+beta)=sin*alph
a*cos*beta+cos*alpha*sin*beta" "6#/-%$sinG6#,&%&alphaG\"\"\"%%betaGF),
&**F%F)F(F)%$cosGF)F*F)F)**F-F)F(F)F%F)F*F)F)" }{TEXT -1 13 " shows th
at  " }{XPPEDIT 18 0 "sin(k*Pi/2)*cos*k*x+cos(k*Pi/2)*sin*k*x=sin(k*Pi
/2+k*x)" "6#/,&**-%$sinG6#*(%\"kG\"\"\"%#PiGF+\"\"#!\"\"F+%$cosGF+F*F+
%\"xGF+F+**-F/6#*(F*F+F,F+F-F.F+F'F+F*F+F0F+F+-F'6#,&*(F*F+F,F+F-F.F+*
&F*F+F0F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence  " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x)=Pi/2 - ``" "6#/-
%\"FG6#%\"xG,&*&%#PiG\"\"\"\"\"#!\"\"F+%!GF-" }{XPPEDIT 18 0 "Sum(``(2
/k)*sin(k*(Pi/2+x)),k=1..infinity)" "6#-%$SumG6$*&-%!G6#*&\"\"#\"\"\"%
\"kG!\"\"F,-%$sinG6#*&F-F,,&*&%#PiGF,F+F.F,%\"xGF,F,F,/F-;F,%)infinity
G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The following picture co
mpares the graphs of some truncated Fourier series with the graph of t
he function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 266 "f := x -> x:\nf_ := x -> f(x-2*Pi*floor(x/(2*Pi)+1/4)):\nFS := \+
(x,n) -> Pi/2-sum(2/k*sin(k*Pi/2)*cos(k*x)+2/k*cos(k*Pi/2)*sin(k*x),k=
1..n);\nplot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5)],x=-4..10,
\n   color=[black,red,blue,green,magenta,coral],linestyle=[3,1$5]);" }
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lating the Fourier coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 50 "The \+
constant coefficient in the Fourier series of " }{XPPEDIT 18 0 "f(x)" 
"6#-%\"fG6#%\"xG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c = 1/(2*Pi)" "6#/%
\"cG*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I
nt(exp(-x),x = 0 .. 2*Pi);" "6#-%$IntG6$-%$expG6#,$%\"xG!\"\"/F*;\"\"!
*&\"\"#\"\"\"%#PiGF1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 8 "
        " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#
PiGF&!\"\"" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "``(-exp(-x));" "6#-%!G6#,
$-%$expG6#,$%\"xG!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2
*Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$*&\"\"#\"\"\"%#PiGF)%
!G7$F+F+7$\"\"!F+" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = (1-exp(-2*Pi))/(2*Pi);" "6#/%!G*&,&\"\"\"F'-%$e
xpG6#,$*&\"\"#F'%#PiGF'!\"\"F/F'*&F-F'F.F'F/" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "The coefficients of the cosine
 and sine terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 58 "The Fourier coefficients of the cosine te
rms are given by " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 
"a[k] = 1/Pi" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(exp(-x)*cos*k*x,x = 0 .. 2*Pi);" "6#-%$IntG6$**-%$e
xpG6#,$%\"xG!\"\"\"\"\"%$cosGF-%\"kGF-F+F-/F+;\"\"!*&\"\"#F-%#PiGF-" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 52 "and  the coefficients o
f the sine terms are given by" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "b[k] = 1/Pi;" "6#/&%\"bG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-x)*sin*k*x,x = 0 .. 2*Pi);" "6
#-%$IntG6$**-%$expG6#,$%\"xG!\"\"\"\"\"%$sinGF-%\"kGF-F+F-/F+;\"\"!*&
\"\"#F-%#PiGF-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }
{XPPEDIT 18 0 "I[c] = Int(exp(-x)*cos*k*x,x);" "6#/&%\"IG6#%\"cG-%$Int
G6$**-%$expG6#,$%\"xG!\"\"\"\"\"%$cosGF2%\"kGF2F0F2F0" }{TEXT -1 7 "  \+
and  " }{XPPEDIT 18 0 "I[s] = Int(exp(-x)*sin*k*x,x);" "6#/&%\"IG6#%\"
sG-%$IntG6$**-%$expG6#,$%\"xG!\"\"\"\"\"%$sinGF2%\"kGF2F0F2F0" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 
"Now  " }{XPPEDIT 18 0 "I[c]" "6#&%\"IG6#%\"cG" }{TEXT -1 2 " =" }
{XPPEDIT 18 0 "Int(exp(-x)*cos*k*x,x) = Int(u*``(dv/dx),x);" "6#/-%$In
tG6$**-%$expG6#,$%\"xG!\"\"\"\"\"%$cosGF.%\"kGF.F,F.F,-F%6$*&%\"uGF.-%
!G6#*&%#dvGF.%#dxGF-F.F," }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = e
xp(-x)" "6#/%\"uG-%$expG6#,$%\"xG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "v = sin*k*x/k;" "6#/%\"vG**%$sinG\"\"\"%\"kGF'%\"xGF'F(!\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "The integration by parts
 formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x
);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)
%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 9 ", gives: " 
}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[c]=Int(exp(-x)*co
s*k*x,x)" "6#/&%\"IG6#%\"cG-%$IntG6$**-%$expG6#,$%\"xG!\"\"\"\"\"%$cos
GF2%\"kGF2F0F2F0" }{TEXT -1 9 "   ...   " }{XPPEDIT 18 0 "PIECEWISE([u
 = exp(-x),v = sin*k*x/k],[du/dx=-exp(-x),dv/dx=cos*k*x])" "6#-%*PIECE
WISEG6$7$/%\"uG-%$expG6#,$%\"xG!\"\"/%\"vG**%$sinG\"\"\"%\"kGF3F-F3F4F
.7$/*&%#duGF3%#dxGF.,$-F*6#,$F-F.F./*&%#dvGF3F9F.*(%$cosGF3F4F3F-F3" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "`` = exp(-x)*sin*k*x/k-Int(``(sin*k*x/
k)*``(-exp(-x)),x);" "6#/%!G,&*,-%$expG6#,$%\"xG!\"\"\"\"\"%$sinGF-%\"
kGF-F+F-F/F,F--%$IntG6$*&-F$6#**F.F-F/F-F+F-F/F,F--F$6#,$-F(6#,$F+F,F,
F-F+F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}
{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "I[c] = exp(-x)*sin*
k*x/k+I[s]/k;" "6#/&%\"IG6#%\"cG,&*,-%$expG6#,$%\"xG!\"\"\"\"\"%$sinGF
0%\"kGF0F.F0F2F/F0*&&F%6#%\"sGF0F2F/F0" }{TEXT -1 16 "   ------- (i). \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Simil
arly,  " }{XPPEDIT 18 0 "I[s]" "6#&%\"IG6#%\"sG" }{TEXT -1 2 " =" }
{XPPEDIT 18 0 "Int(exp(-x)*sin*k*x,x) = Int(u*``(dv/dx),x);" "6#/-%$In
tG6$**-%$expG6#,$%\"xG!\"\"\"\"\"%$sinGF.%\"kGF.F,F.F,-F%6$*&%\"uGF.-%
!G6#*&%#dvGF.%#dxGF-F.F," }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = e
xp(-x)" "6#/%\"uG-%$expG6#,$%\"xG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "v = -cos*k*x/k;" "6#/%\"vG,$**%$cosG\"\"\"%\"kGF(%\"xGF(F)!\"\"F
+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "The integration by p
arts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/d
x),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&
F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 9 ", gives
: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[s] = Int(exp(
-x)*sin*k*x,x);" "6#/&%\"IG6#%\"sG-%$IntG6$**-%$expG6#,$%\"xG!\"\"\"\"
\"%$sinGF2%\"kGF2F0F2F0" }{TEXT -1 9 "   ...   " }{XPPEDIT 18 0 "PIECE
WISE([u = exp(-x), v = -cos*k*x/k],[du/dx = -exp(-x), dv/dx = sin*k*x]
);" "6#-%*PIECEWISEG6$7$/%\"uG-%$expG6#,$%\"xG!\"\"/%\"vG,$**%$cosG\"
\"\"%\"kGF4F-F4F5F.F.7$/*&%#duGF4%#dxGF.,$-F*6#,$F-F.F./*&%#dvGF4F:F.*
(%$sinGF4F5F4F-F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "I[s] = -exp(-x)*cos*
k*x/k-Int(``(-cos*k*x/k)*``(-exp(-x)),x);" "6#/&%\"IG6#%\"sG,&*,-%$exp
G6#,$%\"xG!\"\"\"\"\"%$cosGF0%\"kGF0F.F0F2F/F/-%$IntG6$*&-%!G6#,$**F1F
0F2F0F.F0F2F/F/F0-F86#,$-F+6#,$F.F/F/F0F.F/" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 3 "   " 
}{XPPEDIT 18 0 "I[s] = -exp(-x)*cos*k*x/k-I[c]/k;" "6#/&%\"IG6#%\"sG,&
*,-%$expG6#,$%\"xG!\"\"\"\"\"%$cosGF0%\"kGF0F.F0F2F/F/*&&F%6#%\"cGF0F2
F/F/" }{TEXT -1 15 "  ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 27 "Fr
om (i) and (ii) we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "k*I[c] = exp(-x)*sin*k*x+I[s];" "6#/*&%\"kG\"\"\"&%\"IG
6#%\"cGF&,&**-%$expG6#,$%\"xG!\"\"F&%$sinGF&F%F&F1F&F&&F(6#%\"sGF&" }
{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k*I
[s] = -exp(-x)*cos*k*x-I[c];" "6#/*&%\"kG\"\"\"&%\"IG6#%\"sGF&,&**-%$e
xpG6#,$%\"xG!\"\"F&%$cosGF&F%F&F1F&F2&F(6#%\"cGF2" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "k*I[c]-I[s] = exp(-x)*sin*k*x;" "6#/,&*&%\"kG\"\"\"&%\"
IG6#%\"cGF'F'&F)6#%\"sG!\"\"**-%$expG6#,$%\"xGF/F'%$sinGF'F&F'F5F'" }
{TEXT -1 17 "  ------- (iii). " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "I[c
]+k*I[s] = -exp(-x)*cos*k*x;" "6#/,&&%\"IG6#%\"cG\"\"\"*&%\"kGF)&F&6#%
\"sGF)F),$**-%$expG6#,$%\"xG!\"\"F)%$cosGF)F+F)F5F)F6" }{TEXT -1 15 " \+
 ------- (iv)." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 30 "Multiplying equation (iii) by " }{TEXT 271 1 "k" }{TEXT 
-1 31 " and adding equation (iv) gives" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "(k^2+1)*I[c] = exp(-x)*(k*sin*k*x-cos*k*x);" "6#
/*&,&*$%\"kG\"\"#\"\"\"F)F)F)&%\"IG6#%\"cGF)*&-%$expG6#,$%\"xG!\"\"F),
&**F'F)%$sinGF)F'F)F3F)F)*(%$cosGF)F'F)F3F)F4F)" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "I[c] = exp(-x)*(k*sin*k*x-cos*k*x)/(k^2+1);" "6#/&%\"IG
6#%\"cG*(-%$expG6#,$%\"xG!\"\"\"\"\",&**%\"kGF/%$sinGF/F2F/F-F/F/*(%$c
osGF/F2F/F-F/F.F/,&*$F2\"\"#F/F/F/F." }{TEXT -1 15 "  ------- (v). " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Multiply
ing equation (iv) by " }{TEXT 272 1 "k" }{TEXT -1 37 " and subtracting
 equation (iii) gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(k^2+1)*I[s] = -exp(-x)*(k*cos*k*x
+sin*k*x);" "6#/*&,&*$%\"kG\"\"#\"\"\"F)F)F)&%\"IG6#%\"sGF),$*&-%$expG
6#,$%\"xG!\"\"F),&**F'F)%$cosGF)F'F)F4F)F)*(%$sinGF)F'F)F4F)F)F)F5" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "I[s] = -exp(-x)*(k*cos*k*x+sin*k*x)/(k
^2+1);" "6#/&%\"IG6#%\"sG,$*(-%$expG6#,$%\"xG!\"\"\"\"\",&**%\"kGF0%$c
osGF0F3F0F.F0F0*(%$sinGF0F3F0F.F0F0F0,&*$F3\"\"#F0F0F0F/F/" }{TEXT -1 
17 "  ------- (vi).  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 16 "From (v) we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "a[k] = 1/Pi;" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }
{XPPEDIT 18 0 "``(exp(-x)*(k*sin*k*x-cos*k*x)/(k^2+1));" "6#-%!G6#*(-%
$expG6#,$%\"xG!\"\"\"\"\",&**%\"kGF-%$sinGF-F0F-F+F-F-*(%$cosGF-F0F-F+
F-F,F-,&*$F0\"\"#F-F-F-F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([
2*Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$*&\"\"#\"\"\"%#PiGF)
%!G7$F+F+7$\"\"!F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-exp(-2*Pi))/(
Pi*(k^2+1));" "6#/%!G*&,&\"\"\"F'-%$expG6#,$*&\"\"#F'%#PiGF'!\"\"F/F'*
&F.F',&*$%\"kGF-F'F'F'F'F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "From (vi) we have" }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k] = 1/Pi;" "6#/&%\"bG6#%\"kG
*&\"\"\"F)%#PiG!\"\"" }{XPPEDIT 18 0 "``(-exp(-x)*(k*cos*k*x+sin*k*x)/
(k^2+1));" "6#-%!G6#,$*(-%$expG6#,$%\"xG!\"\"\"\"\",&**%\"kGF.%$cosGF.
F1F.F,F.F.*(%$sinGF.F1F.F,F.F.F.,&*$F1\"\"#F.F.F.F-F-" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "PIECEWISE([2*Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECE
WISEG6%7$*&\"\"#\"\"\"%#PiGF)%!G7$F+F+7$\"\"!F+" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = k*(1-exp(-2*Pi))/(Pi*(k^2+1));" "6#/%!G*(%\"kG\"\"
\",&F'F'-%$expG6#,$*&\"\"#F'%#PiGF'!\"\"F0F'*&F/F',&*$F&F.F'F'F'F'F0" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 71 "Note that the Fourier coefficients can be found using Map
le as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 70 "c=1/(2*Pi)*Int(exp(-x),x=0..2*Pi);\n``=simplify(
expand(value(rhs(%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*&#
\"\"\"\"\"#F(,$-%$IntG6$-%$expG6#,$%\"xG!\"\"/F2;\"\"!,$*&F)F(%#PiGF(F
(*$F9F3F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"\"#F(*
&,&F(!\"\"-%$expG6#,$*&F)F(%#PiGF(F,F(F(F2F,F(F," }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "interface(s
howassumed=0): k := 'k': assume(k,integer):\na[k]=1/Pi*Int(exp(-x)*cos
(k*x),x = 0 .. 2*Pi);\n``=simplify(expand(value(rhs(%))));\nb[k]=1/Pi*
Int(exp(-x)*sin(k*x),x = 0 .. 2*Pi);\n``=simplify(expand(value(rhs(%))
));\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%#k|irG,$-%
$IntG6$*&-%$expG6#,$%\"xG!\"\"\"\"\"-%$cosG6#*&F'F3F1F3F3/F1;\"\"!,$*&
\"\"#F3%#PiGF3F3*$F>F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,&\"
\"\"!\"\"-%$expG6#,$*&\"\"#F(%#PiGF(F)F(F(F0F),&F(F(*$)%#k|irGF/F(F(F)
F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$-%$IntG6$*&-%$
expG6#,$%\"xG!\"\"\"\"\"-%$sinG6#*&F'F3F1F3F3/F1;\"\"!,$*&\"\"#F3%#PiG
F3F3*$F>F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**%#k|irG\"\"\",&F
(!\"\"-%$expG6#,$*&\"\"#F(%#PiGF(F*F(F(F1F*,&F(F(*$)F'F0F(F(F*F*" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
22 "The Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "F
(x) = (1-exp(-2*Pi))/Pi;" "6#/-%\"FG6#%\"xG*&,&\"\"\"F*-%$expG6#,$*&\"
\"#F*%#PiGF*!\"\"F2F*F1F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/2+Sum(
(k*sin*k*x+cos*k*x)/(k^2+1),k = 1 .. infinity));" "6#-%!G6#,&*&\"\"\"F
(\"\"#!\"\"F(-%$SumG6$*&,&**%\"kGF(%$sinGF(F1F(%\"xGF(F(*(%$cosGF(F1F(
F3F(F(F(,&*$F1F)F(F(F(F*/F1;F(%)infinityGF(" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "(1-exp(-2*Pi))/Pi " "6#*&
,&\"\"\"F%-%$expG6#,$*&\"\"#F%%#PiGF%!\"\"F-F%F,F-" }{XPPEDIT 18 0 "``
(1/2+sin*x/2+cos*x/2+2/5*sin*2*x+cos*2*x/5+3/10*sin*3*x+cos*3*x/10+4/1
7*sin*4*x+cos*4*x/17+` . . . . `);" "6#-%!G6#,6*&\"\"\"F(\"\"#!\"\"F(*
(%$sinGF(%\"xGF(F)F*F(*(%$cosGF(F-F(F)F*F(*,F)F(\"\"&F*F,F(F)F(F-F(F(*
*F/F(F)F(F-F(F1F*F(*,\"\"$F(\"#5F*F,F(F4F(F-F(F(**F/F(F4F(F-F(F5F*F(*,
\"\"%F(\"#<F*F,F(F8F(F-F(F(**F/F(F8F(F-F(F9F*F(%*~.~.~.~.~GF(" }{TEXT 
-1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
106 "The following picture compares the graphs of some truncated Fouri
er series with the graph of the function " }{XPPEDIT 18 0 "f(x)" "6#-%
\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 268 "f := x -> exp(-x):\nf_ := x -> f(x
-2*Pi*floor(x/(2*Pi))):\nFS := (x,n) ->\n (1-exp(-2*Pi))/Pi*(1/2+sum((
k*sin(k*x)+cos(k*x))/(k^2+1),k=1..n));\nplot([f_(x),FS(x,1),FS(x,2),FS
(x,3),FS(x,4),FS(x,5)],x=-4..10,\n   color=[black,red,blue,green,magen
ta,coral],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FS
Gf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF)*(,&\"\"\"F/-%$expG6#,$*&\"\"
#F/%#PiGF/!\"\"F7F/F6F7,&#F/F5F/-%$sumG6$*&,&*&%\"kGF/-%$sinG6#*&F@F/9
$F/F/F/-%$cosGFCF/F/,&*$)F@F5F/F/F/F/F7/F@;F/9%F/F/F)F)F)" }}{PARA 13 
"" 1 "" {GLPLOT2D 556 294 294 {PLOTDATA 2 "6*-%'CURVESG6%7`p7$$!\"%\"
\"!$\"3Kv4W%=*e>5!#=7$$!3cmmmT)R[p$!#<$\"3!o\"='ogEW^(!#>7$$!3PLLe>;KH
MF1$\"3SHBmof7idF47$$!3ymm;4'=28$F1$\"3QHEF(QbYF%F47$$!3vmm;ki8IGF1$\"
3'GEm\")**)*[;$F47$$!3+LLeMD)4`#F1$\"3yeFXi)*fYBF47$$!3)om;HtGOD#F1$\"
3!)pHnH8Ay<F47$$!3%)***\\i$\\Wm>F1$\"3.fXJ'pIVL\"F47$$!3Mmm\"H/R%p;F1$
\"3%*=I#Q\"fg9**!#?7$$!3+++D^cQt8F1$\"3q9ol8b'RP(FX7$$!3%HLL$[$e)o5F1$
\"3nB3*\\I#4QaFX7$$!33kmmT`I1!)F-$\"3%>mc;g\"peTFX7$$!3#Q*****\\ap')\\
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{PARA 0 "" 0 "" {TEXT -1 91 "Even with a fairly large number of terms \+
there is a poor match near the discontinuities of " }{XPPEDIT 18 0 "f(
x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "f_ := x -> exp(2*Pi*floor
(x/(2*Pi))-x):\nFS := (x,n) ->\n (1-exp(-2*Pi))/Pi*(1/2+sum((k*sin(k*x
)+cos(k*x))/(k^2+1),k=1..n)): \nplot([f_(x),FS(x,50)],x=-2..8,numpoint
s=80,\n        color=[black,COLOR(RGB,.8,0,1)],linestyle=[3,1]);" }}
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\"3G>8pP%f;-\"F17$$\"3xm75B[rTjF1$\"3#Q19y#orJ5F17$$\"3%***=lfi#*\\jF1
$\"398yND#pD-\"F17$Fegl$\"3mL0*43%G*)**FV7$$\"3)**3n=wjKP'F1$\"3m?\">M
K\"*)e$*FV7$$\"3'okJv#)*Q)Q'F1$\"3q644CA4^()FV7$$\"3t.i>$*e^.kF1$\"3u-
YMhuc(Q)FV7$Fjgl$\"3j^!38z,HK)FV7$$\"3[<`_C!oPV'F1$\"3A)ej\">%\\*f%)FV
7$$\"3Ou)*=!4%*)[kF1$\"3$z)*pz,y`i)FV7$$\"3!G:AI7dkX'F1$\"3/f[PZ*Hvm)F
V7$$\"3CJW&e:?SY'F1$\"3m_MM/&H\\m)FV7$$\"3o4no)=$erkF1$\"3Q&G(3^\"yGh)
FV7$F_hl$\"3Jshaj.!Q^)FV7$$\"3e3')3U&G,^'F1$\"31bZ30pUiyFV7$Fdhl$\"3w!
z#*\\!z=.vFV7$$\"3=M1!GWc)[lF1$\"3:p3@\\T/-vFV7$$\"3GRI%H--mb'F1$\"34S
-,@)eD_(FV7$$\"3SWa3.wMklF1$\"3Y))ylCz%=b(FV7$$\"3_\\yA$=$4slF1$\"3#H#
)*3$)\\_wvFV7$$\"3ka-Pj(Q)zlF1$\"3quL#GgbYe(FV7$$\"3wfE^VVe(e'F1$\"3eh
]$*fKSnvFV7$$\"3'[1bO#*H`f'F1$\"3\")GSSp%[,_(FV7$Fihl$\"3YO&)\\Dz%HW(F
V7$F^il$\"3$)3I^o]X-kFV7$Fcil$\"3mu?&)43t^cFV7$Fhil$\"3I?p\\tBKm\\FV7$
F]jl$\"39:r92u6[VFV7$Fbjl$\"3)p=1*3@i\")QFV7$Fgjl$\"3\"3[Ug8IjT$FV7$F
\\[m$\"3Ow3f51@7IFV7$Fa[m$\"3p*)Reza#3h#FV7$Ff[m$\"3))3FtE;#QL#FV7$F[
\\m$\"3>J;M5@Y5?FV7$F`\\m$\"3Gf-cXnV$y\"FV-%&COLORG6&Fg\\m$Fa\\m!\"\"$
F*F*$\"\"\"F*-Fi\\m6#F``p-%+AXESLABELSG6$Q\"x6\"Q!Fg`p-%%VIEWG6$;F(F`
\\m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We che
ck the convergence numerically when " }{XPPEDIT 18 0 "x = Pi" "6#/%\"x
G%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }
{XPPEDIT 18 0 "x = Pi" "6#/%\"xG%#PiG" }{TEXT -1 29 " the Fourier seri
es becomes: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "(1-e
xp(-2*Pi))/Pi;" "6#*&,&\"\"\"F%-%$expG6#,$*&\"\"#F%%#PiGF%!\"\"F-F%F,F
-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/2+Sum((k*sin*k*Pi+cos*k*Pi)/(k
^2+1),k = 1 .. infinity));" "6#-%!G6#,&*&\"\"\"F(\"\"#!\"\"F(-%$SumG6$
*&,&**%\"kGF(%$sinGF(F1F(%#PiGF(F(*(%$cosGF(F1F(F3F(F(F(,&*$F1F)F(F(F(
F*/F1;F(%)infinityGF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 
"  " }{XPPEDIT 18 0 "`` = (1-exp(-2*Pi))/Pi;" "6#/%!G*&,&\"\"\"F'-%$ex
pG6#,$*&\"\"#F'%#PiGF'!\"\"F/F'F.F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`
`(1/2+Sum((-1)^k/(k^2+1),k = 1 .. infinity));" "6#-%!G6#,&*&\"\"\"F(\"
\"#!\"\"F(-%$SumG6$*&),$F(F*%\"kGF(,&*$F1F)F(F(F(F*/F1;F(%)infinityGF(
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 24 "The alternating series  " }{XPPEDIT 18 0 "Sum((-1)^k/(1+k
^2),k = 1 .. infinity" "6#-%$SumG6$*&),$\"\"\"!\"\"%\"kGF),&F)F)*$F+\"
\"#F)F*/F+;F)%)infinityG" }{TEXT -1 51 "  can be summed numerically in
 an efficient manner." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 44 "Sum((-1)^k/(1+k^2),k=1..infinity);\nevalf
(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\"%\"kG\"\"\",&
*$)F)\"\"#F*F*F*F*F(/F);F*%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$!+DZ&)RO!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "The following " }{TEXT 0 3 "add" }{TEXT -1 69 " command t
akes a while to execute because it uses eaxact arithmetic. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ad
d((-1)^k/(1+k^2),k=1..50000):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$!+BZ&)RO!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 99 "It requires around 100000 terms to obtain a value which
 agrees to 10 digits with the infinite sum. " }}{PARA 0 "" 0 "" {TEXT 
-1 137 "This calculation can be carried out by the following \"for-loo
p\" which performs the addition \"backwards\" starting with the smalle
st term. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 167 "Digits := 10:\npartial_sum := 0.:\nfor k from 100000
 to 1 by -1 do\n   term := evalf((-1)^k/(1+k^2));\n   partial_sum := p
artial_sum + term;\nend do:\npartial_sum;\nk := 'k':" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$!+DZ&)RO!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 34 "The sum of the Fourier series for " }
{XPPEDIT 18 0 "x = Pi" "6#/%\"xG%#PiG" }{TEXT -1 37 " correct to about
 10 digits is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 79 "(1-exp(-2*Pi))*(1/2+Sum((-1)^k/(1+k^2),k=
1..infinity))/Pi;\nevalf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*(,&\"\"\"F%-%$expG6#,$*&\"\"#F%%#PiGF%!\"\"F-F%,&#F%F+F%-%$SumG
6$*&)F-%\"kGF%,&*$)F5F+F%F%F%F%F-/F5;F%%)infinityGF%F%F,F-" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+E=R@V!#6" }}}{PARA 0 "" 0 "" {TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 50 ". . . which agrees with the value o
f the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 "
 at " }{XPPEDIT 18 0 "x=Pi" "6#/%\"xG%#PiG" }{TEXT -1 11 ", which is \+
" }{XPPEDIT 18 0 "exp(-Pi)" "6#-%$expG6#,$%#PiG!\"\"" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "evalf(evalf[14](exp(-Pi)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+E=R@V!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "In fact this an
alysis suggests that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Pi*exp(-Pi)/(1-exp(-2*Pi))=1/2+``" "6#/*(%#PiG\"\"\"-%$expG6#,$F
%!\"\"F&,&F&F&-F(6#,$*&\"\"#F&F%F&F+F+F+,&*&F&F&F1F+F&%!GF&" }
{XPPEDIT 18 0 "Sum((-1)^k/(k^2+1),k = 1 .. infinity)" "6#-%$SumG6$*&),
$\"\"\"!\"\"%\"kGF),&*$F+\"\"#F)F)F)F*/F+;F)%)infinityG" }{TEXT -1 2 "
, " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^k/(1+k^2),k = 1 .. infinity) =
 (Pi*csch(Pi)-1)/2;" "6#/-%$SumG6$*&),$\"\"\"!\"\"%\"kGF*,&F*F**$F,\"
\"#F*F+/F,;F*%)infinityG*&,&*&%#PiGF*-%%cschG6#F6F*F*F*F+F*F/F+" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "evalf((Pi*csch(Pi)-1)/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$!+DZ&)RO!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 61 "Truncated Fourier series can be obtained using \+
the procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "f
 := x -> exp(-x):\n'f(x)'=f(x);\nFourierSeries(f(x),x=0..2*Pi,numterms
=8,info=1):\nF := unapply(%,x):\n'F(x)'=F(x);\nplot(F(x),x=-2..8,color
=red,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%
$expG6#,$F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coeffic
ient~-->G,$*&#\"\"\"\"\"#F'*(,&-%$expG6#,$*&F(F'%#PiGF'F'F'F'!\"\"F'-F
,6#,$*&F(F'F0F'F1F'F0F1F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%<k~th~
cosine~coefficient~..~G/,$**,*-%$expG6#,$*&\"\"#\"\"\"%#PiGF.F.!\"\"*&
F-F.)-%$cosG6#*&F/F.%\"kGF.F-F.F.F.F0**F-F.F7F.-%$sinGF5F.F3F.F0F.,&F.
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{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 "f(x) = exp(sin*x)
;" "6#/-%\"fG6#%\"xG-%$expG6#*&%$sinG\"\"\"F'F-" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
101 "f := x -> exp(sin(x)):\n'f(x)'=f(x);\nplot(f(x),x=-7..13,y=0..2.8
,color=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 
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7$$\"3eLLL`v&Q<\"F^hm$\"3K#zcSMZ\")y%F-7$$\"3#om\"zW?)\\>\"F^hm$\"33[+
b,l$*3cF-7$$\"30++DOl5;7F^hm$\"3oSPeCD_TnF-7$$\"39+++q`KO7F^hm$\"3Ig!o
dw6K<)F-7$$\"3A++v.Uac7F^hm$\"3)>G#oU&=2***F-7$$\"33+D\"G:3uE\"F^hm$\"
3JCoUQR\\86F17$$\"37+](=5s#y7F^hm$\"3_lIqH&[%R7F17$$\"3;+v$40O\"*G\"F^
hm$\"3]Oh$[-jhP\"F17$$\"#8F*$\"30!p![+e@A:F1-%*THICKNESSG6#\"\"#-%+AXE
SLABELSG6$Q\"x6\"Q\"yFg_n-%&COLORG6&%$RGBG$\"\"%!\"\"$F*F*$\"\"\"F*-%%
VIEWG6$;F(F[_n;F``n$\"#GF_`n" 1 2 0 1 10 2 2 6 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 37 "The Fourier series for  the function " }{XPPEDIT 18 0 "f(
x) = exp(sin*x);" "6#/-%\"fG6#%\"xG-%$expG6#*&%$sinG\"\"\"F'F-" }
{TEXT -1 5 " is  " }{XPPEDIT 18 0 "c+``  " "6#,&%\"cG\"\"\"%!GF%" }
{XPPEDIT 18 0 "Sum(a[k]*cos*k*x+b[k]*sin*k*x,k = 1 .. infinity);" "6#-
%$SumG6$,&**&%\"aG6#%\"kG\"\"\"%$cosGF,F+F,%\"xGF,F,**&%\"bG6#F+F,%$si
nGF,F+F,F.F,F,/F+;F,%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 3 " = " }{XPPEDIT 18 0 "c+a[1]*cos*x+b[1]*sin*x+a[2]*cos*2*x+
b[2]*sin*2*x+a[3]*cos*3*x+b[3]*sin*3*x+` . . . `;" "6#,2%\"cG\"\"\"*(&
%\"aG6#F%F%%$cosGF%%\"xGF%F%*(&%\"bG6#F%F%%$sinGF%F+F%F%**&F(6#\"\"#F%
F*F%F4F%F+F%F%**&F.6#F4F%F0F%F4F%F+F%F%**&F(6#\"\"$F%F*F%F;F%F+F%F%**&
F.6#F;F%F0F%F;F%F+F%F%%(~.~.~.~GF%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 10 "  where   " }
{XPPEDIT 18 0 "c = 1/(2*Pi);" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"
" }{XPPEDIT 18 0 "Int(exp(sin*x),x = -Pi .. Pi);" "6#-%$IntG6$-%$expG6
#*&%$sinG\"\"\"%\"xGF+/F,;,$%#PiG!\"\"F0" }{TEXT -1 2 ", " }}{PARA 
256 "" 0 "" {XPPEDIT 18 0 "a[k] = 1/Pi;" "6#/&%\"aG6#%\"kG*&\"\"\"F)%#
PiG!\"\"" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(exp(sin*x)*cos*k*x,x = \+
-Pi .. Pi);" "6#-%$IntG6$**-%$expG6#*&%$sinG\"\"\"%\"xGF,F,%$cosGF,%\"
kGF,F-F,/F-;,$%#PiG!\"\"F3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT 
-1 6 "and   " }{XPPEDIT 18 0 "b[k] = 1/Pi;" "6#/&%\"bG6#%\"kG*&\"\"\"F
)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(sin*x)*sin*k*x,x \+
= -Pi .. Pi);" "6#-%$IntG6$**-%$expG6#*&%$sinG\"\"\"%\"xGF,F,F+F,%\"kG
F,F-F,/F-;,$%#PiG!\"\"F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "Some of the integrals which occur
 here are 0, which leads to the following results for the Fourier coef
ficients:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "PIECEWISE([a[k] = 0, `when k is odd`],[b[k] =
 0, `when k is even`]);" "6#-%*PIECEWISEG6$7$/&%\"aG6#%\"kG\"\"!%.when
~k~is~oddG7$/&%\"bG6#F+F,%/when~k~is~evenG" }{TEXT -1 1 " " }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 40 "Details concerning the zero coefficients
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 46 "One integral which can be found immediately is" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(sin*x)*cos(x),x) = \+
exp(sin*x)+c;" "6#/-%$IntG6$*&-%$expG6#*&%$sinG\"\"\"%\"xGF-F--%$cosG6
#F.F-F.,&-F)6#*&F,F-F.F-F-%\"cGF-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a
[1]=1/Pi" "6#/&%\"aG6#\"\"\"*&F'F'%#PiG!\"\"" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "exp(sin(x));" "6#-%$expG6#-%$sinG6#%\"xG" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[-Pi, ``]);" "6#-%*PIEC
EWISEG6%7$%#PiG%!G7$F(F(7$,$F'!\"\"F(" }{TEXT -1 2 ". " }}{PARA 256 "
" 0 "" {TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 3 "Now" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(sin*x)*sin*2*x,x) =
 Int(2*exp(sin*x)*cos*x*sin*x,x);" "6#/-%$IntG6$**-%$expG6#*&%$sinG\"
\"\"%\"xGF-F-F,F-\"\"#F-F.F-F.-F%6$*.F/F--F)6#*&F,F-F.F-F-%$cosGF-F.F-
F,F-F.F-F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }
{XPPEDIT 18 0 "Int(u*``(dv/dx),x);" "6#-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%
#dvGF(%#dxG!\"\"F(%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = si
n*x;" "6#/%\"uG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "v = 2*exp(sin*x);" "6#/%\"vG*&\"\"#\"\"\"-%$expG6#*&%$sinGF'%\"x
GF'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "Using the integr
ation by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int
(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F
)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 
9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2
*exp(sin*x)*cos*x*sin*x,x) = 2*exp(sin*x)*sin*x-Int(2*exp(sin*x)*cos*x
,x);" "6#/-%$IntG6$*.\"\"#\"\"\"-%$expG6#*&%$sinGF)%\"xGF)F)%$cosGF)F/
F)F.F)F/F)F/,&**F(F)-F+6#*&F.F)F/F)F)F.F)F/F)F)-F%6$**F(F)-F+6#*&F.F)F
/F)F)F0F)F/F)F/!\"\"" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 4 
"  = " }{XPPEDIT 18 0 "2*exp(sin*x)*sin*x-2*exp(sin*x)+c;" "6#,(**\"\"
#\"\"\"-%$expG6#*&%$sinGF&%\"xGF&F&F+F&F,F&F&*&F%F&-F(6#*&F+F&F,F&F&!
\"\"%\"cGF&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "b[2]=1/Pi" "6#/&%\"bG6#\"\"#*&\"\"\"F)%#PiG!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "2*exp(sin*x)*sin*x-2*exp(sin*x);" "6#,&
**\"\"#\"\"\"-%$expG6#*&%$sinGF&%\"xGF&F&F+F&F,F&F&*&F%F&-F(6#*&F+F&F,
F&F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``]
,[-Pi, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$,$F'!\"\"F(" }{TEXT 
-1 3 "   " }}{PARA 256 "" 0 "" {TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Now consider  " }
{XPPEDIT 18 0 "Int(exp(sin*x)*cos(3*x),x);" "6#-%$IntG6$*&-%$expG6#*&%
$sinG\"\"\"%\"xGF,F,-%$cosG6#*&\"\"$F,F-F,F,F-" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "2*cos*2*x*cos
*x = cos*3*x+cos*x;" "6#/*.\"\"#\"\"\"%$cosGF&F%F&%\"xGF&F'F&F(F&,&*(F
'F&\"\"$F&F(F&F&*&F'F&F(F&F&" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "cos*
3*x = 2*cos*2*x*cos*x-cos*x;" "6#/*(%$cosG\"\"\"\"\"$F&%\"xGF&,&*.\"\"
#F&F%F&F+F&F(F&F%F&F(F&F&*&F%F&F(F&!\"\"" }{TEXT -1 4 " and" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(sin*x)*cos*3*x,x) =
 2*Int(exp(sin*x)*cos*2*x*cos*x,x)-Int(exp(sin*x)*cos*x,x);" "6#/-%$In
tG6$**-%$expG6#*&%$sinG\"\"\"%\"xGF-F-%$cosGF-\"\"$F-F.F-F.,&*&\"\"#F-
-F%6$*.-F)6#*&F,F-F.F-F-F/F-F3F-F.F-F/F-F.F-F.F-F--F%6$*(-F)6#*&F,F-F.
F-F-F/F-F.F-F.!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 58 "Applying the integration by parts formul
a to the integral " }{XPPEDIT 18 0 "Int(exp(sin*x)*cos*2*x*cos*x,x);" 
"6#-%$IntG6$*.-%$expG6#*&%$sinG\"\"\"%\"xGF,F,%$cosGF,\"\"#F,F-F,F.F,F
-F,F-" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "u = cos*2*x;" "6#/%\"uG*(%
$cosG\"\"\"\"\"#F'%\"xGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = exp
(sin*x);" "6#/%\"vG-%$expG6#*&%$sinG\"\"\"%\"xGF*" }{TEXT -1 13 ", we \+
see that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(s
in*x)*cos*2*x*cos*x,x) = exp(sin(x))*cos*2*x+Int(exp(sin*x)*2*sin*2*x,
x);" "6#/-%$IntG6$*.-%$expG6#*&%$sinG\"\"\"%\"xGF-F-%$cosGF-\"\"#F-F.F
-F/F-F.F-F.,&**-F)6#-F,6#F.F-F/F-F0F-F.F-F--F%6$*,-F)6#*&F,F-F.F-F-F0F
-F,F-F0F-F.F-F.F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "It \+
follows that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(e
xp(sin*x)*cos*3*x,x) = 2*exp(sin*x)*cos*2*x+4*Int(exp(sin*x)*sin*2*x,x
)-Int(exp(sin*x)*cos*x,x);" "6#/-%$IntG6$**-%$expG6#*&%$sinG\"\"\"%\"x
GF-F-%$cosGF-\"\"$F-F.F-F.,(*,\"\"#F--F)6#*&F,F-F.F-F-F/F-F3F-F.F-F-*&
\"\"%F--F%6$**-F)6#*&F,F-F.F-F-F,F-F3F-F.F-F.F-F--F%6$*(-F)6#*&F,F-F.F
-F-F/F-F.F-F.!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = 2*exp(sin*x)*cos*2*x+8*exp(sin*x)*sin*x-8*exp(s
in*x)-exp(sin*x)+c;" "6#/%!G,,*,\"\"#\"\"\"-%$expG6#*&%$sinGF(%\"xGF(F
(%$cosGF(F'F(F.F(F(**\"\")F(-F*6#*&F-F(F.F(F(F-F(F.F(F(*&F1F(-F*6#*&F-
F(F.F(F(!\"\"-F*6#*&F-F(F.F(F9%\"cGF(" }{TEXT -1 3 "   " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*exp(sin*x)*cos*2*x+8*exp(
sin*x)*sin*x-9*exp(sin*x)+c;" "6#/%!G,**,\"\"#\"\"\"-%$expG6#*&%$sinGF
(%\"xGF(F(%$cosGF(F'F(F.F(F(**\"\")F(-F*6#*&F-F(F.F(F(F-F(F.F(F(*&\"\"
*F(-F*6#*&F-F(F.F(F(!\"\"%\"cGF(" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Since this function is pe
riodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }
{TEXT -1 13 ", we see that" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "a[3]=1/Pi" "6#/&%\"aG6#\"\"$*&\"\"\"F)%#PiG!\"\"" }
{XPPEDIT 18 0 "``(2*exp(sin*x)*cos*2*x+8*exp(sin*x)*sin*x-9*exp(sin*x)
);" "6#-%!G6#,(*,\"\"#\"\"\"-%$expG6#*&%$sinGF)%\"xGF)F)%$cosGF)F(F)F/
F)F)**\"\")F)-F+6#*&F.F)F/F)F)F.F)F/F)F)*&\"\"*F)-F+6#*&F.F)F/F)F)!\"
\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[-Pi, \+
``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$,$F'!\"\"F(" }{TEXT -1 2 " \+
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0" "6#/%!G
\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 " A pattern seem
s to be emerging whereby the integral " }}{PARA 256 "" 0 "" {TEXT -1 
2 "  " }{XPPEDIT 18 0 "Int(exp(sin*x)*cos*k*x,x);" "6#-%$IntG6$**-%$ex
pG6#*&%$sinG\"\"\"%\"xGF,F,%$cosGF,%\"kGF,F-F,F-" }{TEXT -1 8 ",  with
 " }{TEXT 273 1 "k" }{TEXT -1 5 " odd," }}{PARA 0 "" 0 "" {TEXT -1 16 
"and the integral" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 
"Int(exp(sin*x)*sin*k*x,x);" "6#-%$IntG6$**-%$expG6#*&%$sinG\"\"\"%\"x
GF,F,F+F,%\"kGF,F-F,F-" }{TEXT -1 8 ",  with " }{TEXT 274 1 "k" }
{TEXT -1 6 " even," }}{PARA 0 "" 0 "" {TEXT -1 83 "can be expressed an
alytically in terms of a function which is periodic with period " }
{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 28 "This leads to the fact that " }}{PARA 256 "" 0 
"" {TEXT -1 6 "      " }{XPPEDIT 18 0 "PIECEWISE([Int(exp(sin*x)*cos*k
*x,x = -Pi .. Pi) = 0, `when k is odd`],[Int(exp(sin(x))*sin*k*x,x = -
Pi .. Pi) = 0, `when k is even`]);" "6#-%*PIECEWISEG6$7$/-%$IntG6$**-%
$expG6#*&%$sinG\"\"\"%\"xGF1F1%$cosGF1%\"kGF1F2F1/F2;,$%#PiG!\"\"F8\"
\"!%.when~k~is~oddG7$/-F)6$**-F-6#-F06#F2F1F0F1F4F1F2F1/F2;,$F8F9F8F:%
/when~k~is~evenG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Rega
rding the Fourier coefficients, we have " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "PIECEWISE([a[k] = 0, `when k is odd`],[b[k] =
 0, `when k is even`]);" "6#-%*PIECEWISEG6$7$/&%\"aG6#%\"kG\"\"!%.when
~k~is~oddG7$/&%\"bG6#F+F,%/when~k~is~evenG" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT 277 4 "Note" }{TEXT -1 103 ": These results can be pro
ved by general methods involving certain symmetry properties of the fu
nction " }{XPPEDIT 18 0 "f(x)=exp(sin*x)" "6#/-%\"fG6#%\"xG-%$expG6#*&
%$sinG\"\"\"F'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 61 "We procede a bit further with these compu
tations using Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 300 "Int(exp(sin(x))*cos(x),x)=int(exp(sin(x)
)*cos(x),x)+'c';\nInt(exp(sin(x))*sin(2*x),x)=int(exp(sin(x))*sin(2*x)
,x)+'c';\nInt(exp(sin(x))*cos(3*x),x)=int(exp(sin(x))*cos(3*x),x)+'c';
\nInt(exp(sin(x))*sin(4*x),x)=int(exp(sin(x))*sin(4*x),x)+'c';\nInt(ex
p(sin(x))*cos(5*x),x)=int(exp(sin(x))*cos(5*x),x)+'c';" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"\"\"-%$cosGF-
F/F.,&F(F/%\"cGF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$ex
pG6#-%$sinG6#%\"xG\"\"\"-F,6#,$F.\"\"#F/F.,(*&F(F/F+F/F3F(!\"#%\"cGF/
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#-%$sinG6#%\"x
G\"\"\"-%$cosG6#,$F.\"\"$F/F.,**&F(F/F+F/\"\")*&F(F/-F16#,$F.\"\"#F/F<
F(!\"*%\"cGF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#
-%$sinG6#%\"xG\"\"\"-F,6#,$F.\"\"%F/F.,,*&F(F/F+F/!#]*&F(F/-%$cosG6#,$
F.\"\"#F/!#7*&F(F/-F,6#,$F.\"\"$F/F<F(\"#c%\"cGF/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"\"\"-%$cosG6#,$F.
\"\"&F/F.,.*&F(F/F+F/!$3%*&F(F/-F16#,$F.\"\"%F/\"\"#*&F(F/-F16#,$F.F=F
/!#)**&F(F/-F,6#,$F.\"\"$F/\"#;F(\"$d%%\"cGF/" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The corresponding definit
e integrals between the limits " }{XPPEDIT 18 0 "x=-Pi" "6#/%\"xG,$%#P
iG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=Pi" "6#/%\"xG%#PiG" }
{TEXT -1 11 " are all 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 481 "Int(exp(sin(x))*cos(x),x=-Pi..Pi)=
\n       int(exp(sin(x))*cos(x),x=-Pi..Pi);\nInt(exp(sin(x))*sin(2*x),
x=-Pi..Pi)=\n       int(exp(sin(x))*sin(2*x),x=-Pi..Pi);\nInt(exp(sin(
x))*cos(3*x),x=-Pi..Pi)=\n       int(exp(sin(x))*cos(3*x),x=-Pi..Pi);
\nInt(exp(sin(x))*sin(4*x),x=-Pi..Pi)=\n       int(exp(sin(x))*sin(4*x
),x=-Pi..Pi);\nInt(exp(sin(x))*cos(5*x),x=-Pi..Pi)=\n       int(exp(si
n(x))*cos(5*x),x=-Pi..Pi);\nInt(exp(sin(x))*sin(6*x),x=-Pi..Pi)=\n    \+
   int(exp(sin(x))*sin(6*x),x=-Pi..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"\"\"-%$cosGF-F//F.;,$%#PiG!
\"\"F5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#-%
$sinG6#%\"xG\"\"\"-F,6#,$F.\"\"#F//F.;,$%#PiG!\"\"F7\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"\"\"-%$cos
G6#,$F.\"\"$F//F.;,$%#PiG!\"\"F8\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"\"\"-F,6#,$F.\"\"%F//F.;,$%#Pi
G!\"\"F7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#
-%$sinG6#%\"xG\"\"\"-%$cosG6#,$F.\"\"&F//F.;,$%#PiG!\"\"F8\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"
\"\"-F,6#,$F.\"\"'F//F.;,$%#PiG!\"\"F7\"\"!" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The coefficients whi
ch are not zero can be calculated numerically." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "Digits := 1
4:\na := NULL:\nfor k from 1 to 9 do\n   if irem(k,2)=1 then ak := 0\n
   else ak := evalf(1/Pi*Int(exp(sin(x))*cos(k*x),x=-Pi..Pi)) end if;
\n   a := a,ak;\nend do:\na := [a];\nDigits := 10: k := 'k':" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"aG7+\"\"!$!/2M&R`\\r#!#9F&$\"/z4U/Cua!#;
F&$!/G(\\HKx\\%!#=F&$\"/Cz[[7#*>!#?F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "Digits := 14:\nb := NUL
L:\nfor k from 1 to 9 do\n   if irem(k,2)=0 then bk := 0\n   else bk :
= evalf(1/Pi*Int(exp(sin(x))*sin(k*x),x=-Pi..Pi)) end if;\n   b := b,b
k;\nend do:\nb := [b];\nDigits := 10: k := 'k':" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"bG7+$\"/])z?=.8\"!#8\"\"!$!/l'[)\\oLW!#:F)$\"/79>JE
Ha!#<F)$!/cOhkV)>$!#>F)$\"/YgTrn.6!#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 34 "The initial constant term is . . ." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "1/(2*Pi)*Int(exp(sin(x)),x=-Pi..Pi);\nc := evalf[14](%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&,$-%$IntG6$-%$expG6#
-%$sinG6#%\"xG/F2;,$%#PiG!\"\"F6*$F6F7F&F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"cG$\"/?vxe1m7!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "f := x -> exp(sin(x));\nF
 := x -> c+b[1]*sin(x)+a[2]*cos(2*x)+b[3]*sin(3*x)+a[4]*cos(4*x)+\n  b
[5]*sin(5*x)+a[6]*cos(6*x)+b[7]*sin(7*x)+a[8]*cos(8*x)+b[9]*sin(9*x):
\n'F(x)'=F(x);\nplot([f(x),F(x)],x=-7..13,0..2.8,color=[red,blue],thic
kness=[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%
)operatorG%&arrowGF(-%$expG6#-%$sinG6#9$F(F(F(" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/-%\"FG6#%\"xG,6$\"/?vxe1m7!#8\"\"\"*&$\"/])z?=.8\"F+F,
-%$sinGF&F,F,*&$\"/2M&R`\\r#!#9F,-%$cosG6#,$*&\"\"#F,F'F,F,F,!\"\"*&$
\"/l'[)\\oLW!#:F,-F16#,$*&\"\"$F,F'F,F,F,F<*&$\"/z4U/Cua!#;F,-F76#,$*&
\"\"%F,F'F,F,F,F,*&$\"/79>JEHa!#<F,-F16#,$*&\"\"&F,F'F,F,F,F,*&$\"/G(
\\HKx\\%!#=F,-F76#,$*&\"\"'F,F'F,F,F,F<*&$\"/cOhkV)>$!#>F,-F16#,$*&\"
\"(F,F'F,F,F,F<*&$\"/Cz[[7#*>!#?F,-F76#,$*&\"\")F,F'F,F,F,F,*&$\"/YgTr
n.6!#@F,-F16#,$*&\"\"*F,F'F,F,F,F," }}{PARA 13 "" 1 "" {GLPLOT2D 672 
295 295 {PLOTDATA 2 "6&-%'CURVESG6%7au7$$!\"(\"\"!$\"3Od*4)R;6%=&!#=7$
$!3YLLLe%G?y'!#<$\"3E\"*pTw/o(>'F-7$$!3!pmmm\"p0klF1$\"3[[%)p'[!3zvF-7
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" {TEXT -1 78 "We can plot the difference so as to check the error inv
olved in approximating " }{XPPEDIT 18 0 "f(x) = exp(sin(x));" "6#/-%\"
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{MPLTEXT 1 0 36 "plot(f(x)-F(x),x=-7..13,color=blue);" }}{PARA 13 "" 
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^7F^`_l$!3_2!Rbhs-l%F-7$$\"3=]7`\\o,a7F^`_l$!3)*p@bg9LP_F-7$$\"3A++v.U
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7$$\"3?voal@pk7F^`_l$!3T`*oS/P:+%F-7$$\"3k(oz\"f,0m7F^`_l$!37$o&R\")*Q
;X$F-7$$\"33+D\"G:3uE\"F^`_l$!3O,oe6/SPGF-7$$\"3_7`WYhwo7F^`_l$!3)=AfF
6_+<#F-7$$\"39D\"y+9C,F\"F^`_l$!3%4\\5u<@=Y\"F-7$$\"3wP4rL@[r7F^`_l$!3
!3lZ#e)yrD(Fjo7$$\"3=]PMF,%GF\"F^`_l$\"3o_*zKoc@Z#Fe^m7$$\"3iil(47)>u7
F^`_l$\"3AHyj9-scxFjo7$$\"31v$4Y6cbF\"F^`_l$\"3?>A:6%zK^\"F-7$$\"3](=U
#3T\"pF\"F^`_l$\"3a(*f2hh\"RA#F-7$$\"37+](=5s#y7F^`_l$\"3CwyO[OU%*GF-7
$$\"3u7y]&4I'z7F^`_l$\"3?:)pJEpB^$F-7$$\"3=D19*3))4G\"F^`_l$\"3'eiw\"H
8FmSF-7$$\"3gPMx#3YBG\"F^`_l$\"3CZ-UR/\"ea%F-7$$\"3/]iSwSq$G\"F^`_l$\"
3,.()3s0.U\\F-7$$\"35v=nj+U'G\"F^`_l$\"3Q!)z(>iKkX&F-7$$\"3;+v$40O\"*G
\"F^`_l$\"3s$)oe'=*fqbF-7$$\"3-DJ?Q?&=H\"F^`_l$\"3_6Z6)4=]F&F-7$$\"33]
(oa-oXH\"F^`_l$\"3sg$4?%*G.f%F-7$$\"3oi:5>g#fH\"F^`_l$\"3&z35j>hk6%F-7
$$\"37vVt7SG(H\"F^`_l$\"3om?Lc`!fc$F-7$$\"3c(=nj+U')H\"F^`_l$\"3=A0:a`
u[HF-7$$\"#8F*$\"3#f7\">mSIwAF--%+AXESLABELSG6$Q\"x6\"Q!F`gbl-%'COLOUR
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1 10 0 2 6 1 4 2 1.000000 45.000000 42.000000 0 0 "Curve 1" }}}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Truncated Fouri
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0 "> " 0 "" {MPLTEXT 1 0 200 "f := x -> exp(sin(x)):\n'f(x)'=f(x);\nDi
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 := unapply(%,x):\n'F(x)'=F(x);\nDigits := 10:\nplot(F(x),x=-7..13,0..
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\"37+](=5s#y7F^hm$\"3\\BOTH&[%R7F17$$\"3;+v$40O\"*G\"F^hm$\"3aw!zU-jhP
\"F17$$\"#8F*$\"3ZfID+e@A:F1-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6
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n$\"#G!\"\"" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "
Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T
asks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 63 "(a) Find form
ulas for the Fourier coefficients of the function " }{XPPEDIT 18 0 "f(
x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " defined by: " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"fG6#%\"xGF'" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 
0 "`` < 2*Pi;" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ", " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with
 period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(b)
 Compare the graphs of some truncated Fourier series with the graph of
 " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}
{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c = Pi;" "6#/%\"cG%#Pi
G" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "a[k] = 0;" "6#/&%\"aG6#%\"kG\"\"
!" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "b[k] = -2/k;" "6#/&%\"bG6#%\"kG,
$*&\"\"#\"\"\"F'!\"\"F," }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 
17 " Fourier series: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "Pi-2*sin*x-sin*2*x-2/3*sin*3*x-sin*4*x/2-2/5*sin*5*x-sin*6*x/3-2/7*
sin*7*x-sin*8*x/4+` . . . `" "6#,6%#PiG\"\"\"*(\"\"#F%%$sinGF%%\"xGF%!
\"\"*(F(F%F'F%F)F%F**,F'F%\"\"$F*F(F%F-F%F)F%F***F(F%\"\"%F%F)F%F'F*F*
*,F'F%\"\"&F*F(F%F1F%F)F%F***F(F%\"\"'F%F)F%F-F*F**,F'F%\"\"(F*F(F%F5F
%F)F%F***F(F%\"\")F%F)F%F/F*F*%(~.~.~.~GF%" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "______________
____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
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" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
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0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________
______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT 
-1 63 "(a) Find formulas for the Fourier coefficients of the function \+
" }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " defined by: \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = sin(x/2);
" "6#/-%\"fG6#%\"xG-%$sinG6#*&F'\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` < 2*Pi;" "6
#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "
6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 
18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Compare the graphs
 of some truncated Fourier series with the graph of " }{XPPEDIT 18 0 "
f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "c = 2/Pi;" "6#/%\"cG*&\"\"#\"\"\"%#PiG!\"\"
" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "a[k] = -4/((4*k^2-1)*Pi);" "6#/&%
\"aG6#%\"kG,$*&\"\"%\"\"\"*&,&*&F*F+*$F'\"\"#F+F+F+!\"\"F+%#PiGF+F1F1
" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 18 "  Fourier series: " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "2/Pi-4/(3*Pi)*cos*x-4/(15*Pi)*cos*2*x-4/(35*Pi)*cos*3*x
-4/(63*Pi)*cos*4*x-4/(99*Pi)*cos*5*x-4/(143*Pi)*cos*6*x-4/(195*Pi)*cos
*7*x-4/(255*Pi)*cos*8*x+` . . . `" "6#,6*&\"\"#\"\"\"%#PiG!\"\"F&**\"
\"%F&*&\"\"$F&F'F&F(%$cosGF&%\"xGF&F(*,F*F&*&\"#:F&F'F&F(F-F&F%F&F.F&F
(*,F*F&*&\"#NF&F'F&F(F-F&F,F&F.F&F(*,F*F&*&\"#jF&F'F&F(F-F&F*F&F.F&F(*
,F*F&*&\"#**F&F'F&F(F-F&\"\"&F&F.F&F(*,F*F&*&\"$V\"F&F'F&F(F-F&\"\"'F&
F.F&F(*,F*F&*&\"$&>F&F'F&F(F-F&\"\"(F&F.F&F(*,F*F&*&\"$b#F&F'F&F(F-F&
\"\")F&F.F&F(%(~.~.~.~GF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 
34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}
{PARA 0 "" 0 "" {TEXT -1 63 "(a) Find formulas for the Fourier coeffic
ients of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 13 " defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` \+
< 2*Pi;" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 
0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " }
{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Compa
re the graphs of some truncated Fourier series with the graph of " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 
18 0 "c = (exp(2*Pi)-1)/(2*Pi);" "6#/%\"cG*&,&-%$expG6#*&\"\"#\"\"\"%#
PiGF,F,F,!\"\"F,*&F+F,F-F,F." }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "a[k]=
(exp(2*Pi)-1)/((k^2+1) *Pi)" "6#/&%\"aG6#%\"kG*&,&-%$expG6#*&\"\"#\"\"
\"%#PiGF/F/F/!\"\"F/*&,&*$F'F.F/F/F/F/F0F/F1" }{TEXT -1 4 ",   " }
{XPPEDIT 18 0 "b[k]=-k*(exp(2*Pi)-1)/((k^2+1) *Pi)" "6#/&%\"bG6#%\"kG,
$*(F'\"\"\",&-%$expG6#*&\"\"#F*%#PiGF*F*F*!\"\"F**&,&*$F'F0F*F*F*F*F1F
*F2F2" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 19 "   Fourier series: " }}{PARA 0 "" 0 "" {TEXT -1 2 " \+
 " }{XPPEDIT 18 0 "(exp(2*Pi)-1)/(2*Pi)+(exp(2*Pi)-1)*cos*x/(2*Pi)-(ex
p(2*Pi)-1)*sin*x/(2*Pi)+(exp(2*Pi)-1)*cos*2*x/(5*Pi)-2*(exp(2*Pi)-1)*s
in*2*x/(5*Pi)+(exp(2*Pi)-1)*cos*3*x/(10*Pi)-3*(exp(2*Pi)-1)*sin*3*x/(1
0*Pi)+` . . . `;" "6#,2*&,&-%$expG6#*&\"\"#\"\"\"%#PiGF+F+F+!\"\"F+*&F
*F+F,F+F-F+**,&-F'6#*&F*F+F,F+F+F+F-F+%$cosGF+%\"xGF+*&F*F+F,F+F-F+**,
&-F'6#*&F*F+F,F+F+F+F-F+%$sinGF+F5F+*&F*F+F,F+F-F-*,,&-F'6#*&F*F+F,F+F
+F+F-F+F4F+F*F+F5F+*&\"\"&F+F,F+F-F+*.F*F+,&-F'6#*&F*F+F,F+F+F+F-F+F<F
+F*F+F5F+*&FDF+F,F+F-F-*,,&-F'6#*&F*F+F,F+F+F+F-F+F4F+\"\"$F+F5F+*&\"#
5F+F,F+F-F+*.FPF+,&-F'6#*&F*F+F,F+F+F+F-F+F<F+FPF+F5F+*&FRF+F,F+F-F-%(
~.~.~.~GF+" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "`` = (exp(2*Pi)-1)/Pi;" "6
#/%!G*&,&-%$expG6#*&\"\"#\"\"\"%#PiGF,F,F,!\"\"F,F-F." }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "``(1/2+cos*x/2-sin*x/2+cos*2*x/5-2*sin*2*x/5+cos*3*x
/10-3*sin*3*x/10+` . . . `)" "6#-%!G6#,2*&\"\"\"F(\"\"#!\"\"F(*(%$cosG
F(%\"xGF(F)F*F(*(%$sinGF(F-F(F)F*F***F,F(F)F(F-F(\"\"&F*F(*,F)F(F/F(F)
F(F-F(F1F*F***F,F(\"\"$F(F-F(\"#5F*F(*,F4F(F/F(F4F(F-F(F5F*F*%(~.~.~.~
GF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "______________________
____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________
______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 63 
"(a) Find formulas for the Fourier coefficients of the function " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " defined by: " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x*(2*Pi-x);" "
6#/-%\"fG6#%\"xG*&F'\"\"\",&*&\"\"#F)%#PiGF)F)F'!\"\"F)" }{TEXT -1 4 "
,   " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` < 2*
Pi;" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f
(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " }
{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Compa
re the graphs of some truncated Fourier series with the graph of " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 
"" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5
" }}{PARA 0 "" 0 "" {TEXT -1 54 "Obtain a few non-zero terms in the Fo
urier series for " }{XPPEDIT 18 0 "f(x) = exp(cos(x));" "6#/-%\"fG6#%
\"xG-%$expG6#-%$cosG6#F'" }{TEXT -1 48 "and compare the resulting trun
cated series with " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "________________________________
__" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_______________________________
___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 63 "(a) Find form
ulas for the Fourier coefficients of the function " }{XPPEDIT 18 0 "f(
x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " defined by: " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x*exp(-x);" "6#/-%\"fG6#%\"xG*
&F'\"\"\"-%$expG6#,$F'!\"\"F)" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "0 <= x
;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` < 2*Pi;" "6#2%!G*&\"\"#\"\"\"%#P
iGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"
\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 74 "(b) Compare the graphs of some truncated \+
Fourier series with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%
\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________
_________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________
______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 
{PARA 257 "" 0 "" {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 21 "C
ode for 1st picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 639 "f := x -> 4+sin(x/2)*cos(3*x):\npp := 4*
evalf(Pi):\naa := pp+1: bb := pp+10:\np1 := plot(f(x),x=-1..25,color=C
OLOR(RGB,1,0,0)):\np2 := plot(f(x),x=1..10,\n   color=COLOR(RGB,.95,.8
5,.95),filled=true,style=patchnogrid):\np3 := plot(f(x),x=aa..bb,\n   \+
color=COLOR(RGB,.95,.85,.95),filled=true,style=patchnogrid):\np4 := pl
ot([[[1,0],[1,f(1)]],[[10,0],[10,f(10)]],[[aa,0],[aa,f(aa)]],\n     [[
bb,0],[bb,f(bb)]],[[-1,0],[25,0]]],color=brown):\nt1 := plots[textplot
]([13,4.9,`y = f(x)`],color=red):\nt2 := plots[textplot]([[1,-.3,`a`],
[10,-.3,`b`],\n       [aa,-.3,`a + p`],[bb,-.3,`b + p`]]):\nplots[disp
lay]([p2,p1,p3,p4,t1,t2],tickmarks=[0,0],axes=none);" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 257 "" 0 "" {TEXT -1 21 "Code for 2nd picture " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 762 "f
 := x -> 2+sin(x)*cos(3*x):\npi := evalf(Pi):\npp := pi+1:\np1 := plot
(f(x),x=-1..7,color=COLOR(RGB,1,0,0)):\np2 := plot(f(x),x=0..1,\n   co
lor=COLOR(RGB,.95,.85,.95),filled=true,style=patchnogrid):\np3 := plot
(f(x),x=pi..pp,\n   color=COLOR(RGB,.95,.85,.95),filled=true,style=pat
chnogrid):\np4 := plot(f(x),x=1..pi,\n   color=COLOR(RGB,.95,.95,.95),
filled=true,style=patchnogrid):\np5 := plot([[[1,0],[1,f(1)]],[[pi,0],
[pi,f(pi)]],\n     [[pp,0],[pp,f(pp)]]],color=black):\nt1 := plots[tex
tplot]([4.4,2.8,`y = f(x)`],color=red):\nt2 := plots[textplot]([[6.9,-
.2,`x`],[-.2,2.85,`y`],\n               [-.2,-.2,`0`],[pi,-.2,`p`],[pp
,-.2,`p + `]]):\nt3 := plots[textplot]([[1,-.2,`a`],[pp+.35,-.2,`a`]],
font=[SYMBOL,9]):\nplots[display]([p2,p1,p3,p4,p5,t1,t2,t3],tickmarks=
[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
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2 33 1 1 }