{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 
0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 
0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 
0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 
128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 
1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 "
Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 262 "" 1 
18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 
0 1 }{CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 
18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 18 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 
1 }{CSTYLE "" -1 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 
-1 271 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 18 
0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 280 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
281 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 
1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1
" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }
1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 
-1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 
2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 
0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple
 Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 
1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 
3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times
" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 "The complex Fourier series" }}
{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 59 "Formulas for the coefficients in the complex Fourier se
ries" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 34 "By making use of the Euler formula" }}{PARA 256 "" 0 "
" {TEXT -1 2 "  " }{XPPEDIT 18 0 "exp(i*theta) = cos*theta+i*sin*theta
;" "6#/-%$expG6#*&%\"iG\"\"\"%&thetaGF),&*&%$cosGF)F*F)F)*(F(F)%$sinGF
)F*F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 74 "it is possible
 to express Fourier series in terms of complex exponentials." }}{PARA 
0 "" 0 "" {TEXT -1 15 "Note that since" }}{PARA 256 "" 0 "" {TEXT -1 
2 "  " }{XPPEDIT 18 0 "exp(-i*theta) = cos*theta-i*sin*theta;" "6#/-%$
expG6#,$*&%\"iG\"\"\"%&thetaGF*!\"\",&*&%$cosGF*F+F*F**(F)F*%$sinGF*F+
F*F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 15 "it follows that" 
}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta = (exp(i*
theta)+exp(-i*theta))/2;" "6#/*&%$cosG\"\"\"%&thetaGF&*&,&-%$expG6#*&%
\"iGF&F'F&F&-F+6#,$*&F.F&F'F&!\"\"F&F&\"\"#F3" }{TEXT -1 20 "     ----
------ (i) " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*theta = (exp(i*theta)-exp(-i*theta)
)/(2*i);" "6#/*&%$sinG\"\"\"%&thetaGF&*&,&-%$expG6#*&%\"iGF&F'F&F&-F+6
#,$*&F.F&F'F&!\"\"F3F&*&\"\"#F&F.F&F3" }{TEXT -1 22 "      ---------- \+
(ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "
Suppose that the Fourier series of a periodic function " }{XPPEDIT 18 
0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = c" "6#/-%\"FG6#%\"xG%\"cG" }
{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(``(a[k]*cos(k*Pi*x/L)+b[k]*sin(k*
Pi*x/L)),k = 1 .. infinity);" "6#-%$SumG6$-%!G6#,&*&&%\"aG6#%\"kG\"\"
\"-%$cosG6#**F.F/%#PiGF/%\"xGF/%\"LG!\"\"F/F/*&&%\"bG6#F.F/-%$sinG6#**
F.F/F4F/F5F/F6F7F/F//F.;F/%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 20 "where the period of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f
G6#%\"xG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"L
GF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "Then, using the fo
rmulas (i) and (ii), we have" }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }
{XPPEDIT 18 0 "F(x)= c" "6#/-%\"FG6#%\"xG%\"cG" }{TEXT -1 2 " +" }
{XPPEDIT 18 0 "Sum(a[k]*``((exp(k*Pi*i*x/L)+exp(-k*Pi*i*x/L))/2)+b[k]*
``((exp(k*Pi*i*x/L)-exp(-k*Pi*i*x/L))/(2*i)),k = 1 .. infinity);" "6#-
%$SumG6$,&*&&%\"aG6#%\"kG\"\"\"-%!G6#*&,&-%$expG6#*,F+F,%#PiGF,%\"iGF,
%\"xGF,%\"LG!\"\"F,-F36#,$*,F+F,F6F,F7F,F8F,F9F:F:F,F,\"\"#F:F,F,*&&%
\"bG6#F+F,-F.6#*&,&-F36#*,F+F,F6F,F7F,F8F,F9F:F,-F36#,$*,F+F,F6F,F7F,F
8F,F9F:F:F:F,*&F?F,F7F,F:F,F,/F+;F,%)infinityG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 12 "Noting that " }{XPPEDIT 18 0 "1/i=-i" "6#
/*&\"\"\"F%%\"iG!\"\",$F&F'" }{TEXT -1 12 ", this gives" }}{PARA 256 "
" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "F(x)= c" "6#/-%\"FG6#%\"xG%\"cG
" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(``((a[k]-i*b[k])/2)*exp(k*Pi*i
*x/L)+``((a[k]+i*b[k])/2)*exp(-k*Pi*i*x/L),k = 1 .. infinity);" "6#-%$
SumG6$,&*&-%!G6#*&,&&%\"aG6#%\"kG\"\"\"*&%\"iGF1&%\"bG6#F0F1!\"\"F1\"
\"#F7F1-%$expG6#*,F0F1%#PiGF1F3F1%\"xGF1%\"LGF7F1F1*&-F)6#*&,&&F.6#F0F
1*&F3F1&F56#F0F1F1F1F8F7F1-F:6#,$*,F0F1F=F1F3F1F>F1F?F7F7F1F1/F0;F1%)i
nfinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 10 "If we let " }{XPPEDIT 18 0 "c[0]=c,c[k]=(a[k]-i*b[
k])/2" "6$/&%\"cG6#\"\"!F%/&F%6#%\"kG*&,&&%\"aG6#F+\"\"\"*&%\"iGF1&%\"
bG6#F+F1!\"\"F1\"\"#F7" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c[-k]=(a[k
]+i*b[k])/2" "6#/&%\"cG6#,$%\"kG!\"\"*&,&&%\"aG6#F(\"\"\"*&%\"iGF/&%\"
bG6#F(F/F/F/\"\"#F)" }{TEXT -1 6 ", then" }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "F(x) = c[0]" "6#/-%\"FG6#%\"xG&%\"cG6#\"\"!" 
}{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(c[k]*exp(k*Pi*i*x/L)+c[-k]*exp(-
k*Pi*i*x/L),k = 1 .. infinity);" "6#-%$SumG6$,&*&&%\"cG6#%\"kG\"\"\"-%
$expG6#*,F+F,%#PiGF,%\"iGF,%\"xGF,%\"LG!\"\"F,F,*&&F)6#,$F+F5F,-F.6#,$
*,F+F,F1F,F2F,F3F,F4F5F5F,F,/F+;F,%)infinityG" }{TEXT -1 3 ",  " }}
{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 3 "
   " }{XPPEDIT 18 0 "F(x) = c[0]" "6#/-%\"FG6#%\"xG&%\"cG6#\"\"!" }
{TEXT -1 2 " +" }{XPPEDIT 18 0 "Sum(c[k]*exp(k*Pi*i*x/L),k = 1 .. infi
nity);" "6#-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#*,F*F+%#PiGF+%\"iGF+
%\"xGF+%\"LG!\"\"F+/F*;F+%)infinityG" }{TEXT -1 3 " + " }{XPPEDIT 18 
0 "Sum(c[k]*exp(k*Pi*i*x/L),k = -1 .. -infinity);" "6#-%$SumG6$*&&%\"c
G6#%\"kG\"\"\"-%$expG6#*,F*F+%#PiGF+%\"iGF+%\"xGF+%\"LG!\"\"F+/F*;,$F+
F4,$%)infinityGF4" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" 
}}{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "F(x) = Sum(c[k]*e
xp(k*Pi*i*x/L),k = -infinity .. infinity);" "6#/-%\"FG6#%\"xG-%$SumG6$
*&&%\"cG6#%\"kG\"\"\"-%$expG6#*,F/F0%#PiGF0%\"iGF0F'F0%\"LG!\"\"F0/F/;
,$%)infinityGF8F<" }{TEXT -1 4 ".   " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{TEXT 262 16 "________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "This is called the " }
{TEXT 261 22 "complex Fourier series" }{TEXT -1 5 " for " }{XPPEDIT 
18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We have" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "c[k]=(a[k]-i*b[k])/2" "6#/&%\"cG6#%\"k
G*&,&&%\"aG6#F'\"\"\"*&%\"iGF-&%\"bG6#F'F-!\"\"F-\"\"#F3" }{TEXT -1 1 
" " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*L);" 
"6#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "``(Int(f(x)*cos(k*Pi*i*x/L),x = -L .. L)-i*Int(f(x)*sin(k*Pi*i*x/L)
,x = -L .. L));" "6#-%!G6#,&-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#*,%
\"kGF/%#PiGF/%\"iGF/F.F/%\"LG!\"\"F//F.;,$F7F8F7F/*&F6F/-F(6$*&-F,6#F.
F/-%$sinG6#*,F4F/F5F/F6F/F.F/F7F8F//F.;,$F7F8F7F/F8" }{TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "``(Int(f(x)*cos(k*Pi*i*x/L)-i*f(x)*sin
(k*Pi*i*x/L),x = -L .. L));" "6#-%!G6#-%$IntG6$,&*&-%\"fG6#%\"xG\"\"\"
-%$cosG6#*,%\"kGF/%#PiGF/%\"iGF/F.F/%\"LG!\"\"F/F/*(F6F/-F,6#F.F/-%$si
nG6#*,F4F/F5F/F6F/F.F/F7F8F/F8/F.;,$F7F8F7" }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"L
GF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*exp(-k*Pi*i*x/L),x \+
= -L .. L);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*,%\"kGF+%#Pi
GF+%\"iGF+F*F+%\"LG!\"\"F5F+/F*;,$F4F5F4" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 11 "Similarly, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c[-k] = (a[k]+i*b[k]
)/2;" "6#/&%\"cG6#,$%\"kG!\"\"*&,&&%\"aG6#F(\"\"\"*&%\"iGF/&%\"bG6#F(F
/F/F/\"\"#F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "``(Int(f(x)*cos(k*Pi*i*x/L),x = -L .. \+
L)+i*Int(f(x)*sin(k*Pi*i*x/L),x = -L .. L));" "6#-%!G6#,&-%$IntG6$*&-%
\"fG6#%\"xG\"\"\"-%$cosG6#*,%\"kGF/%#PiGF/%\"iGF/F.F/%\"LG!\"\"F//F.;,
$F7F8F7F/*&F6F/-F(6$*&-F,6#F.F/-%$sinG6#*,F4F/F5F/F6F/F.F/F7F8F//F.;,$
F7F8F7F/F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\"
\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(Int(f(
x)*cos(k*Pi*i*x/L)+i*f(x)*sin(k*Pi*i*x/L),x = -L .. L));" "6#-%!G6#-%$
IntG6$,&*&-%\"fG6#%\"xG\"\"\"-%$cosG6#*,%\"kGF/%#PiGF/%\"iGF/F.F/%\"LG
!\"\"F/F/*(F6F/-F,6#F.F/-%$sinG6#*,F4F/F5F/F6F/F.F/F7F8F/F//F.;,$F7F8F
7" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
`` = 1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(f(x)*exp(k*Pi*i*x/L),x = -L .. L);" "6#-%$IntG6$
*&-%\"fG6#%\"xG\"\"\"-%$expG6#*,%\"kGF+%#PiGF+%\"iGF+F*F+%\"LG!\"\"F+/
F*;,$F3F4F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 5 "Also " }{XPPEDIT 18 0 "c[0] =1/(2*L)" "6#/&%\"cG
6#\"\"!*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{XPPEDIT 18 0 "Int(f(x),x = -
L .. L);" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$%\"LG!\"\"F-" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "It fo
llows that" }}{PARA 256 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "c[k]
=1/(2*L)" "6#/&%\"cG6#%\"kG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "Int(f(x)*exp(-k*Pi*i*x/L),x = -L .. L);" "6#-%$I
ntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*,%\"kGF+%#PiGF+%\"iGF+F*F+%\"LG!
\"\"F5F+/F*;,$F4F5F4" }{TEXT -1 4 ",   " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{TEXT 263 18 "__________________" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 " for all integer val
ues of " }{TEXT 273 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 261 5 "Notes" }{TEXT -1 2 ": " }}{PARA 
15 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*L" "6#*&
\"\"#\"\"\"%\"LGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[k]" "6#&%\"cG6#
%\"kG" }{TEXT -1 35 " can also be found from the formula" }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[k] = 1/(2*L);" "6#/&%\"cG6#%
\"kG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I
nt(f(x)*exp(-k*Pi*i*x/L),x = 0 .. 2*L);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"
\"\"-%$expG6#,$*,%\"kGF+%#PiGF+%\"iGF+F*F+%\"LG!\"\"F5F+/F*;\"\"!*&\"
\"#F+F4F+" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 3 "If " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 33 " is a real valued
 function, then " }{XPPEDIT 18 0 "c[-k] = conjugate(c[k]);" "6#/&%\"cG
6#,$%\"kG!\"\"-%*conjugateG6#&F%6#F(" }{TEXT -1 9 ", where  " }
{XPPEDIT 18 0 "conjugate(z)" "6#-%*conjugateG6#%\"zG" }{TEXT -1 35 "  \+
denotes the complex conjugate of " }{TEXT 274 1 "z" }{TEXT -1 7 ", and
  " }{XPPEDIT 18 0 "abs(c[k]) = sqrt(a[k]^2+b[k]^2);" "6#/-%$absG6#&%
\"cG6#%\"kG-%%sqrtG6#,&*$&%\"aG6#F*\"\"#\"\"\"*$&%\"bG6#F*F3F4" }
{TEXT -1 5 " for " }{XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" }{TEXT -1 
127 ".\nIn this case the coefficients in the trigonometric form of the
 Fourier series can be recovered from the complex coefficients " }
{XPPEDIT 18 0 "c[k]" "6#&%\"cG6#%\"kG" }{TEXT -1 27 " by means of the \+
formulas: " }}{PARA 256 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "a[k]
=c[k]+c[-k]" "6#/&%\"aG6#%\"kG,&&%\"cG6#F'\"\"\"&F*6#,$F'!\"\"F," }
{XPPEDIT 18 0 "`` = c[k]+conjugate(c[k]);" "6#/%!G,&&%\"cG6#%\"kG\"\"
\"-%*conjugateG6#&F'6#F)F*" }{XPPEDIT 18 0 " ``= 2*Re(c[k])" "6#/%!G*&
\"\"#\"\"\"-%#ReG6#&%\"cG6#%\"kGF'" }{TEXT -1 2 ", " }}{PARA 256 "" 0 
"" {TEXT -1 2 "  " }{XPPEDIT 18 0 "b[k]=i*(c[k]-c[-k])" "6#/&%\"bG6#%
\"kG*&%\"iG\"\"\",&&%\"cG6#F'F*&F-6#,$F'!\"\"F2F*" }{XPPEDIT 18 0 "`` \+
= i*(c[k]-conjugate(c[k]));" "6#/%!G*&%\"iG\"\"\",&&%\"cG6#%\"kGF'-%*c
onjugateG6#&F*6#F,!\"\"F'" }{XPPEDIT 18 0 "``= -2*Im(c[k])" "6#/%!G,$*
&\"\"#\"\"\"-%#ImG6#&%\"cG6#%\"kGF(!\"\"" }{TEXT -1 2 ". " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 61 "Orthogonality of the functions in the complex Fourier ser
ies " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 14 "The functions " }{XPPEDIT 18 0 "eta[k](x) = exp(k*Pi*i*
x/L);" "6#/-&%$etaG6#%\"kG6#%\"xG-%$expG6#*,F(\"\"\"%#PiGF/%\"iGF/F*F/
%\"LG!\"\"" }{TEXT -1 6 ", for " }{XPPEDIT 18 0 "k = ` . . . `,-3,-2,-
1,0,1,2,3,` . . . `" "6+/%\"kG%(~.~.~.~G,$\"\"$!\"\",$\"\"#F(,$\"\"\"F
(\"\"!F,F*F'F%" }{TEXT -1 52 ", which appear in the complex complex Fo
urier series" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "F(x)
 = Sum(c[k]*exp(k*Pi*i*x/L),k = -infinity .. infinity);" "6#/-%\"FG6#%
\"xG-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#*,F/F0%#PiGF0%\"iGF0F'F0%\"
LG!\"\"F0/F/;,$%)infinityGF8F<" }{TEXT -1 4 ",   " }}{PARA 0 "" 0 "" 
{TEXT -1 30 "that is, the set of functions " }{XPPEDIT 18 0 "\{` . . .
 `,exp(-k*Pi*i*x/L),` . . . `,exp(-2*Pi*i*x/L),exp(-Pi*i*x/L),1,exp(Pi
*i*x/L),exp(2*Pi*i*x/L),` . . . `,exp(k*Pi*i*x/L),` . . . `\}" "6#<-%(
~.~.~.~G-%$expG6#,$*,%\"kG\"\"\"%#PiGF+%\"iGF+%\"xGF+%\"LG!\"\"F0F$-F&
6#,$*,\"\"#F+F,F+F-F+F.F+F/F0F0-F&6#,$**F,F+F-F+F.F+F/F0F0F+-F&6#**F,F
+F-F+F.F+F/F0-F&6#*,F5F+F,F+F-F+F.F+F/F0F$-F&6#*,F*F+F,F+F-F+F.F+F/F0F
$" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "are " }{TEXT 261 19 "
mutually orthogonal" }{TEXT -1 35 " with respect to the \"dot product
\" " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)*`.`*g(x)=
Int(f(x)*conjugate(g(x)),x=-L..L)" "6#/*(-%\"fG6#%\"xG\"\"\"%\".GF)-%
\"gG6#F(F)-%$IntG6$*&-F&6#F(F)-%*conjugateG6#-F,6#F(F)/F(;,$%\"LG!\"\"
F<" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 8 "In fact
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "eta[k](x)*`.`*et
a[m](x) = Int(exp(k*Pi*i*x/L)*exp(-m*Pi*i*x/L),x = -L .. L);" "6#/*(-&
%$etaG6#%\"kG6#%\"xG\"\"\"%\".GF,-&F'6#%\"mG6#F+F,-%$IntG6$*&-%$expG6#
*,F)F,%#PiGF,%\"iGF,F+F,%\"LG!\"\"F,-F86#,$*,F1F,F;F,F<F,F+F,F=F>F>F,/
F+;,$F=F>F=" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = PIECEWISE([0, m <> k],[2*L, m = k]);" "6#/%!G-%*PI
ECEWISEG6$7$\"\"!0%\"mG%\"kG7$*&\"\"#\"\"\"%\"LGF0/F+F," }{TEXT -1 18 
"     ------- (ii)." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 15 
"__________     " }{TEXT -1 6 "      " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note
" }{TEXT -1 112 ": Taking the conjugate of the second function in the \+
integral in (i) ensures that the dot product of a function " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 100 " with itself is \+
a real number. This leads to the possibility of defining the length or
 magnitude of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 "
 as a \"vector\" by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "abs(abs(f(x))) = sqrt(f(x)*`.`*f(x));" "6#/-%$absG6#-F%6#-%\"fG6#%
\"xG-%%sqrtG6#*(-F*6#F,\"\"\"%\".GF3-F*6#F,F3" }{XPPEDIT 18 0 "`` = In
t(f(x)*conjugate(f(x)),x = -L .. L);" "6#/%!G-%$IntG6$*&-%\"fG6#%\"xG
\"\"\"-%*conjugateG6#-F*6#F,F-/F,;,$%\"LG!\"\"F6" }{XPPEDIT 18 0 "``= \+
Int(abs(f(x))^2,x = -L .. L)" "6#/%!G-%$IntG6$*$-%$absG6#-%\"fG6#%\"xG
\"\"#/F/;,$%\"LG!\"\"F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 
44 "The dot product has the following properties" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " (a) " }
{XPPEDIT 18 0 "f(x)*`.`*g(x) = conjugate(g(x)*`.`*f(x))" "6#/*(-%\"fG6
#%\"xG\"\"\"%\".GF)-%\"gG6#F(F)-%*conjugateG6#*(-F,6#F(F)F*F)-F&6#F(F)
" }{TEXT -1 28 "  -----  the dot product is " }{TEXT 261 20 "conjugate
 symmetric " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 8 " (b) If " }{TEXT 275 1 "r" }{TEXT -1 5 " and " }{TEXT 276 1 "s" 
}{TEXT -1 27 " are complex numbers, then " }}{PARA 0 "" 0 "" {TEXT -1 
8 "        " }{XPPEDIT 18 0 "[r*f(x)+s*g(x)]*`.`*h(x) = r*[f(x)*`.`*h(
x)]+s*[f(x)*`.`*h(x)]" "6#/*(7#,&*&%\"rG\"\"\"-%\"fG6#%\"xGF)F)*&%\"sG
F)-%\"gG6#F-F)F)F)%\".GF)-%\"hG6#F-F),&*&F(F)7#*(-F+6#F-F)F3F)-F56#F-F
)F)F)*&F/F)7#*(-F+6#F-F)F3F)-F56#F-F)F)F)" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 9 "     and " }}{PARA 0 "" 0 "" {TEXT -1 8 "        " 
}{XPPEDIT 18 0 "h(x)*`.`*[r*f(x)+s*g(x)] = r*[h(x)*`.`*f(x)]+s*[h(x)*`
.`*f(x)];" "6#/*(-%\"hG6#%\"xG\"\"\"%\".GF)7#,&*&%\"rGF)-%\"fG6#F(F)F)
*&%\"sGF)-%\"gG6#F(F)F)F),&*&F.F)7#*(-F&6#F(F)F*F)-F06#F(F)F)F)*&F3F)7
#*(-F&6#F(F)F*F)-F06#F(F)F)F)" }{TEXT -1 28 "  -----  the dot product \+
is " }{TEXT 261 8 "bilinear" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " (c) " }{XPPEDIT 18 0 "f(x)*`.`*
g(x)" "6#*(-%\"fG6#%\"xG\"\"\"%\".GF(-%\"gG6#F'F(" }{TEXT -1 34 " is r
eal and non-negative and, if " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 15 " is continuous," }}{PARA 0 "" 0 "" {TEXT -1 9 "       \+
  " }{XPPEDIT 18 0 "f(x)*`.`*f(x)=0" "6#/*(-%\"fG6#%\"xG\"\"\"%\".GF)-
F&6#F(F)\"\"!" }{TEXT -1 14 " exactly when " }{XPPEDIT 18 0 "f(x)" "6#
-%\"fG6#%\"xG" }{TEXT -1 48 " is identically zero  -----  the dot prod
uct is " }{TEXT 261 17 "positive definite" }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{TEXT 265 16 "________________" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "To check (ii) firs
t suppose that  " }{XPPEDIT 18 0 "m <> k;" "6#0%\"mG%\"kG" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "eta[k](x)*`.`*eta[m](x) = Int(eta[k](x)*conj
ugate(eta[m](x)),x = -L .. L)" "6#/*(-&%$etaG6#%\"kG6#%\"xG\"\"\"%\".G
F,-&F'6#%\"mG6#F+F,-%$IntG6$*&-&F'6#F)6#F+F,-%*conjugateG6#-&F'6#F16#F
+F,/F+;,$%\"LG!\"\"FE" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "`` = Int(exp(k*Pi*i*x/L)*exp(-m*Pi*i*x/L),x = -L .
. L);" "6#/%!G-%$IntG6$*&-%$expG6#*,%\"kG\"\"\"%#PiGF.%\"iGF.%\"xGF.%
\"LG!\"\"F.-F*6#,$*,%\"mGF.F/F.F0F.F1F.F2F3F3F./F1;,$F2F3F2" }{TEXT 
-1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(
exp((k-m)*Pi*i*x/L),x = -L .. L);" "6#/%!G-%$IntG6$-%$expG6#*,,&%\"kG
\"\"\"%\"mG!\"\"F.%#PiGF.%\"iGF.%\"xGF.%\"LGF0/F3;,$F4F0F4" }{TEXT -1 
1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = L/((k-m)
*Pi*i);" "6#/%!G*&%\"LG\"\"\"*(,&%\"kGF'%\"mG!\"\"F'%#PiGF'%\"iGF'F," 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "exp((k-m)*Pi*i*x/L)" "6#-%$expG6#*,,&%
\"kG\"\"\"%\"mG!\"\"F)%#PiGF)%\"iGF)%\"xGF)%\"LGF+" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([L,``],[-L,``])" "6#-%*PIECEWISEG6$7$%\"LG%!G
7$,$F'!\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = L/((k-m)*Pi*i);
" "6#/%!G*&%\"LG\"\"\"*(,&%\"kGF'%\"mG!\"\"F'%#PiGF'%\"iGF'F," }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "[exp((k-m)*Pi*i)-exp(-(k-m)*Pi*i)];" "6#7#,&-
%$expG6#*(,&%\"kG\"\"\"%\"mG!\"\"F+%#PiGF+%\"iGF+F+-F&6#,$*(,&F*F+F,F-
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 "`` = L/((k-m)*Pi*i);
" "6#/%!G*&%\"LG\"\"\"*(,&%\"kGF'%\"mG!\"\"F'%#PiGF'%\"iGF'F," }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "[(-1)^(k-m)-(-1)^(m-k)]" "6#7#,&),$\"\"\"!\"
\",&%\"kGF'%\"mGF(F'),$F'F(,&F+F'F*F(F(" }{TEXT -1 1 " " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 0" "6#/%!G\"\"!" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Also" }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "eta[k](x)*`.`*eta[k](x) = Int(eta[k](x)*conj
ugate(eta[k](x)),x = -L .. L);" "6#/*(-&%$etaG6#%\"kG6#%\"xG\"\"\"%\".
GF,-&F'6#F)6#F+F,-%$IntG6$*&-&F'6#F)6#F+F,-%*conjugateG6#-&F'6#F)6#F+F
,/F+;,$%\"LG!\"\"FD" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = Int(exp(k*Pi*i*x/L)*exp(-k*Pi*i*x/L),x = -L .. \+
L);" "6#/%!G-%$IntG6$*&-%$expG6#*,%\"kG\"\"\"%#PiGF.%\"iGF.%\"xGF.%\"L
G!\"\"F.-F*6#,$*,F-F.F/F.F0F.F1F.F2F3F3F./F1;,$F2F3F2" }{TEXT -1 2 "  \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(1,x = -L
 .. L);" "6#/%!G-%$IntG6$\"\"\"/%\"xG;,$%\"LG!\"\"F-" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*L;" "6#/%!G*&\"\"#\"\"\"%\"
LGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
59 "The complex Fourier series via the orthogonality properties" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 13 "Suppose that " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 
-1 45 " can be expanded as a complex Fourier series " }}{PARA 256 "" 
0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x) = Sum(c[k]*exp(k*Pi*i*x/L),k
 = -infinity .. infinity);" "6#/-%\"fG6#%\"xG-%$SumG6$*&&%\"cG6#%\"kG
\"\"\"-%$expG6#*,F/F0%#PiGF0%\"iGF0F'F0%\"LG!\"\"F0/F/;,$%)infinityGF8
F<" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = Sum(c[k]*eta[
k](x),k = -infinity .. infinity);" "6#/-%\"fG6#%\"xG-%$SumG6$*&&%\"cG6
#%\"kG\"\"\"-&%$etaG6#F/6#F'F0/F/;,$%)infinityG!\"\"F9" }{TEXT -1 1 ",
" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "eta[k](x) = ex
p(k*Pi*i*x/L)" "6#/-&%$etaG6#%\"kG6#%\"xG-%$expG6#*,F(\"\"\"%#PiGF/%\"
iGF/F*F/%\"LG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 63 "Tak
ing the dot product (on the right) of each side of (i) with " }
{XPPEDIT 18 0 "eta[m](x)" "6#-&%$etaG6#%\"mG6#%\"xG" }{TEXT -1 7 " giv
es " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)*`.`*eta[m
](x) = ``(Sum(c[k]*eta[k](x),k = -infinity .. infinity))*`.`*eta[m](x)
;" "6#/*(-%\"fG6#%\"xG\"\"\"%\".GF)-&%$etaG6#%\"mG6#F(F)*(-%!G6#-%$Sum
G6$*&&%\"cG6#%\"kGF)-&F-6#F<6#F(F)/F<;,$%)infinityG!\"\"FDF)F*F)-&F-6#
F/6#F(F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = Sum(c[k]*[eta[k](x)*`.`*eta[m](x)],k = -infinity .
. infinity);" "6#/%!G-%$SumG6$*&&%\"cG6#%\"kG\"\"\"7#*(-&%$etaG6#F,6#%
\"xGF-%\".GF--&F26#%\"mG6#F5F-F-/F,;,$%)infinityG!\"\"F?" }{TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = c[m]*``(eta[m](x)*`.`*eta[m](x));" "6#/%!G*&&%
\"cG6#%\"mG\"\"\"-F$6#*(-&%$etaG6#F)6#%\"xGF*%\".GF*-&F06#F)6#F3F*F*" 
}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = c[m]*``(2*L);" "6#/%!G*&&%\"cG6#%
\"mG\"\"\"-F$6#*&\"\"#F*%\"LGF*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
c[m] = 1/(2*L)" "6#/&%\"cG6#%\"mG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)*`.`*eta[m](x);" "6#*(-%\"fG6#%\"xG
\"\"\"%\".GF(-&%$etaG6#%\"mG6#F'F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+
1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(f(x)*exp(-m*Pi*i*x/L),x = -L .. L);" "6#-%$IntG6$*&
-%\"fG6#%\"xG\"\"\"-%$expG6#,$*,%\"mGF+%#PiGF+%\"iGF+F*F+%\"LG!\"\"F5F
+/F*;,$F4F5F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 38 "The (double) sequence of coefficients " }
{XPPEDIT 18 0 "c[m]" "6#&%\"cG6#%\"mG" }{TEXT -1 20 " may be regarded \+
as " }{TEXT 261 11 "coordinates" }{TEXT -1 17 " of the function " }
{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 65 " with respect to
 the coordinate system determined by the vectors " }{XPPEDIT 18 0 "eta
[k](x)" "6#-&%$etaG6#%\"kG6#%\"xG" }{TEXT -1 105 " in the \"space\" of
 all functions which can be expanded in the form of a complex Fourier \+
series as in (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Exa
mples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 58 "We shall find the complex Fo
urier series for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 8 " where  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "f(x)=x" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "0
<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#Pi
GF'" }{TEXT -1 2 ", " }}{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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
82 "f := x -> x-2*Pi*floor(x/(2*Pi));\nplot(f(x),x=-2*Pi..20,color=COL
OR(RGB,.4,0,.9));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6
\"6$%)operatorG%&arrowGF(,&9$\"\"\"*&%#PiGF.-%&floorG6#,$*&F-F.F0!\"\"
#F.\"\"#F.!\"#F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 586 192 192 
{PLOTDATA 2 "6&-%'CURVESG6#7_q7$$!1++i%H&=$G'!#:$\"11(*)fiefD\"!#B7$$!
1X!)>`mG5dF*$\"18a\")Rl)*Gd!#;7$$!1\">p&31\"=@&F*$\"1&R5')pu82\"F*7$$!
15?Si@A^YF*$\"1wvxWJ'>j\"F*7$$!1&QA@x<p3%F*$\"1,s0NvE'>#F*7$$!1\"z0@U&
HDNF*$\"1'zt]))*)yv#F*7$$!1>a(4h*f/IF*$\"1nT?'p&eyKF*7$$!1i$G%e#\\aY#F
*$\"1C7v[gt<QF*7$$!1X;c#>fy!>F*$\"1Tzh9hKvVF*7$$!1Vx5Dt0_8F*$\"1V=2#)z
7J\\F*7$$!1\"3yp(yY.yF3$\"1y<[>&QG]&F*7$$!1n?G9Jl&G&F3$\"1!Q^d**>Yv&F*
7$$!1age^$Qyw#F3$\"1\")4-s9S1gF*7$$!1%G)e\"eA#f?F3$\"1e2-\\IExgF*7$$!1
80f6og]8F3$\"1N0-EY7[hF*7$$!1$G;fE*)H'**!#<$\"1B/_9ab$='F*7$$!1Ku#fT5*
>kFbp$\"17.-.i)*=iF*7$$!12I$4*4P[YFbp$\"1c-F(f,nB'F*7$$!1\"eQfcJo(GFbp
$\"1+-_\"*pTaiF*7$$!1cT%49#H06Fbp$\"1X,x&QK@F'F*7$$\"1)p-0%GZim!#=Fjq7
$$\"1n0Soi&Q[\"F3F^r7$$\"13hR3y3,HF3Far7$$\"1RK7m%*=ZdF3Fdr7$$\"1r.&Q7
\"H$f)F3Fgr7$$\"13;zv$yyS\"F*Fjr7$$\"1\"\\dt,Ypl\"F*F]s7$$\"1vL#*eO,1>
F*F`s7$$\"1rK3v\"z@?#F*Fcs7$$\"1mJC\"pW$)\\#F*Ffs7$$\"14`_JdB\\FF*Fis7
$$\"1_u!=xE,+$F*F\\t7$$\"1A%*o>'=Qe$F*F_t7$$\"1BC%y![[+TF*Fbt7$$\"1\\>
V.&\\tm%F*Fet7$$\"1(RcauQr?&F*Fht7$$\"1[wnoau)[&F*F[u7$$\"1)*))*=>_.x&
F*F^u7$$\"13*4>j`'**eF*Fau7$$\"1<4#>2b*GgF*Fdu7$$\"1@k#>z0O4'F*Fgu7$$
\"1E>$>^c#ehF*Fju7$$\"1yY$>(=e!>'F*F]v7$$\"1Iu$>B2HA'F*F`v7$$\"12)Q>\"
*p!RiF*Fcv7$$\"1$=S>fK_D'F*Ffv7$$\"1f:%>F&RriF*Fiv7$$\"1OH%>&zb(G'F*$
\"1?%\\LwWEP%F\\r7$$\"1\"G#3%4'\\mlF*$\"1_p-py5LGF37$$\"1F;AOUVXoF*$\"
14/U!H*[AcF37$$\"1_QJ24\"\\U(F*$\"1mU8+csT6F*7$$\"1Te(H0Z$HzF*$\"1bizX
<;Y;F*7$$\"1!eq(4>:u%)F*$\"1%*4f-m'4>#F*7$$\"1WJ.O'))p.*F*$\"1dN&)GL!Q
v#F*7$$\"1TAwmXh(e*F*$\"1bEef#HWI$F*7$$\"1p83<S*R&)*F*$\"1$y,*4(33d$F*
7$$\"1]+uYt.75!#9$\"164Ag\")=PQF*7$$\"1!p_IC9;/\"F[z$\"17tMBr&H8%F*7$$
\"1I`OR6>r5F[z$\"18PZ'3E(GWF*7$$\"1Vy='Hnx4\"F[z$\"1\\))paw[%p%F*7$$\"
1d.,`MMC6F[z$\"1%)R#HA\\-'\\F*7$$\"141'H*[4\"=\"F[z$\"14lUAOwFbF*7$$\"
15$>F^2o?\"F[z$\"1;N,?)*)[y&F*7$$\"16!yC8?DB\"F[z$\"1C0g<g,UgF*7$$\"1%
HDP!yaR7F[z$\"1^L2IFH7hF*7$$\"1wD(\\ZvlC\"F[z$\"1yhaU%pD='F*7$$\"1=if5
$*3]7F[z$\"1$f#y)z2x@'F*7$$\"1f)>i9.OD\"F[z$\"12!>]:YGD'F*7$$\"1!oJS1g
`D\"F[z$\"19s8L`TqiF*7$$\"1,N%=)p6d7F[z$\"1&yLe2/#*z%F\\r7$$\"1@`l**Q(
)e7F[z$\"1\">/%>#QoB#Fbp7$$\"1UrY<3jg7F[z$\"1F[AJgv$*RFbp7$$\"13<'*fho
u7F[z$\"1W!zD&)4\\!=F37$$\"1tiX-:u)G\"F[z$\"1)eN?5W/@$F37$$\"1PTJm9>:8
F[z$\"1m>#y[Sa&eF37$$\"1,?<I9kT8F[z$\"1:%3O(oV+&)F37$$\"1ji0(4PpR\"F[z
$\"1gM?c.+.9F*7$$\"1ULc=()*4X\"F[z$\"1^UFrlhV>F*7$$\"1P46Eif2:F[z$\"1&
>]nk\"f4DF*7$$\"1/El_j5i:F[z$\"1ro;7HpaIF*7$$\"1$[8,$3&yh\"F[z$\"1acx'
oP@h$F*7$$\"11i::P8t;F[z$\"1$)G?Pl'\\;%F*7$$\"1N@0XH$Rs\"F[z$\"1u@;O)e
Hn%F*7$$\"1AI?&fa@y\"F[z$\"1^5nP`<b_F*7$$\"1kJ=k;>3=F[z$\"1nCZFga:bF*7
$$\"11L;L(GU$=F[z$\"1\"*QF<n\"fx&F*7$$\"1%4Mi?4\"[=F[z$\"1m<)zW@Z\"fF*
7$$\"1#)[Iz'*)>'=F[z$\"1W'*oyh_`gF*7$$\"1v-%e\"*H*o=F[z$\"1#eVSaGH7'F*
7$$\"1pcP_,(e(=F[z$\"1>vR44L#>'F*7$$\"1mLkq-Mz=F[z$\"1)[u?4KqA'F*7$$\"
1j5\"*)Q5G)=F[z$\"1d9vuKthiF*7$$\"16\\/[aa%)=F[z$\"1T**3mQ3ziF*7$$\"1g
(yr]!G')=F[z$\"1@R)[-:\\K\"Fbp7$$\"13EJmb,))=F[z$\"1fBteT(*fIFbp7$$\"1
dkWD1v*)=F[z$\"1(z!e#HL]z%Fbp7$$\"1d37$R@j\">F[z$\"1$3)p'4!eOJF37$$\"1
c_zg@*G%>F[z$\"1zz8kol$z&F37$$\"1+++)*******>F[z$\"1S7YeSW]6F*-%+AXESL
ABELSG6$Q\"x6\"%!G-%&COLORG6&%$RGBG$\"\"%!\"\"\"\"!$\"\"*F\\hl-%%VIEWG
6$;$!+3`=$G'!\"*$\"#?F]hl%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 
45.000000 44.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 27 "The constant coefficient is" }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "c[0] = 1/(2*Pi);" "6#/&%\"cG6#\"\"!*&\"\"\"F
)*&\"\"#F)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x,x = 0 ..
 2*Pi);" "6#-%$IntG6$%\"xG/F&;\"\"!*&\"\"#\"\"\"%#PiGF," }{TEXT -1 1 "
 " }}{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#*&%\"xG\"\"#F%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIEC
EWISE([2*Pi ,`` ],[0 ,`` ])" "6#-%*PIECEWISEG6$7$*&\"\"#\"\"\"%#PiGF)%
!G7$\"\"!F+" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = P
i;" "6#/%!G%#PiG" }{TEXT -1 4 ".   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 41 "The coefficients of the exponential terms" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c[k]=1/(2*Pi)" "6#
/&%\"cG6#%\"kG*&\"\"\"F)*&\"\"#F)%#PiGF)!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(x*exp(-i*k*x),x = 0 .. 2*Pi);" "6#-%$IntG6$*&%\"xG
\"\"\"-%$expG6#,$*(%\"iGF(%\"kGF(F'F(!\"\"F(/F';\"\"!*&\"\"#F(%#PiGF(
" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 
"k<>0" "6#0%\"kG\"\"!" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*exp(-i*k*x),x) = Int(u*``(dv/dx),
x);" "6#/-%$IntG6$*&%\"xG\"\"\"-%$expG6#,$*(%\"iGF)%\"kGF)F(F)!\"\"F)F
(-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxGF1F)F(" }{TEXT -1 9 ",  where " }
{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "v = exp(-i*k*x)/(-i*k);" "6#/%\"vG*&-%$expG6#,$*(%\"iG\"\"\"%\"k
GF,%\"xGF,!\"\"F,,$*&F+F,F-F,F/F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 46 "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 9 ", we have" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[k]=1/(2*Pi)" "6#/&%\"cG6#%\"kG
*&\"\"\"F)*&\"\"#F)%#PiGF)!\"\"" }{XPPEDIT 18 0 "``(x*exp(-i*k*x)/(-i*
k));" "6#-%!G6#*(%\"xG\"\"\"-%$expG6#,$*(%\"iGF(%\"kGF(F'F(!\"\"F(,$*&
F.F(F/F(F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2*Pi, ``],[0,
 ``]);" "6#-%*PIECEWISEG6$7$*&\"\"#\"\"\"%#PiGF)%!G7$\"\"!F+" }
{XPPEDIT 18 0 "-1/(2*Pi)" "6#,$*&\"\"\"F%*&\"\"#F%%#PiGF%!\"\"F)" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-i*k*x)/(-i*k),x = 0 .. 2*Pi);
" "6#-%$IntG6$*&-%$expG6#,$*(%\"iG\"\"\"%\"kGF-%\"xGF-!\"\"F-,$*&F,F-F
.F-F0F0/F/;\"\"!*&\"\"#F-%#PiGF-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*Pi*exp(-2*k*Pi*i)/(-2*k*Pi*i)" "6#
/%!G**\"\"#\"\"\"%#PiGF'-%$expG6#,$**F&F'%\"kGF'F(F'%\"iGF'!\"\"F',$**
F&F'F.F'F(F'F/F'F0F0" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "-1/(2*Pi)" "6#,
$*&\"\"\"F%*&\"\"#F%%#PiGF%!\"\"F)" }{XPPEDIT 18 0 "``(exp(-i*k*x)/(-k
^2));" "6#-%!G6#*&-%$expG6#,$*(%\"iG\"\"\"%\"kGF-%\"xGF-!\"\"F-,$*$F.
\"\"#F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2*Pi, ``],[0, ``
]);" "6#-%*PIECEWISEG6$7$*&\"\"#\"\"\"%#PiGF)%!G7$\"\"!F+" }{TEXT -1 
1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(-k*i)
+0;" "6#/%!G,&*&\"\"\"F',$*&%\"kGF'%\"iGF'!\"\"F,F'\"\"!F'" }{TEXT -1 
2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = i/k;" "
6#/%!G*&%\"iG\"\"\"%\"kG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "assume(k_,i
nteger):\n1/(2*Pi)*Int(x*exp(-I*k*x),x=0..2*Pi);\nc[k]=subs(k_=k,simpl
ify(value(subs(k=k_,%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$I
ntG6$*&%\"xG\"\"\"-%$expG6#,$*(%\"IGF*%\"kGF*F)F*!\"\"F*/F);\"\"!,$%#P
iG\"\"#F*F7F2#F*F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#%\"kG*&
%\"IG\"\"\"F'!\"\"" }}}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 
30 "The complex Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%
\"xG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "F(x) = Sum(c[k]*exp(k*i*x),x = -infinity .. infinity);" "6#/-%\"
FG6#%\"xG-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#*(F/F0%\"iGF0F'F0F0/F'
;,$%)infinityG!\"\"F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "w
here " }{XPPEDIT 18 0 "c[0]=Pi" "6#/&%\"cG6#\"\"!%#PiG" }{TEXT -1 5 " \+
and " }{XPPEDIT 18 0 "c[k]=i/k" "6#/&%\"cG6#%\"kG*&%\"iG\"\"\"F'!\"\"
" }{TEXT -1 6 "  for " }{XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "F(x) = Pi+Sum(i*exp(k*i*x)/k,k = -infi
nity .. infinity*`/`);" "6#/-%\"FG6#%\"xG,&%#PiG\"\"\"-%$SumG6$*(%\"iG
F*-%$expG6#*(%\"kGF*F/F*F'F*F*F4!\"\"/F4;,$%)infinityGF5*&F9F*%\"/GF*F
*" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"`` = Pi+Sum(exp(i*(k*x+Pi/2))/k,k = -infinity .. infinity*`/`);" "6#/
%!G,&%#PiG\"\"\"-%$SumG6$*&-%$expG6#*&%\"iGF',&*&%\"kGF'%\"xGF'F'*&F&F
'\"\"#!\"\"F'F'F'F3F7/F3;,$%)infinityGF7*&F;F'%\"/GF'F'" }{TEXT -1 1 "
," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Sum(``,k=-inf
inity..infinity*`/`)" "6#-%$SumG6$%!G/%\"kG;,$%)infinityG!\"\"*&F+\"\"
\"%\"/GF." }{TEXT -1 56 " means that the summation is over all non-zer
o integers." }}{PARA 0 "" 0 "" {TEXT -1 17 "The coefficients " }
{XPPEDIT 18 0 "a[k]" "6#&%\"aG6#%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "b[k]" "6#&%\"bG6#%\"kG" }{TEXT -1 47 " in the corresponding trig
onometric series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 
-1 15 " are given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "a[k] = 2*Re(c[k]);" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"-%#ReG6#&%\"cG
6#F'F*" }{TEXT -1 9 " = 0 and " }{XPPEDIT 18 0 "b[k] = -2*Im(c[k])" "6
#/&%\"bG6#%\"kG,$*&\"\"#\"\"\"-%#ImG6#&%\"cG6#F'F+!\"\"" }{TEXT -1 3 "
 = " }{XPPEDIT 18 0 " -2/k" "6#,$*&\"\"#\"\"\"%\"kG!\"\"F(" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "T
he trigonometric series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" 
}{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi-2*``(sin(x)+sin(2*x)/2+sin(3*x)/3+si
n(4*x)/4+sin(5*x)/5+sin(6*x)/6+sin(7*x)/7+`. . .`);" "6#,&%#PiG\"\"\"*
&\"\"#F%-%!G6#,2-%$sinG6#%\"xGF%*&-F-6#*&F'F%F/F%F%F'!\"\"F%*&-F-6#*&
\"\"$F%F/F%F%F9F4F%*&-F-6#*&\"\"%F%F/F%F%F>F4F%*&-F-6#*&\"\"&F%F/F%F%F
CF4F%*&-F-6#*&\"\"'F%F/F%F%FHF4F%*&-F-6#*&\"\"(F%F/F%F%FMF4F%%&.~.~.GF
%F%F4" }{TEXT -1 2 ". " }}{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 58 "We shall find the complex Fourier serie
s for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 
-1 8 " where  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x
) = PIECEWISE([1, abs(x) <= a],[0, a < abs(x) and abs(x) <= L]);" "6#/
-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"\"1-%$absG6#F'%\"aG7$\"\"!32F1-F/6#F
'1-F/6#F'%\"LG" }{TEXT -1 2 ", " }}{PARA 0 "" 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*L;" "
6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 13 "" 1 "" {GLPLOT2D 547 264 264 {PLOTDATA 2 "6--%'CURVESG6
$7$7$$!\"\"\"\"!F*7$F($\"\"\"F*-%&COLORG6&%$RGBG$\"\"%F)F*$\"\"*F)-F$6
$7$7$F,F*7$F,F,F.-F$6$7$F+F:F.-F$6$7$7$$\"\"$F*F*7$FBF,F.-F$6$7$7$$\"
\"&F*F*7$FIF,F.-F$6$7$FDFKF.-%%TEXTG6%7$$\"$N'!\"#$!\"&FU%\"xG-%'COLOU
RG6&F1F*F*F*-FP6%7$$F)F)$\"$X\"FU%\"yGFY-%+AXESLABELSG6$%!GF`o-%*AXEST
ICKSG6$7)/FU%#-LG/F)%#-aG/F-%\"aG/\"\"#%\"LG/FC%%2L-aG/F3%#2LG/FJ%%2L+
aG7$F*F--%%VIEWG6$;$FUF*$\"#lF);FV$\"#:F)" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 27 "The constant coefficient is" 
}}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c[0] = 1/(2*L);" "
6#/&%\"cG6#\"\"!*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(f(x),x = -L .. L);" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$
%\"LG!\"\"F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1,x = -a .. a);" "6#-%$IntG6$\"\"
\"/%\"xG;,$%\"aG!\"\"F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "`` = a/L;" "6#/%!G*&%\"aG\"\"\"%\"LG!\"\"" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "The co
efficients of the exponential terms" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 
0 "c[k] = 1/(2*L);" "6#/&%\"cG6#%\"kG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*exp(-i*k*Pi*x/L),x = -L .. L)
;" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*,%\"iGF+%\"kGF+%#PiGF+
F*F+%\"LG!\"\"F5F+/F*;,$F4F5F4" }{TEXT -1 3 "   " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"#
F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-i*k*Pi*x/L),x
 = -a .. a);" "6#-%$IntG6$-%$expG6#,$*,%\"iG\"\"\"%\"kGF,%#PiGF,%\"xGF
,%\"LG!\"\"F1/F/;,$%\"aGF1F5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*L
);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "``(-L*exp(-i*k*Pi*x/L)/(k*Pi*i));" "6#-%!G6#,$*(%\"LG\"\"\"-%$ex
pG6#,$*,%\"iGF)%\"kGF)%#PiGF)%\"xGF)F(!\"\"F3F)*(F0F)F1F)F/F)F3F3" }
{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([a, ``],[``, ``],[-a, ``]);
" "6#-%*PIECEWISEG6%7$%\"aG%!G7$F(F(7$,$F'!\"\"F(" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/(2*k*Pi*i);" "6#/%!G*&\"\"\"F&**\"\"#F&%\"kGF&%#
PiGF&%\"iGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(exp(i*k*Pi*a/L)-
exp(-i*k*Pi*a/L));" "6#-%!G6#,&-%$expG6#*,%\"iG\"\"\"%\"kGF,%#PiGF,%\"
aGF,%\"LG!\"\"F,-F(6#,$*,F+F,F-F,F.F,F/F,F0F1F1F1" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/(k*Pi);" "6#/%!G*&\"\"\"F&*&%\"kGF&%#PiGF&!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(k*Pi*a/L)" "6#-%$sinG6#**%\"kG\"
\"\"%#PiGF(%\"aGF(%\"LG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "assume(k_,integer
):\n1/(2*L)*Int(exp(-I*k*Pi*x/L),x=-a..a);\nsubs(k_=k,value(subs(k=k_,
%)));\nc[k]=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"
\"\"\"#F&,$-%$IntG6$-%$expG6#*,^#!\"\"F&%\"kGF&%#PiGF&%\"xGF&%\"LGF1/F
4;,$%\"aGF1F9*$F5F1F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**^##\"\"\"
\"\"#F&,&-%$expG6#*,^#!\"\"F&%\"kGF&%#PiGF&%\"aGF&%\"LGF.F&-F*6#*,F/F&
F0F&F1F&F2F.^#F&F&F.F&F/F.F0F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%
\"cG6#%\"kG*(-%$sinG6#**F'\"\"\"%#PiGF-%\"aGF-%\"LG!\"\"F-F'F1F.F1" }}
}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The complex Fourier \+
series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 " is" 
}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = Sum(c[k]*exp
(k*Pi*i*x/L),k = -infinity .. infinity);" "6#/-%\"FG6#%\"xG-%$SumG6$*&
&%\"cG6#%\"kG\"\"\"-%$expG6#*,F/F0%#PiGF0%\"iGF0F'F0%\"LG!\"\"F0/F/;,$
%)infinityGF8F<" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where \+
" }{XPPEDIT 18 0 "c[0] = a/L;" "6#/&%\"cG6#\"\"!*&%\"aG\"\"\"%\"LG!\"
\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c[k] = 1/(k*Pi);" "6#/&%\"cG6#
%\"kG*&\"\"\"F)*&F'F)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin
(k*Pi*a/L) " "6#-%$sinG6#**%\"kG\"\"\"%#PiGF(%\"aGF(%\"LG!\"\"" }
{TEXT -1 5 " for " }{XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 32 "We can write the expression for " }
{XPPEDIT 18 0 "c[k]" "6#&%\"cG6#%\"kG" }{TEXT -1 13 " in the form " }
{XPPEDIT 18 0 "c[k]" "6#&%\"cG6#%\"kG" }{TEXT -1 3 " = " }{XPPEDIT 18 
0 "a/L" "6#*&%\"aG\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "si
n(k*Pi*a/L)/``(k*Pi*a/L);" "6#*&-%$sinG6#**%\"kG\"\"\"%#PiGF)%\"aGF)%
\"LG!\"\"F)-%!G6#**F(F)F*F)F+F)F,F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 
"" {TEXT -1 20 "Define the function " }{XPPEDIT 18 0 "S(x)" "6#-%\"SG6
#%\"xG" }{TEXT -1 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "S(x) = PIECEWISE([sin*x/x, x <> 0],[1, x = 0]);" "6#/-%
\"SG6#%\"xG-%*PIECEWISEG6$7$*(%$sinG\"\"\"F'F.F'!\"\"0F'\"\"!7$F./F'F1
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 36 "Then S is a continuous function and " }{XPPEDIT 18 0 "c[k
] = a/L" "6#/&%\"cG6#%\"kG*&%\"aG\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "S(k*Pi*a/L)" "6#-%\"SG6#**%\"kG\"\"\"%#PiGF(%\"aGF(%\"L
G!\"\"" }{TEXT -1 19 ", for all integers " }{TEXT 277 1 "k" }{TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "T
he complex Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(x)=a/L " "6#/
-%\"fG6#%\"xG*&%\"aG\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
Sum(S(k*Pi*a/L)*exp(k*Pi*i*x/L),k = -infinity .. infinity);" "6#-%$Sum
G6$*&-%\"SG6#**%\"kG\"\"\"%#PiGF,%\"aGF,%\"LG!\"\"F,-%$expG6#*,F+F,F-F
,%\"iGF,%\"xGF,F/F0F,/F+;,$%)infinityGF0F:" }{TEXT -1 2 "  " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The coefficient
s " }{XPPEDIT 18 0 "a[k]" "6#&%\"aG6#%\"kG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" }{TEXT -1 47 " in the correspon
ding trigonometric series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 15 " are given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "a[k] = 2*Re(c[k]);" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"-%#Re
G6#&%\"cG6#F'F*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*a/L" "6#*(\"\"#\"
\"\"%\"aGF%%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "S(k*Pi*a/L);" "
6#-%\"SG6#**%\"kG\"\"\"%#PiGF(%\"aGF(%\"LG!\"\"" }{TEXT -1 7 "  and  \+
" }{XPPEDIT 18 0 "b[k] = -2*Im(c[k])" "6#/&%\"bG6#%\"kG,$*&\"\"#\"\"\"
-%#ImG6#&%\"cG6#F'F+!\"\"" }{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "L=3" "6#/%\"LG\"\"$" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT -1 
31 ", the trigonometric series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#
%\"xG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "F(x) = 1/3" "6#/-%\"FG6#%\"xG*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 3 "
 + " }{XPPEDIT 18 0 "Sum(``(2*sin(k*Pi/3)/(k*Pi))*cos(k*Pi*x/3),k = 1 \+
.. infinity);" "6#-%$SumG6$*&-%!G6#*(\"\"#\"\"\"-%$sinG6#*(%\"kGF,%#Pi
GF,\"\"$!\"\"F,*&F1F,F2F,F4F,-%$cosG6#**F1F,F2F,%\"xGF,F3F4F,/F1;F,%)i
nfinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 2 "  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "fr
 := x -> piecewise(x<-1,0,x<1,1,0);\nf := x -> fr(x-6*floor(x/6+1/2));
\nFS := (x,n) -> 1/3+Sum(2*sin(k*Pi/3)/(k*Pi)*cos(k*Pi*x/3),k=1..n);\n
plot([f(x),FS(x,4),FS(x,20)],x=-3..9,color=[black,red,blue],\n        \+
          linestyle=[3,1],ytickmarks=[0,0.5,1]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#frGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6
'29$!\"\"\"\"!2F0\"\"\"F4F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%#frG6#,&9$\"\"\"-%&floorG6#,
&F0#F1\"\"'#F1\"\"#F1!\"'F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&#\"\"\"\"\"$F/-%$SumG6$,$
*&*&-%$sinG6#,$*&%\"kGF/%#PiGF/F.F/-%$cosG6#,$*(F<F/F=F/9$F/F.F/F/*&F<
F/F=F/!\"\"\"\"#/F<;F/9%F/F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 515 
201 201 {PLOTDATA 2 "6(-%'CURVESG6%7ip7$$!\"$\"\"!F*7$$!1+++]TVQF!#:F*
7$$!1++]-r%3^#F.F*7$$!1+++l;!\\D#F.F*7$$!1+++b'fs*>F.F*7$$!1++]s@%3u\"
F.F*7$$!1++]U.6.:F.F*7$$!1++]-G&pD\"F.F*7$$!1++]iXmH6F.F*7$$!1++]AjP-5
F.F*7$$!1+]P4KYW**!#;$\"\"\"F*7$$!1++v$>j^')*FJFK7$$!1**\\7yJ'ey*FJFK7
$$!1****\\iJc1(*FJFK7$$!1***\\78jza*FJFK7$$!1++++JO*Q*FJFK7$$!1,+]PI;s
!*FJFK7$$!1+++vH'\\v)FJFK7$$!1+++]Gc?\")FJFK7$$!1,++DF;'[(FJFK7$$!1++]
7d/\"='FJFK7$$!1)******pGf([FJFK7$$!1+++]J$od#FJFK7$$\"18-+++RS6!#=FK7
$$\"1*******\\o-h#FJFK7$$\"1-+++hv9^FJFK7$$\"1,+]PM\">D'FJFK7$$\"1+++v
22*Q(FJFK7$$\"1++D1e;l!)FJFK7$$\"1++]P3ET()FJFK7$$\"1+]7`$3$z!*FJFK7$$
\"1++voeN<%*FJFK7$$\"1+DcE'zje*FJFK7$$\"1,]P%Q.av*FJFK7$$\"1^7Gj_\"*R)
*FJFK7$$\"1,v=UrUC**FJFK7$$\"1v$4@!R*3+\"F.F*7$$\"1+++!4X$45F.F*7$$\"1
+++DL*Q7\"F.F*7$$\"1+++g:WQ7F.F*7$$\"1++]<_$\\]\"F.F*7$$\"1+++gs#3u\"F
.F*7$$\"1++]<#Q'**>F.F*7$$\"1++]_u3YAF.F*7$$\"1+++v8B.DF.F*7$$\"1++]n(
p$RFF.F*7$$\"1+++Dp2%*HF.F*7$$\"1++]xgkeKF.F*7$$\"1++]-V&*)[$F.F*7$$\"
1+++&\\$pPPF.F*7$$\"1+++?am%*RF.F*7$$\"1+++:B1YUF.F*7$$\"1++]P<I*[%F.F
*7$$\"1++D\"pRVi%F.F*7$$\"1+++XwPfZF.F*7$$\"1++v[k/?[F.F*7$$\"1++]__r!
)[F.F*7$$\"1+]Pa'\\5\"\\F.F*7$$\"1++DcSQT\\F.F*7$$\"1*\\(=d7bc\\F.F*7$
$\"1**\\7e%=<(\\F.F*7$$\"1\\Pfe?Iz\\F.F*7$$\"1+D1fc)o)\\F.F*7$$\"1\\7`
f#pW*\\F.F*7$$\"1******fG0-]F.FK7$$\"1+++bmgJ^F.FK7$$\"1+++]/;h_F.FK7$
$\"1****\\P/&f\\&F.FK7$$\"1,++5zj_dF.FK7$$\"1****\\<3;%*fF.FK7$$\"1++]
Z=iYiF.FK7$$\"1******\\'[M\\'F.FK7$$\"1***\\P/^Ei'F.FK7$$\"1****\\PM&=
v'F.FK7$$\"1+]7o?29oF.FK7$$\"1++v)p!HwoF.FK7$$\"1+D19+S2pF.FK7$$\"1+]P
H$4&QpF.FK7$$\"1]7.()R1apF.FK7$$\"1+voW'='ppF.FK7$$\"1Dc^tfRxpF.FK7$$
\"1^PM-L<&)pF.FK7$$\"1w=<J1&H*pF.FK7$$\"1,++gzs+qF.F*7$$\"1++]KI)z7(F.
F*7$$\"1+++0\"Q_D(F.F*7$$\"1,+]x2k2vF.F*7$$\"1,++?EdRxF.F*7$$\"1+++&o#
R0!)F.F*7$$\"1+++?`9V#)F.F*7$$\"1++]<#Rm\\)F.F*7$$\"1++]A_ER()F.F*7$$
\"\"*F*F*-%'COLOURG6&%$RGBGF*F*F*-%*LINESTYLEG6#\"\"$-F$6%7cv7$F($!1k5
IBQL;!)!#<7$F,$!1([g'f\\G<DF\\^l7$$!1++DE1kCEF.$\"15*QuR?O!>F\\^l7$F0$
\"1Byx(*G6TfF\\^l7$$!1+]7V2'oW#F.$\"1<wIbo4(f(F\\^l7$$!1++v$QuGQ#F.$\"
1+2;%[CRf)F\\^l7$$!1+D1/7)3N#F.$\"1t_VKP*p!))F\\^l7$$!1+]PC!)))=BF.$\"
1?;j(>9o\"))F\\^l7$$!1+voW[*oG#F.$\"1?-!zg\"p=')F\\^l7$F3$\"1vo+7\"3A@
)F\\^l7$$!1+++g13E@F.$\"1/A#>&pL9YF\\^l7$F6$!1an24)\\=E\"F\\^l7$$!1++v
840p=F.$!1KjtUMeCvF\\^l7$F9$!1.%H`pE->\"FJ7$$!1+]7LR)fs\"F.$!1k#*34.S=
7FJ7$$!1++v$pD6r\"F.$!1=EhmdwS7FJ7$$!1+]PauE'p\"F.$!1SfZbK/d7FJ7$$!1++
+:#49o\"F.$!1Z!f\"p%opE\"FJ7$$!1++DOFp^;F.$!1c#G<I)ym7FJ7$$!1++]di(>i
\"F.$!15k]\"Qd%Q7FJ7$$!1++++Lai:F.$!1\\jqbJ#=4\"FJ7$F<$!1D_0@X:%>)F\\^
l7$$!1++]d4dT9F.$!1u]VFu7GSF\\^l7$$!1++]s:.!Q\"F.$\"1]^c-GXh9F\\^l7$$!
1++](=#\\=8F.$\"1$fZ68.N;)F\\^l7$F?$\"1sj69lp!f\"FJ7$$!1++]#o3L>\"F.$
\"1?V:(HluZ#FJ7$FB$\"1-1I**3:BMFJ7$$!1++]U/-m5F.$\"1x7W7+S'R%FJ7$FE$\"
1vr`0l9l`FJ7$FZ$\"1Jc*))4:cH'FJ7$Fjn$\"1+@1]YKjrFJ7$F]o$\"1y!4V]m^%zFJ
7$F`o$\"1A8(yJ]Pi)FJ7$Fco$\"1*y3:v_ll*FJ7$Ffo$\"1\"*z*)R-&4-\"F.7$$!1*
**\\ivT&)e%FJ$\"1B)49=@v-\"F.7$$!1****\\7[:,VFJ$\"1qp&[=\"RK5F.7$$!1**
*\\(oyw8SFJ$\"1OJ8C_tN5F.7$$!1*****\\#4QEPFJ$\"1F4%G`Px.\"F.7$$!1***\\
7)R**QMFJ$\"1RR.`$)eQ5F.7$$!1****\\Pqg^JFJ$\"1>uwG1[Q5F.7$$!1++v$4?U'G
FJ$\"1_*)4%p1w.\"F.7$Fio$\"1&*[:0]:O5F.7$$!1*****\\i9FG\"FJ$\"1e`^i91F
5F.7$F\\p$\"1(zXd%[\\A5F.7$$\"1,+++i$3J\"FJ$\"1jI!QGXs-\"F.7$F`p$\"1dA
ciwMO5F.7$$\"1+++]%HL#HFJ$\"1m-UU(Ry.\"F.7$$\"1++++/ROKFJ$\"1\"*=mv1gQ
5F.7$$\"1+++]8X\\NFJ$\"1uNv-`QQ5F.7$$\"1,+++B^iQFJ$\"1gp%RNWp.\"F.7$$
\"1,+++Uj)[%FJ$\"1b/R)Q)RH5F.7$Fcp$\"1aa)*4$yS,\"F.7$Ffp$\"1y4jWfE8'*F
J7$Fip$\"1^(z<w8xr)FJ7$F\\q$\"1Wi+x+u3!)FJ7$F_q$\"1JP?%3#>\"=(FJ7$Feq$
\"1%35&HLpbiFJ7$Fgr$\"13%G71g-E&FJ7$$\"1++]2#>m1\"F.$\"1O)[qr9sQ%FJ7$F
jr$\"1pRAi)\\2^$FJ7$$\"1++]Uu;\"=\"F.$\"1Dc@9'fUl#FJ7$F]s$\"1k42MZyS=F
J7$$\"1+]Pu\\108F.$\"1d#*>'\\)=r(*F\\^l7$$\"1++v)Q)or8F.$\"1*p$*4&HC-B
F\\^l7$$\"1+]7.=JQ9F.$!1+U2yf))pPF\\^l7$F`s$!1_[#4$)\\oH)F\\^l7$$\"1+]
7G#3Rc\"F.$!1nl?*4$e'4\"FJ7$$\"1++vQ7)Gi\"F.$!1hHtL2vR7FJ7$$\"1+D1WxO_
;F.$!1s;U;()4n7FJ7$$\"1+]P\\U&=o\"F.$!1@7Bxewm7FJ7$$\"1+voa2M6<F.$!1lO
Z4[[S7FJ7$Fcs$!10HW'yd->\"FJ7$$\"1++vQFBq=F.$!1B>f()*[=Z(F\\^l7$Ffs$!1
&Rbcs8G9\"F\\^l7$$\"1+++NG'G7#F.$\"1SDYhm9*[%F\\^l7$Fis$\"1#f7E+KS1)F
\\^l7$$\"1+D\"GWI#yAF.$\"1Uub\\\\4H&)F\\^l7$$\"1+]7LMP5BF.$\"1M\"4\"*
\\([%y)F\\^l7$$\"1+vVBk^UBF.$\"1bfk(G\\%H))F\\^l7$$\"1++v8%fYP#F.$\"1q
L^,/hn')F\\^l7$$\"1+]P%RX*QCF.$\"1Jd2af**exF\\^l7$F\\t$\"1.>n6DtnhF\\^
l7$$\"1++Dr0I@EF.$\"1Y07!=*oL?F\\^l7$F_t$!1!eEVxn<b#F\\^l7$$\"1+](obYI
!GF.$!19JbB?dHZF\\^l7$$\"1++DYLsmGF.$!1r9Cr?cakF\\^l7$$\"1+v$4uh&)*GF.
$!1Uu&y$Gs*4(F\\^l7$$\"1+]iN,SIHF.$!1V$yf]-2e(F\\^l7$$\"1+DJI&QA'HF.$!
1Jiq'GOt)yF\\^l7$Fbt$!1>w]dB:8!)F\\^l7$$\"1+D1p![r-$F.$!1/*=2@z&\\zF\\
^l7$$\"1+]78#>-1$F.$!1!o=&3f_*o(F\\^l7$$\"1+v=d.H$4$F.$!1k8F4?\"*QsF\\
^l7$$\"1++D,:OEJF.$!1u=9En+3mF\\^l7$$\"1+]P*y.D>$F.$!1nq&yStn'[F\\^l7$
Fet$!1A5&o:-Zi#F\\^l7$$\"1+++!>+QP$F.$\"1wBUWhyU=F\\^l7$Fht$\"1xj=#y2^
$fF\\^l7$$\"1+]i+\"R6b$F.$\"1v/lQ(oXb(F\\^l7$$\"1++v)*QK8OF.$\"1i%p0)o
Yb&)F\\^l7$$\"1+D\"yH;Wk$F.$\"1tn2;7D)y)F\\^l7$$\"1+](op3bn$F.$\"1+h%G
>O+$))F\\^l7$$\"1+v$f4,mq$F.$\"17HBCD*en)F\\^l7$F[u$\"1#f**pP%oC$)F\\^
l7$$\"1++]d%zh'QF.$\"1Wt_([*Q4\\F\\^l7$F^u$!19`Zh1]%e)F^p7$$\"1++]nQO?
TF.$!1#GrJ7Pn/(F\\^l7$Fau$!1G\"Rh093;\"FJ7$$\"1*\\7GCnkF%F.$!1!=Hnn(\\
A7FJ7$$\"1+]iq@(oI%F.$!1W!H%=%y'f7FJ7$$\"1+vV)4xsL%F.$!1\"Q$*f__+F\"FJ
7$$\"1++DE?onVF.$!1\\cWj'3;D\"FJ7$$\"1+](=)=\\GWF.$!1>Sf(fr=7\"FJ7$Fdu
$!1!*\\@O&GKh)F\\^l7$$\"1+]P92#ob%F.$!1,(pw\\\">aTF\\^l7$Fgu$\"14ye:??
**=F\\^l7$$\"1,]7o'e=p%F.$\"1Pyr(ff&)R*F\\^l7$Fju$\"1kmWQq&4\"=FJ7$F]v
$\"1Y;>\"zX?n#FJ7$F`v$\"1%)**He$[3e$FJ7$Ffv$\"1S08,g%*4XFJ7$Fhw$\"1^A%
eJ!fJaFJ7$$\"1****\\d(Ho1&F.$\"1\"4%y$yduP'FJ7$F[x$\"1.Std\\]bsFJ7$$\"
1++]_NQ'>&F.$\"1*3Tx<.</)FJ7$F^x$\"1R2$4\"yN=()FJ7$$\"1***\\PWb&y`F.$
\"1#=!H&*=LO'*FJ7$Fax$\"1I6!fDej,\"F.7$$\"1)\\i:PO!GbF.$\"1!Qx@([uC5F.
7$$\"1**\\i0B7gbF.$\"1vb)H-<4.\"F.7$$\"1*\\(oR#3Af&F.$\"1&z&yG<6N5F.7$
$\"1++vtTHCcF.$\"1ex)G2#eP5F.7$$\"1,D\"y5!QccF.$\"1W4*zg\"fQ5F.7$$\"1+
](=/m%)o&F.$\"1*Q1Yj3%Q5F.7$$\"1+v$f(>b?dF.$\"1a*\\%e/IP5F.7$Fdx$\"16;
-Mx_N5F.7$$\"1++vj$*RteF.$\"1_%p2T`p-\"F.7$Fgx$\"1=s2eZ]A5F.7$$\"1++]K
8R?hF.$\"1G\"\\i%)fl-\"F.7$Fjx$\"1c]E,5[N5F.7$$\"1+D\"y>vuF'F.$\"1#fhq
-3s.\"F.7$$\"1+]7[&G$3jF.$\"1\\EHd^LQ5F.7$$\"1+vV)*==RjF.$\"1f`v$GJ'Q5
F.7$$\"1++v[_.qjF.$\"1Uq*Q8fy.\"F.7$$\"1*\\i!*f))3S'F.$\"19C?3*zd.\"F.
7$$\"1**\\P\\>uJkF.$\"1g-R(*z:K5F.7$$\"1)\\(o*H&fikF.$\"1.d+)pln-\"F.7
$F]y$\"1gF$[%yQ>5F.7$F`y$\"1gE0*p[*G'*FJ7$Fcy$\"1wTquK#=f)FJ7$Ffy$\"1W
`x%\\O=#zFJ7$Fiy$\"1V\\c'GZH:(FJ7$F_z$\"1\\&\\Yr$p,jFJ7$Fa[l$\"1>Hi&*[
*)*Q&FJ7$$\"1++D'\\bV1(F.$\"1\"ol8tQ>U%FJ7$Fd[l$\"1x=.*o^'[MFJ7$$\"1,+
vo0h\">(F.$\"1))Q.Wx1-DFJ7$Fg[l$\"1wUZ6&fNh\"FJ7$$\"1+]7t(Q$=tF.$\"1EF
u3[f\"=)F\\^l7$$\"1++DT%R9Q(F.$\"1pnD)pq<K\"F\\^l7$$\"1,]P4,aWuF.$!1ps
*HiX,E%F\\^l7$Fj[l$!1.8Yf5)oW)F\\^l7$$\"1,]7QPilvF.$!1x8\"\\GrC5\"FJ7$
$\"1,+v)p1Oi(F.$!1!*Gm(pn2C\"FJ7$$\"1,D1z\")f_wF.$!1QG#4O,sE\"FJ7$$\"1
,]Pf'*e\"o(F.$!16/PRq)oE\"FJ7$$\"1+voR6e5xF.$!1&H3$p9ZT7FJ7$F]\\l$!16y
LiO&G>\"FJ7$$\"1++]_E[syF.$!1^T\"\\U;5P(F\\^l7$F`\\l$!1Tw>a5bb&)F^p7$$
\"1++]-!pU7)F.$\"10A\"fXLSa%F\\^l7$Fc\\l$\"197Z&*f86!)F\\^l7$$\"1*\\(=
d?$[F)F.$\"1w?*)RJy*[)F\\^l7$$\"1+]P%z=lI)F.$\"1tR485*\\w)F\\^l7$$\"1,
DcJb?Q$)F.$\"1%4g=&3bN))F\\^l7$$\"1++voA*)p$)F.$\"1qw'>E8Wq)F\\^l7$$\"
1+]7VdEL%)F.$\"1b.r,KwoyF\\^l7$Ff\\l$\"1K,(eVozN'F\\^l7$$\"1******>A&z
h)F.$\"1(oM;`8P;#F\\^l7$Fi\\l$!1)f+%ov\"za#F\\^l7$F\\]lFj]l-F_]l6&Fa]l
$\"*++++\"!\")F*F*-Fc]l6#FL-F$6%7gdl7$F($\"1*[a\"4:6Q()F^p7$$!1++]P&3Y
$HF.$\"1C<xC!>(R6F^p7$$!1+++vq@pGF.$!1GS=[0e%\\)F^p7$$!1+]P4#pG&GF.$!1
ZKZK')Rk()F^p7$$!1++vV8_OGF.$!1\\<r4by5zF^p7$$!1+]7yM<?GF.$!1J.XW)Q(Qg
F^p7$$!1++]7c#Q!GF.$!1aqWT)pNQ$F^p7$$!1++D\"))H6x#F.$\"1(p/*pDdsGF^p7$
F,$\"1Q)frO'RlxF^p7$$!1+D1pd)*4FF.$\"1:[\"pAhI'*)F^p7$$!1+]7)QP:o#F.$
\"15c%4t!G`nF^p7$$!1+v=2!*3`EF.$\"1`^T3pRL>F^p7$Fa^l$!12_dwo5(p$F^p7$$
!1+DJXA>'f#F.$!1S#p7r6R+)F^p7$$!1+]PkQunDF.$!1*4PK*Q^@$*F^p7$$!1]i!Rn>
Nb#F.$!1x%R4EN@j)F^p7$$!1+vV$[&HRDF.$!17#fmNVE4(F^p7$$!1](oHHr]_#F.$!1
NfevCgW[F^p7$F0$!1Z;\\Y@y*4#F^p7$F3$!1zblVz'oB)F^p7$F6$!1U$R4%z[/8F\\^
l7$$!1+]P%GbJ$>F.$!1*=twM0$ex!#>7$F]al$\"1<jv!z#ec9F\\^l7$$!1+]7Vl%\\!
=F.$\"1$oOvi?Tx&F^p7$F9$!1NzxgET2:F\\^l7$Feal$!1*Q0)HR7U<F\\^l7$Fjal$!
1S2j$4loz\"F\\^l7$F_bl$!1)fb92f!e;F\\^l7$Fdbl$!1Z&3s!RDK8F\\^l7$Fibl$!
1J$=v<QUX#F^p7$F^cl$\"1EvC\")y!Q0\"F\\^l7$$!1+]7=!=rg\"F.$\"1>,)*R:Q8;
F\\^l7$$!1++vy(fAf\"F.$\"1A%)Q&f#\\C?F\\^l7$$!1+]PR:Sx:F.$\"1\\0S>)*pM
AF\\^l7$Fccl$\"1<C%\\D(H5AF\\^l7$$!1++D@o#G`\"F.$\"1nR')[6tT9F\\^l7$F<
$!1i!3:YJ/)fF]eo7$$!1+++]1Ms9F.$!1uM8#))HEy\"F\\^l7$F[dl$!1R`?wwH'*GF
\\^l7$$!1++D6h=E9F.$!1yfx'Q2Z,$F\\^l7$$!1+++l7!3T\"F.$!1a352\\GzFF\\^l
7$$!1++v=kT&R\"F.$!1\"H4Fo;$*=#F\\^l7$F`dl$!1&*)QmrV[G\"F\\^l7$$!1+++!
)=E\\8F.$\"1$3sE/(y96F\\^l7$Fedl$\"1s`%*yC^LMF\\^l7$$!1++DTt5.8F.$\"1)
H8,!)zf>%F\\^l7$$!1+++&\\AxG\"F.$\"1r'>jMV$>XF\\^l7$$!1+](=2I+G\"F.$\"
1z^%)o%)\\'[%F\\^l7$$!1++v[wLs7F.$\"1(Qou!G\"QJ%F\\^l7$$!1+]iD_kk7F.$
\"1\"e_CSVz*RF\\^l7$F?$\"1%y-JMC$RNF\\^l7$F]el$!1Z!QDCY0=%F\\^l7$FB$!1
*y`()e*QI\")F\\^l7$$!1++]KNv86F.$!1o(HsD')zJ'F\\^l7$$!1++]-D%y4\"F.$!1
@FxHf)=$HF\\^l7$$!1++]s9$>3\"F.$\"1\"[P)f\"RV6#F\\^l7$Feel$\"1jAXp/k,)
)F\\^l7$$!1++]7%4,0\"F.$\"1]!y4>D,q\"FJ7$$!1++]#Q)>M5F.$\"1_F7t>%zk#FJ
7$$!1++]nGCE5F.$\"123h\\q$*fJFJ7$$!1++]_tG=5F.$\"1aS!p\"Q4\"p$FJ7$$!1+
+]P=L55F.$\"1?'Q;TpiB%FJ7$FE$\"1!\\=,v^+z%FJ7$FH$\"1uvJ&*>2X`FJ7$FN$\"
1$z>T$R\\(*eFJ7$FQ$\"1g3([5Z<W'FJ7$FT$\"1h!GB#HRspFJ7$FW$\"1!))>im`B(z
FJ7$FZ$\"1yn#o5:,'))FJ7$$!1,+voIwI#*FJ$\"1:SJ^n<1'*FJ7$Fgn$\"1n1.S%y!>
5F.7$$!1++D1Ic8*)FJ$\"133=)*3^g5F.7$Fjn$\"1fZLQD:&3\"F.7$F]o$\"1!e@%fo
*R0\"F.7$F`o$\"1aRj#e,Bi*FJ7$$!1+vV)4BIK(FJ$\"1S_)**G()G]*FJ7$$!1+](=Z
$))frFJ$\"1u(pa2pzX*FJ7$$!1^PfeOJyqFJ$\"1ewQxS;j%*FJ7$$!1,DJXQu'*pFJ$
\"1\"o%ernd&[*FJ7$$!1]7.KS<:pFJ$\"1Y0:<kqB&*FJ7$$!1++v=UgLoFJ$\"1DmS,_
lv&*FJ7$$!1+]il\\K2lFJ$\"11t)*=E)Q()*FJ7$Fco$\"1W1Z#yl*=5F.7$$!1*\\Pf3
1z,'FJ$\"1KtkCycI5F.7$$!1**\\PfkwaeFJ$\"12\\.%*RgP5F.7$$!1+D\"G$oi\"p&
FJ$\"1c%fK<#\\R5F.7$$!1***\\i?([GbFJ$\"1,LmD6IO5F.7$$!1**\\7`z?-_FJ$\"
10]\"H3uy,\"F.7$Ffo$\"1P5X?\")RE**FJ7$F_gl$\"1GVJ;$Ruu*FJ7$Fdgl$\"1op*
=f0gn*FJ7$Figl$\"1s*)Q/p_J(*FJ7$F^hl$\"1m'Ggq'e&))*FJ7$Fchl$\"1'3;f94v
+\"F.7$Fhhl$\"1;]a8brA5F.7$$!1**\\ilN\"z+$FJ$\"1!QRg`\\q-\"F.7$F]il$\"
1_3/ejiG5F.7$$!1+](=iE0s#FJ$\"1zSSqpMF5F.7$Fio$\"1><I$)**QB5F.7$$!1++]
()QxH>FJ$\"15O$GM(y\"))*FJ7$Feil$\"1K.h(3Fku*FJ7$$!1&)***\\i`lN'F\\^l$
\"1_7\"RRbV+\"F.7$F\\p$\"1Q;cS#Hh-\"F.7$$\"19+++0?6mF\\^l$\"1qwkaS*G+
\"F.7$F]jl$\"1];f;2qT(*FJ7$$\"1+++]Bbg>FJ$\"1m*eEF'f)*)*FJ7$F`p$\"1Gwi
A2`C5F.7$$\"1*****\\(*)zmFFJ$\"1wi$GJl!G5F.7$Fejl$\"1(4,%[UKG5F.7$$\"1
+++D*f)zIFJ$\"1%R$3aq?D5F.7$Fjjl$\"1zgq5_+>5F.7$F_[m$\"1a/YX<G+5F.7$Fd
[m$\"1z\\8#4wP!)*FJ7$$\"1,++vF/>SFJ$\"1)Get(zYH(*FJ7$$\"1,++]KdvTFJ$\"
1Vmrz\\H&o*FJ7$$\"1,++DP5KVFJ$\"1a:fUZ_x'*FJ7$Fi[m$\"1![JK?#Q3(*FJ7$$
\"1-++]^p,[FJ$\"19UC>wws)*FJ7$Fcp$\"1SI>M'\\7,\"F.7$$\"1-]PMa/*R&FJ$\"
1)zY]@%eI5F.7$$\"1,+voZL$o&FJ$\"1_uMd\"\\%R5F.7$$\"1,]7.TinfFJ$\"1o\\L
!4jK.\"F.7$Ffp$\"1o$[RM(y75F.7$$\"1+](=x-i`'FJ$\"1g!pNg%GW)*FJ7$$\"1++
D1@\\?oFJ$\"1bI?8&\\^e*FJ7$$\"1,vVtnjipFJ$\"1E(4g%*3(*\\*FJ7$$\"1+]iS9
y/rFJ$\"1K'Re;J&f%*FJ7$$\"1](=Ux`e<(FJ$\"1D\"p2Al!f%*FJ7$$\"1*\\7y5EpC
(FJ$\"1q'*H!pzDZ*FJ7$$\"1\\iST%)*zJ(FJ$\"117c,'z.]*FJ7$Fip$\"1'HT'o0XU
&*FJ7$F\\q$\"1l(p(*e@a/\"F.7$F_q$\"1L;O%RGl3\"F.7$$\"1+DJ&f%G5*)FJ$\"1
k!=k[(=h5F.7$Fbq$\"1'*))[c%3o,\"F.7$$\"1,v$46K$[#*FJ$\"1#)H**pU@J&*FJ7
$Feq$\"1oIlF/+8()FJ7$$\"1]ilZx'=]*FJ$\"1'))>[O8MC)FJ7$Fhq$\"1*4p-kL\"R
xFJ7$$\"1^(oa]\"*3n*FJ$\"1wbMR#f_?(FJ7$F[r$\"15m*RUsuk'FJ7$F^r$\"1yf/N
_#>2'FJ7$Far$\"1H&f_-9^[&FJ7$Fdr$\"1l/w(3YP*[FJ7$Fgr$\"1#3*oS;g/VFJ7$$
\"1+]P>OmB5F.$\"1(\\8S$eGILFJ7$$\"1++v[@)z.\"F.$\"13eKbTb7CFJ7$$\"1+]7
y1I_5F.$\"1A%4SZy*y:FJ7$Fc]m$\"1lV'H0e9_)F\\^l7$$\"1+](otP43\"F.$\"1K!
\\2uPZ[#F\\^l7$$\"1++DmiD&4\"F.$!1/g'*RXYDAF\\^l7$$\"1+]i&zu&46F.$!1S6
3GNO'e&F\\^l7$Fjr$!1Jorh$p`k(F\\^l7$F[^m$!1J-uwlw-eF\\^l7$F]s$\"1gr^z$
\\f)=F\\^l7$$\"1]Pf8u4b7F.$\"1S.sCpu2MF\\^l7$$\"1+v=nKvr7F.$\"1G]^!Gg[
H%F\\^l7$$\"1]7y?\"4%)G\"F.$\"1%y@^$Ho:XF\\^l7$Fc^m$\"1CSELg+@TF\\^l7$
$\"1+Dc\"ow$Q8F.$\"1%GlMi$y/?F\\^l7$Fh^m$!1$Q#oz\"3e*oF^p7$$\"1]PMUUM)
Q\"F.$!1wx.<\"o'3=F\\^l7$$\"1+v$f4+]S\"F.$!1C$Hoo0sf#F\\^l7$$\"1]7`\\f
l@9F.$!1?^`v_$H)HF\\^l7$F]_m$!1v\")>r9`\\HF\\^l7$$\"1+DJ5Nir9F.$!1zs%p
$f_==F\\^l7$F`s$\"1VoaT?$fU%F]eo7$$\"1+D\"Gs@W`\"F.$\"1&34G!yG0:F\\^l7
$Fe_m$\"1!4t[X$yAAF\\^l7$$\"1]7y![^'y:F.$\"1wsd</tDAF\\^l7$$\"1+vVLZR$
f\"F.$\"1nm&4#Hh**>F\\^l7$$\"1]P4')z83;F.$\"15$[L2G!z:F\\^l7$Fj_m$\"11
$Gy-Gj,\"F\\^l7$F_`m$!1&G0\"R?8WFF^p7$Fd`m$!1?T\"4'4_W8F\\^l7$Fi`m$!1v
tEbRW(z\"F\\^l7$Fcs$!1:EKb#Hx]\"F\\^l7$$\"1+voH'yJx\"F.$!1?/)y2+sX&F^p
7$$\"1+]P***Hb!=F.$\"1`;OO7?ifF^p7$$\"1](=Uo0<#=F.$\"18o'4l.\"e5F\\^l7
$$\"1+D1p8)y$=F.$\"1n2Bgs!=P\"F\\^l7$$\"1]i!R0dS&=F.$\"1cOufUM/:F\\^l7
$Faam$\"1p!4.iIgW\"F\\^l7$$\"1+]7ya$\\$>F.$!1%eO9)Q4>8F^p7$Ffs$!1O_?ZK
$oI\"F\\^l7$Fis$!1xC[y1GlnF^p7$F\\t$!1%R1eW]s:&F]eo7$$\"1+D1u'[F`#F.$!
1QAFa`gMhF^p7$$\"1+]7tfEiDF.$!1l]?!\\NB;*F^p7$$\"1+v=sKy\"f#F.$!1dU&)Q
=6L%)F^p7$F]dm$!1v_(o#pu5VF^p7$$\"1+DJqy\"3l#F.$\"1(f]6Gga[\"F^p7$$\"1
+]Pp^L!o#F.$\"1Bdz;6[#f'F^p7$$\"1+vVoC&)4FF.$\"1=U)e8C0'*)F^p7$F_t$\"1
R,!y1g-n(F^p7$$\"1+v=i\"37x#F.$\"1CV+EQ*z&GF^p7$Fedm$!1p'pW00HC$F^p7$$
\"1](=Uvl*=GF.$!1p'yD\\hd'eF^p7$$\"1+Dc^\\)[$GF.$!1jMCS+LmxF^p7$$\"1]i
!*[T!3&GF.$!1;A3$Gfxr)F^p7$Fjdm$!1&)Q7C_[3')F^p7$Fdem$\"1KK\\q^\"yJ$F]
eo7$Fbt$\"1L5[9s(Rm)F^p7$Fffm$\"1OuS4\"f\"=@F^p7$F`gm$!179aX;;I#)F^p7$
$\"1+DJXEVfJF.$!1ai,*)QKB#)F^p7$Fegm$!1-qXr_%=.%F^p7$$\"1+vVL\\dDKF.$
\"1MVit))Q_AF^p7$Fet$\"1.ai3:\"zX(F^p7$$\"1+]i0YV(G$F.$\"1P'*Qg6#p**)F
^p7$$\"1++vLJA;LF.$\"1Yxhxu))RqF^p7$$\"1+](=m6]M$F.$\"1+F>'yPRI#F^p7$F
]hm$!1%fu=G1SS$F^p7$$\"1+]7=()e-MF.$!14cplK(>(yF^p7$$\"1++DYsPJMF.$!1v
W:M4WM$*F^p7$$\"1+DJ5:xXMF.$!1Gd))y)[uo)F^p7$$\"1+]Pud;gMF.$!15m.G?*\\
;(F^p7$$\"1+vVQ+cuMF.$!1VH^0/=5\\F^p7$Fht$!1LfIX9PS@F^p7$F[u$!1$)R0Vy(
\\A*F^p7$F^u$!1qu>)>\")yH\"F\\^l7$$\"1++vVY^dSF.$!1Zq=3aapNF^p7$Fijm$
\"19qw$*H+H8F\\^l7$$\"1+vVth2OTF.$\"13/qoP4\"\\\"F\\^l7$$\"1+]Pz%)y^TF
.$\"1p'yOj![y9F\\^l7$$\"1+DJ&y+v;%F.$\"1.@E)*HP'G\"F\\^l7$$\"1++D\"48K
=%F.$\"1y\"[aR!R7$*F^p7$$\"1+]7.xj9UF.$!18o'z(*)e`5F^p7$Fau$!1$[A5/ng<
\"F\\^l7$$\"1\\i!*yZEhUF.$!1k>H35j]:F\\^l7$Fa[n$!1.DI/,:k<F\\^l7$$\"1]
(=nqp;H%F.$!1I#H-[;cy\"F\\^l7$Ff[n$!1\"\\]kc#>/;F\\^l7$F[\\n$!14bJy/0.
qF^p7$F`\\n$\"1!zN#Q6t3hF^p7$$\"1*\\iS&p3)R%F.$\"1%*>;n*4tx\"F\\^l7$Fe
\\n$\"1CNy)QIUD#F\\^l7$$\"1+vo4o*)eWF.$\"1*3lwYgiu\"F\\^l7$Fdu$\"1%4@C
k&*fm$F^p7$$\"1+v$fAhI_%F.$!14b7A='[a\"F\\^l7$F]]n$!1'f2$*pOZ'GF\\^l7$
$\"1]Pfe/qtXF.$!1lmyrP5:IF\\^l7$$\"1+D\"G?!e!f%F.$!1)y%G2a\\SFF\\^l7$$
\"1]7.Z*fug%F.$!1fgI?FCU?F\\^l7$Fgu$!1pi%z_Oj!)*F^p7$$\"1+voz\"*4eYF.$
\"1rT,Piy?<F\\^l7$Fe]n$\"1a[83AH*)RF\\^l7$$\"1wVBS&)H+ZF.$\"1]AMinA4VF
\\^l7$$\"1^PM7%Q(3ZF.$\"1kBn/oC*[%F\\^l7$$\"1EJX%Gyrr%F.$\"1gk[#)pH9XF
\\^l7$$\"1,Dcc\"=cs%F.$\"179V%[*ftVF\\^l7$$\"1^7y+z\\UZF.$\"1O@U&)\\tw
NF\\^l7$Fju$\"1uDU&GAz6#F\\^l7$$\"1+](o/7(*y%F.$!1RES/%yxu\"F\\^l7$F]v
$!1@7$*[iD`fF\\^l7$$\"1+vo\\O@N[F.$!1%[NE7@!fvF\\^l7$$\"1+]i]3Q][F.$!1
Hf6!zC9Y)F\\^l7$$\"1]P4^W'z&[F.$!1[(fY=@Sc)F\\^l7$$\"1+Dc^![b'[F.$!1C#
e%o*=DR)F\\^l7$$\"1]7._;8t[F.$!1S2LFQ.>zF\\^l7$F`v$!1<N\"pSQ\">rF\\^l7
$$\"1*\\PMX#)e*[F.$!1hV&*3%=IY%F\\^l7$Fcv$!1+O^6agwJF^p7$$\"1+DJbo@E\\
F.$\"1KcAcMcR`F\\^l7$Ffv$\"1v;#zMATC\"FJ7$Fiv$\"1$e0/wCI3#FJ7$F\\w$\"1
MGG\\(4n-$FJ7$F_w$\"1\"Gm+[E%GNFJ7$Fbw$\"15N@x(HV/%FJ7$Few$\"186P<yzpX
FJ7$Fhw$\"1h\")[4!Q+5&FJ7$$\"1*\\(=s*\\,,&F.$\"1`/[9g$fm&FJ7$$\"1**\\P
%3Z#=]F.$\"1/kgr%[dA'FJ7$$\"1+Dc'>Wj-&F.$\"1VDKAehtnFJ7$$\"1***\\(38WM
]F.$\"1$e3ue')QI(FJ7$$\"1**\\7Lbj]]F.$\"1f'zzf/2H)FJ7$Fi^n$\"1(pcFN6!
\\\"*FJ7$$\"1**\\(=)R-$3&F.$\"1))H%e=u3&)*FJ7$$\"1***\\i?=#*4&F.$\"1;0
\"zC_z.\"F.7$$\"1+]iICT:^F.$\"1Yn`5*GI2\"F.7$F[x$\"16luk;0\"4\"F.7$Fa_
n$\"1q\"Q$4h!3/\"F.7$F^x$\"1jw43X)>a*FJ7$$\"1v$4rj(\\o_F.$\"1q&y'>'G*)
\\*FJ7$$\"1](=U#[$eF&F.$\"1&[2w$z3r%*FJ7$$\"1D\"G8,sJG&F.$\"1NF'yad%e%
*FJ7$$\"1*\\P%)>40H&F.$\"11.XD7ng%*FJ7$$\"1\\ilsN=0`F.$\"1h3=`d]1&*FJ7
$$\"1+](o%z&)>`F.$\"1(=KgHu%*f*FJ7$$\"1+DJ&p1#\\`F.$\"1f,rbwDt)*FJ7$Fi
_n$\"1;1'==Eh,\"F.7$$\"1\\(oz\")HKR&F.$\"1Qy/9mYF5F.7$$\"1*\\(=#>/zS&F
.$\"1'=Ztf^`.\"F.7$$\"1]iSm&yDU&F.$\"1IT#o]!>R5F.7$$\"1**\\iSHDPaF.$\"
1.0LC%=)Q5F.7$$\"1)\\i!*o,mY&F.$\"10Ypz:$o-\"F.7$Fax$\"1%4rXH>a+\"F.7$
Fa`n$\"1R0%eZ*o=)*FJ7$Ff`n$\"1tO>z:(fo*FJ7$$\"1\\ils_;wbF.$\"1Z%z,'[ex
'*FJ7$F[an$\"1x*)*f*)**)3(*FJ7$$\"1](=n?^#3cF.$\"1Os1bAyu(*FJ7$F`an$\"
1)[EGShh')*FJ7$Fean$\"11%R')G'o25F.7$Fjan$\"1I3aiN0C5F.7$$\"1]i!*3!4Xq
&F.$\"1ErphM'z-\"F.7$F_bn$\"1[YUkuNG5F.7$$\"1](oH%\\fOdF.$\"1\\8Y`RED5
F.7$Fdx$\"1$z')pCH\">5F.7$$\"1+vVt#GGy&F.$\"1cYM`WN-5F.7$$\"1+](oj=I\"
eF.$\"1)e]!4lz])*FJ7$$\"1]Pf=Q6GeF.$\"1n&GXb3gy*FJ7$$\"1+DJ+!4K%eF.$\"
1<AG3*=`u*FJ7$$\"1]7.#=/$eeF.$\"1q$yTGfGt*FJ7$Fgbn$\"1fFSWSp\\(*FJ7$$
\"1+]i!4!yLfF.$\"1d^Pu=$G+\"F.7$Fgx$\"1#zftL@f-\"F.7$$\"1*****\\2ws0'F
.$\"1f:z1@*y+\"F.7$F_cn$\"1xXi,+uk(*FJ7$$\"1**\\(o9qh8'F.$\"1p+2AgmN(*
FJ7$$\"1++Dh*[>:'F.$\"1(Q$>r>8Q(*FJ7$$\"1+]ivxsnhF.$\"1W3[)*f;s(*FJ7$$
\"1+++!f1N='F.$\"1m2E'HlS$)*FJ7$$\"1++v=U1:iF.$\"1J**fXM2,5F.7$Fjx$\"1
z@%)=[y=5F.7$$\"1]ilA&[?E'F.$\"1Rv=[Q&[-\"F.7$Fgcn$\"1is\\1*f\"G5F.7$$
\"1](oH(=!HH'F.$\"1xnFs%p#G5F.7$F\\dn$\"1b=vrD5D5F.7$Fadn$\"1I00JWW55F
.7$Ffdn$\"1\\9?UgR-**FJ7$F[en$\"1a*fs\"HYL(*FJ7$F`en$\"1T;Q/Shw'*FJ7$F
een$\"1*=%3$Q<cw*FJ7$F]y$\"10gOeC&4(**FJ7$$\"1*\\P%[#\\d_'F.$\"1PCe$*)
==-\"F.7$$\"1**\\(o%)\\!elF.$\"1626e,%y.\"F.7$$\"1\\P4Y,?ulF.$\"1&y5X,
n%R5F.7$$\"1*\\7`W].f'F.$\"1GH?!QFg.\"F.7$$\"1\\7`W2]1mF.$\"1x`%\\'\\k
F5F.7$F`y$\"1Y_Y)*y1:5F.7$$\"1)\\(=U;&\\l'F.$\"17047YrI)*FJ7$$\"1**\\i
SAD(o'F.$\"1N]ax*z#\\&*FJ7$$\"1uVB!RF`p'F.$\"1k$H.$G0/&*FJ7$$\"1\\P%)R
DS.nF.$\"1Vy9&[#Gt%*FJ7$$\"1CJX*ox9r'F.$\"183N//ke%*FJ7$$\"1+D1RGb>nF.
$\"1w]y#H)Qh%*FJ7$$\"1]7GQJqNnF.$\"11/NYFt@&*FJ7$Fcy$\"1>(>L;%Ra'*FJ7$
Ffy$\"1C$)ywf,d5F.7$Fiy$\"1&>>Br,V3\"F.7$$\"1]iSc`%=*oF.$\"1GqG$>)[f5F
.7$F\\z$\"1(*GjJ\")\\=5F.7$$\"1](=<naH#pF.$\"13PW%3$H6'*FJ7$F_z$\"1xtl
%4o@)))FJ7$Fbz$\"1+YyEb2;!)FJ7$Fez$\"1lQ.&=Y0/(FJ7$Fhz$\"1O#R:())RAlFJ
7$F[[l$\"12SW&)RR!*fFJ7$F^[l$\"1pILyQj\\aFJ7$Fa[l$\"1/-Ds&e`!\\FJ7$$\"
1]7.-9o3qF.$\"1#fu6KJ1N%FJ7$$\"1,D1W[j;qF.$\"15>J#GfL!QFJ7$$\"1^P4'G)e
CqF.$\"1e@*4=%**oKFJ7$$\"1+]7G<aKqF.$\"1_w'f[QFv#FJ7$$\"1+v=7'[%[qF.$
\"1$*z:rNd$z\"FJ7$F_gn$\"1\"3,7J<?f*F\\^l7$$\"1+DJ!Qi-3(F.$\"1-\"o'z-8
SFF\\^l7$$\"1+]Pk#ph4(F.$!1333k1*)zCF\\^l7$$\"1,vV[h27rF.$!1D$\\^zVv.'
F\\^l7$Fd[l$!1%*)R)z!\\&3!)F\\^l7$Fggn$!1@gATu]<WF\\^l7$Fg[l$\"1]\"f1x
/!=MF\\^l7$$\"1Dc^)oDJE(F.$\"12d+#H\"e=RF\\^l7$$\"1]7.sK,rsF.$\"1up,'p
r'pUF\\^l7$$\"1uoab3!*ysF.$\"1cst#\\E+Z%F\\^l7$$\"1+D1R%)y'G(F.$\"1r(e
d*phAXF\\^l7$$\"1]P41Oc-tF.$\"1F<3:Na:UF\\^l7$F_hn$\"1G&*e1L*GW$F\\^l7
$$\"1,v=2\"*))\\tF.$\"17/rFtui5F\\^l7$Fdhn$!1v2>wy%)y8F\\^l7$$\"1]7G3Y
@(R(F.$!1rT(o>WcF#F\\^l7$$\"1+DJv(*)HT(F.$!1Cu')R=#[$GF\\^l7$$\"1^PMU
\\wGuF.$!1R!))\\&=%*=IF\\^l7$Fihn$!1*)3*Q&pONGF\\^l7$$\"1,vVV/4wuF.$!1
K'>)oTl*e\"F\\^l7$Fj[l$\"1UiJJGTp>F^p7$$\"1,D\"yDKm`(F.$\"1#\\/eaA'*e
\"F\\^l7$Fain$\"1T$3de<bB#F\\^l7$$\"1^7Gy%>,e(F.$\"1?U\"y!z28AF\\^l7$$
\"1+vV=_h%f(F.$\"1>=kqIcr>F\\^l7$$\"1]Pfe464wF.$\"1&o#e/*=ca\"F\\^l7$F
fin$\"1_]1W#)og)*F^p7$F[jn$!1y5x(e?G%GF^p7$F`jn$!18-Yn_CP8F\\^l7$$\"1^
7`*R&3'p(F.$!10Q=bS;b;F\\^l7$Fejn$!1#>.3U;_z\"F\\^l7$$\"1]P%)zo2DxF.$!
1qEPSd&3v\"F\\^l7$F]\\l$!1\\(p5]@Q`\"F\\^l7$$\"1,]7G,!Gx(F.$!1TREJ!*[
\"f&F^p7$$\"1,+DOw-1yF.$\"1&ohh:G;7'F^p7$$\"1+DJ!RTE#yF.$\"1a@P^wq!3\"
F\\^l7$$\"1+]PW^DRyF.$\"1!)oay4S!R\"F\\^l7$$\"1+vV)*)oe&yF.$\"1*p-d+[s
]\"F\\^l7$F][o$\"1>ZH^RJB9F\\^l7$$\"1++vow$*QzF.$!1#>6iJ&zCDF^p7$F`\\l
$!1,CZjZo(H\"F\\^l7$Fc\\l$!18PHDMt5iF^p7$Ff\\l$\"1F9XUtxh()F]eo7$Fi\\l
$\"1St+a40\"o(F^p7$F\\]lF\\_o-F_]l6&Fa]lF*F*Fc^oFb]l-%+AXESLABELSG6$Q
\"x6\"%!G-%*AXESTICKSG6$%(DEFAULTG7%F*$\"\"&!\"\"FL-%%VIEWG6$;F(F\\]lF
hiu" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 44.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "For an
other example, let " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "L = 10" "6#/%\"LG\"#5" }{TEXT -1 64 ". In
 addition, increase the height of each \"pulse\" from 1 to 10." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Then the \+
coefficents in the complex Fourier series are " }{XPPEDIT 18 0 "c[k] =
 S(k*Pi/10);" "6#/&%\"cG6#%\"kG-%\"SG6#*(F'\"\"\"%#PiGF,\"#5!\"\"" }
{TEXT -1 9 ", where  " }{XPPEDIT 18 0 "S(x) = PIECEWISE([sin*x/x, x <>
 0],[1, x = 0]);" "6#/-%\"SG6#%\"xG-%*PIECEWISEG6$7$*(%$sinG\"\"\"F'F.
F'!\"\"0F'\"\"!7$F./F'F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The following graph plots the valu
es of the coefficients " }{XPPEDIT 18 0 "S(k*Pi/10);" "6#-%\"SG6#*(%\"
kG\"\"\"%#PiGF(\"#5!\"\"" }{TEXT -1 46 " against the the associated an
gular frequency " }{XPPEDIT 18 0 "k*Pi/10;" "6#*(%\"kG\"\"\"%#PiGF%\"#
5!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The points lie \+
along the curve " }{XPPEDIT 18 0 "y = sin*x/x;" "6#/%\"yG*(%$sinG\"\"
\"%\"xGF'F(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "S := x -> if x<>0 then sin(
x)/x else 1 fi:\np1 := plot(S(x),x=-13..13,color=grey):\np2 := plots[p
ointplot]([seq([evalf(k*Pi/10),evalf(S(k*Pi/10))],\n    k=-40..40)],sy
mbol=circle,color=COLOR(RGB,.4,0,.9)):\nplots[display]([p1,p2]);" }}
{PARA 13 "" 1 "" {GLPLOT2D 581 319 319 {PLOTDATA 2 "6&-%'CURVESG6$7er7
$$!#8\"\"!$\"1*pN%HT0KK!#<7$$!1nm;*RFLC\"!#9$!1A`Al&Ht1\"F-7$$!1L$3A(o
,%>\"F1$!11g)H]-%3\\F-7$$!1+D\")\\%*Gm6F1$!1%3)py'))[t'F-7$$!1nmTF?cQ6
F1$!1HVE'fIL7)F-7$$!1n;a=kgC6F1$!1TZ9]-\\9')F-7$$!1nmm43l56F1$!1D0%HV&
Q[*)F-7$$!1n;z+_p'4\"F1$!1*[ML-pX6*F-7$$!1nm\">fRF3\"F1$!18(oBrDb5*F-7
$$!1$e9;J])o5F1$!19Mcz!3#=*)F-7$$!1+DJJ5'\\0\"F1$!1:#e51I>b)F-7$$!1</,
^<2T5F1$!1(>*>c#p*3!)F-7$$!1L$32Z#=F5F1$!1jTyMY)\\H(F-7$$!1+]7*=G9+\"F
1$!1t@OD)e:b&F-7$$!1omTv!Rnv*!#:$!1$*)3M!=BSLF-7$$!1++vQxRB#*F]p$\"1RM
b1_jo@F-7$$!1mmTl`\"=n)F]p$\"14v)pUb`)yF-7$$!1MLeW'3pR)F]p$\"1rnk?6m>5
!#;7$$!1++vB>+A\")F]p$\"1y4[U_E(=\"F_q7$$!1MeRZUh!)zF]p$\"1zw%z1.IC\"F
_q7$$!1n;/rlARyF]p$\"15Ik*=(\\v7F_q7$$!1%ekGtK&oxF]p$\"1fW%)3ya#G\"F_q
7$$!1+vo%*)Qyp(F]p$\"1#)>ns?E$G\"F_q7$$!1;/^c]9FwF]p$\"1?M5v&>vF\"F_q7
$$!1MLL=7XcvF]p$\"1rV$y-G_E\"F_q7$$!1++vLDQ2tF]p$\"1-s/Dy3p6F_q7$$!1nm
;\\QJeqF]p$\"1vH!H<hY\"**F-7$$!1LL3-:#zx'F]p$\"1'\\Q\"**)*30qF-7$$!1++
+b\"Hv\\'F]p$\"1*[o0g\\OF$F-7$$!1+++D=WMfF]p$!1y/x`G?edF-7$$!1+++XG!=R
&F]p$!1ht(z$>#GW\"F_q7$$!1M$3_v=a9&F]p$!1UGGM24k<F_q7$$!1omTlY.**[F]p$
!178T!*pw0?F_q7$$!1Mek3ua_ZF]p$!1m\"RtFQC5#F_q7$$!1,](=:ggg%F]p$!1oN,Q
;ze@F_q7$$!1&e*[Bl\"G`%F]p$!1v41Y&f1<#F_q7$$!1nT5&*GdfWF]p$!1Q;<Vl3r@F
_q7$$!1](=nEHjQ%F]p$!1XA^M:pf@F_q7$$!1MLLQc38VF]p$!1XaE&3Kh8#F_q7$$!1M
L$ez(*[1%F]p$!1#ziE&R;i>F_q7$$!1MLL`*4n\"QF]p$!1\\`+-p]P;F_q7$$!1o;/\"
\\3!GNF]p$!1jx&=wD#o5F_q7$$!1-+vGqIRKF]p$!1(H96!Hs6IF-7$$!1o;/;tv$)HF]
p$\"1@>h@A)yE&F-7$$!1NLL.w?GFF]p$\"1yzltvVs9F_q7$$!1o;/m!HyW#F]p$\"1o%
p_yaAh#F_q7$$!1,+vG0Xn@F]p$\"1')G4*[el\"QF_q7$$!1++]dQY+>F]p$\"1%G`Ga@
&y\\F_q7$$!1***\\i=xMj\"F]p$\"1.wr%e())4hF_q7$$!1n;a.Y!\\N\"F]p$\"1FQ*
Rae#4sF_q7$$!1ML$3-Kj2\"F]p$\"1?L=(=))z<)F_q7$$!1TLe*Ga^?)F_q$\"1Jo7Xe
4:*)F_q7$$!1TL$3P))pk&F_q$\"1,vOw_$pZ*F_q7$$!1v;/,YKnUF_q$\"1\\a$pW]#*
p*F_q7$$!15+DJ3m()GF_q$\"163Mm=gh)*F_q7$$!1xTNY*Gy>#F_q$\"1E')y&o'o>**
F_q7$$!1W$e91(*z]\"F_q$\"1$)y5_@9i**F_q7$$!14^il<l\"=)F-$\"17:air%)))*
*F_q7$$!1!ymm\"HL$G\"F-$\"1Vyp6bs****F_q7$$\"1D#3-j\"3#)eF-$\"1;%pn^MU
***F_q7$$\"1B$3xh\\ZI\"F_q$\"1?GP\"G^;(**F_q7$$\"1BeRs5H@?F_q$\"1VnOS`
/K**F_q7$$\"1CL3FD$yt#F_q$\"1so#))fQb()*F_q7$$\"1D$ekV:4<%F_q$\"16lz9$
pDr*F_q7$$\"1DL$eM)*Rg&F_q$\"1O4XlVu%[*F_q7$$\"1fmm;T+*4)F_q$\"13gD4'o
?%*)F_q7$$\"1***\\())4Sf5F]p$\"1\")*pWLc;B)F_q7$$\"1m;HZ%o)G8F]p$\"14U
/?)\\gI(F_q7$$\"1KL$e!fL)f\"F]p$\"1^\"*>HZ8aiF_q7$$\"1**\\(=.Hvt\"F]p$
\"1ic#3g'[vcF_q7$$\"1mm\"z:An(=F]p$\"1H(p=%>.\"3&F_q7$$\"1K$eRG:f,#F]p
$\"1U#HCbprZ%F_q7$$\"1******4%3^:#F]p$\"1;_D'o+.(QF_q7$$\"1***\\i/buU#
F]p$\"19>bu!f\")p#F_q7$$\"1****\\#o,)*p#F]p$\"1pnZ&yoOe\"F_q7$$\"1**\\
(o06L'HF]p$\"1L&*pNqZ%)fF-7$$\"1)**\\7V?oA$F]p$!1X>+!\\M!QEF-7$$\"1(*
\\PkES>NF]p$!1IgjO*[\"[5F_q7$$\"1(***\\(*[)>\"QF]p$!1]&>)3A%)H;F_q7$$
\"1JL3ZI)[2%F]p$!1t#H1qQ?(>F_q7$$\"1lmm'>\"yPVF]p$!1#R+fZaa9#F_q7$$\"1
K$e*oh&zS%F]p$!1Yzx(=2V;#F_q7$$\"1***\\79J\"yWF]p$!11'\\%p@3s@F_q7$$\"
1l;a8hI[XF]p$!1`P,7,4p@F_q7$$\"1LL$e3\"[=YF]p$!1n2)f&Qnb@F_q7$$\"1mmTI
5$)eZF]p$!1ciTG14*4#F_q7$$\"1+++v4=**[F]p$!1T.d^;l0?F_q7$$\"1m;z%zON:&
F]p$!1bY*p&Qka<F_q7$$\"1KLe9E*yS&F]p$!1&z]wF\\'>9F_q7$$\"1,++0)[S'fF]p
$!1T0UDSjg_F-7$$\"1im\"zV[t['F]p$\"1')z/m!z_7$F-7$$\"1JL3()y%3w'F]p$\"
1C7\"zf6&*z'F-7$$\"1,+DOtMMqF]p$\"1Sw/)zHAq*F-7$$\"1mTg(=X!orF]p$\"1fv
V&*)\\&z5F_q7$$\"1K$e*QIu,tF]p$\"1U:SJw&f;\"F_q7$$\"1(\\7.*3WNuF]p$\"1
-lBV:#)G7F_q7$$\"1jmmT(Q\"pvF]p$\"1k$*\\O%>zE\"F_q7$$\"1!e9;PB\"RwF]p$
\"1eIpI7&*y7F_q7$$\"1'\\i:+3\"4xF]p$\"1%Hv#=!yNG\"F_q7$$\"18/^JE4zxF]p
$\"1V>)o)Q*=G\"F_q7$$\"1I$e9Ex!\\yF]p$\"1SgP'y>SF\"F_q7$$\"1jTN@l/*)zF
]p$\"1nr%R'RJS7F_q7$$\"1(**\\7y:!H\")F]p$\"1SVD^n#R=\"F_q7$$\"1m;HZ)H'
)R)F]p$\"1yJ4S=R=5F_q7$$\"1NLL8RCo')F]p$\"15\\wKth=zF-7$$\"1JL$3c#o>#*
F]p$\"1&G\"G^e(*3AF-7$$\"1mm\"z,blw*F]p$!1Fbnt4tJMF-7$$\"1LeR^9y,5F1$!
1$3-g*z()ybF-7$$\"1+++,u!p-\"F1$!1$>fEHt\"zsF-7$$\"1mT&y)fIT5F1$!15V*e
Zk&>!)F-7$$\"1L$3Zd/d0\"F1$!1.G<*=\\gd)F-7$$\"1*\\i:;.,2\"F1$!1TjoWcXU
*)F-7$$\"1mmT[<]%3\"F1$!1z:`dc^;\"*F-7$$\"1Le9;+Q(4\"F1$!1%yC)GMX5\"*F
-7$$\"1+](QGe-6\"F1$!14SE!3%Rb*)F-7$$\"1mTg^l8B6F1$!1e8c\"\\rsl)F-7$$
\"1LLL>[,O6F1$!1%)4jc7:C#)F-7$$\"1m\"H#emZj6F1$!1vF3w,5)*oF-7$$\"1+]7(
\\Q4>\"F1$!1RSSaO;G^F-7$$\"1+]([Y2NC\"F1$!1%ew$*)3#G0\"F-7$$\"#8F*F+-%
'COLOURG6&%$RGBG$\")=THv!\")FeamFeam-%'POINTSG6_p7$$!+iqjc7FgamF*7$$!+
N6AD7Fgam$!+>C8AD!#67$$!+4_!Q>\"Fgam$!+1yiB\\Fcbm7$$!+#G*Qi6Fgam$!+\\)
[*fpFcbm7$$!+bL(48\"Fgam$!+oe=4%)Fcbm7$$!+Hub*4\"Fgam$!+u\"oX4*Fcbm7$$
!+-:9o5Fgam$!+gQ%Q!*)Fcbm7$$!+wbsO5Fgam$!+6!zN!yFcbm7$$!+\\'4`+\"Fgam$
!+?!3o%eFcbm7$$!+Fs$*Q(*!\"*$!+-`+tJFcbm7$$!+izxC%*FjdmF*7$$!+(p=16*Fj
dm$\"+`K$=R$Fcbm7$$!+J%fkz)Fjdm$\"+4j1#o'Fcbm7$$!+m,I#[)Fjdm$\"+#o2x`*
Fcbm7$$!++49o\")Fjdm$\"+7)[V;\"!#57$$!+N;)R&yFjdm$\"+W&RKF\"Fdfm7$$!+q
B#)RvFjdm$\"+\")yPh7Fdfm7$$!+/JmDsFjdm$\"+XRk>6Fdfm7$$!+RQ]6pFjdm$\"+F
![W])Fcbm7$$!+tXM(f'Fjdm$\"+2-'Ro%Fcbm7$$!+3`=$G'FjdmF*7$$!+Vg-pfFjdm$
!+]'3q<&Fcbm7$$!+xn'[l&Fjdm$!+PDVR5Fdfm7$$!+7vqS`Fjdm$!+'R7[^\"Fdfm7$$
!+Y#[l-&Fjdm$!+?o1#*=Fdfm7$$!+\")*)Q7ZFjdm$!+2f1A@Fdfm7$$!+;(H#)R%Fjdm
$!+\"3iB;#Fdfm7$$!+]/2%3%Fjdm$!+=&34)>Fdfm7$$!+&=6*pPFjdm$!+0)[\"f:Fdf
m7$$!+>>vbMFjdm$!+\\e5U*)Fcbm7$$!+aEfTJFjdmF*7$$!+*QLu#GFjdm$\"+[S#H4
\"Fdfm7$$!+BTF8DFjdm$\"+3KsQBFdfm7$$!+e[6*>#Fjdm$\"+2,$)yOFdfm7$$!+#fb
\\)=Fjdm$\"+@:^X]Fdfm7$$!+Fjzq:Fjdm$\"+Ax>mjFdfm7$$F]bmFjdm$\"+\"Gn#ov
Fdfm7$$F_emFdfm$\"+7p$Re)Fdfm7$$F`hmFdfm$\"+LG*[N*Fdfm7$$F`[nFdfm$\"+M
kJO)*Fdfm7$F*$\"\"\"F*7$$\"+aEfTJFdfmFh]n7$$\"+3`=$G'FdfmFd]n7$$\"+izx
C%*FdfmF`]n7$$\"+iqjc7FjdmF\\]n7$$\"+Fjzq:FjdmFh\\n7$$\"+#fb\\)=FjdmFc
\\n7$$\"+e[6*>#FjdmF^\\n7$$\"+BTF8DFjdmFi[n7$$\"+*QLu#GFjdmFd[n7$$F_^n
FjdmF*7$$\"+>>vbMFjdmF\\[n7$$\"+&=6*pPFjdmFgjm7$$\"+]/2%3%FjdmFbjm7$$
\"+;(H#)R%FjdmF]jm7$$\"+\")*)Q7ZFjdmFhim7$$\"+Y#[l-&FjdmFcim7$$\"+7vqS
`FjdmF^im7$$\"+xn'[l&FjdmFihm7$$\"+Vg-pfFjdmFdhm7$$Fb^nFjdmF*7$$\"+tXM
(f'FjdmF\\hm7$$\"+RQ]6pFjdmFggm7$$\"+/JmDsFjdmFbgm7$$\"+qB#)RvFjdmF]gm
7$$\"+N;)R&yFjdmFhfm7$$\"++49o\")FjdmFbfm7$$\"+m,I#[)FjdmF]fm7$$\"+J%f
kz)FjdmFhem7$$\"+(p=16*FjdmFcem7$$Fe^nFjdmF*7$$\"+Fs$*Q(*FjdmF[em7$$\"
+\\'4`+\"FgamFedm7$$\"+wbsO5FgamF`dm7$$\"+-:9o5FgamF[dm7$$\"+Hub*4\"Fg
amFfcm7$$\"+bL(48\"FgamFacm7$$\"+#G*Qi6FgamF\\cm7$$\"+4_!Q>\"FgamFgbm7
$$\"+N6AD7FgamFabm7$$Fh^nFgamF*-%&COLORG6&Fdam$\"\"%!\"\"F*$\"\"*Ffen-
%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F_am%(DEFAU
LTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example \+
3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 58 "We shall find the complex Fourier series for the function
 " }{XPPEDIT 18 0 "f(x) = abs(sin*x);" "6#/-%\"fG6#%\"xG-%$absG6#*&%$s
inG\"\"\"F'F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(abs(sin(x)),x=-Pi..3*Pi
);" }}{PARA 13 "" 1 "" {GLPLOT2D 585 146 146 {PLOTDATA 2 "6%-%'CURVESG
6$7[v7$$!1++JZEfTJ!#:$\"15KT_Kzzi!#C7$$!1=x/o[6tIF*$\"1t%HN.GC%o!#<7$$
!1Nay)3PY+$F*$\"1B%y=Cy_O\"!#;7$$!1_J_4$fh$HF*$\"1$zj/69*R?F97$$!1q3EI
:onGF*$\"1?sPQ\"))\\q#F97$$!1aMn!Q*43GF*$\"1-`I'GeMF$F97$$!1Qg3Js^[FF*
$\"1HrB))4JIQF97$$!1A')\\\"3N*)o#F*$\"1j%3P)*pNP%F97$$!117\">$HNHEF*$
\"1q13csI,\\F97$$!1hw>tlMiDF*$\"12j6b:$RZ&F97$$!1:T[9-M&\\#F*$\"1#f)G0
z)>-'F97$$!1q0xbQLGCF*$\"1w\"z^`;Ia'F97$$!1Dq0(\\F8O#F*$\"1%eR%y*yY.(F
97$$!1(*zMckUEAF*$\"1^?S9#Rm#zF97$$!1p*QcTD:4#F*$\"1l&3HzmXn)F97$$!1Y$
Gg`ls&>F*$\"1g8QwlXi#*F97$$!1AxTcc+B=F*$\"1%pHcgMOo*F97$$!1gx)*3wwg<F*
$\"1hlD$e'4?)*F97$$!1(zd:cH&)p\"F*$\"1&)Q)>'>`=**F97$$!1;G%y`5um\"F*$
\"1[UojZO`**F97$$!1My79:HO;F*$\"1:**4o&f&y**F97$$!1`GT!\\s^g\"F*$\"1'[
6(y>4%***F97$$!1sypmM0u:F*$\"1e^Cbp%*****F97$$!13_7?:$=a\"F*$\"10s8$\\
0e***F97$$!1XDbt&4'4:F*$\"1CT[XnG\")**F97$$!1#))zpi(Qx9F*$\"1Xs7&y0k&*
*F97$$!1>sS!ol^W\"F*$\"1?UeU%)=@**F97$$!1#*=E(y@2Q\"F*$\"12PN\\:!*>)*F
97$$!1ml6%*yF;8F*$\"1\\tDml%yn*F97$$!1*))e6'>)H=\"F*$\"1SL<H!ytD*F97$$
!177?Ggo\\5F*$\"1M5l&fpEn)F97$$!1F/\"*pd<o\"*F9$\"1c%[V='pOzF97$$!1J(3
yD\"\\RyF9$\"1oxg[T\"31(F97$$!1(><%*eLh:(F9$\"1XyMlV!3c'F97$$!1kc-@fxs
kF9$\"1qQB4!p,.'F97$$!1JTj_#=%*y&F9$\"1>R#\\1&QraF97$$!1)fUUegg5&F9$\"
12?c841()[F97$$!1/S'zxeT]%F9$\"1!\\DWs*R`VF97$$!14aorpD-RF9$\"1x5%zarR
!QF97$$!1:oSl^N+LF9$\"1P'=MGm2C$F97$$!1@#G\"fLX)p#F9$\"1<D`FP#em#F97$$
!15OJ&[a3-#F9$\"1b:.dx72?F97$$!1***)\\6cDV8F9$\"1tB$4y>#R8F97$$!1!)Q%o
Pnll'F3$\"1,c6.El^mF37$$\"1*HAIh8U>\"!#=$\"1h%o\"HL@%>\"Fdy7$$\"1j2BgH
BBpF3$\"1t#QIk.x\"pF37$$\"1If\"fX/FP\"F9$\"1QOK(\\(Ro8F97$$\"1$y3eh&3`
?F9$\"1QI93DpQ?F97$$\"1O;qvnYLFF9$\"1lW\"Qn`&*p#F97$$\"1f:#[4S\"*Q$F9$
\"1ZUhI2jCLF97$$\"1#[TRT8[/%F9$\"1@NFq+UNRF97$$\"1/91Ln[+ZF9$\"1K#)=/o
HHXF97$$\"1F8=_+;c`F9$\"1O0FE(3P5&F97$$\"1(>c*G%))pa'F9$\"18L\"p$Q?*3'
F97$$\"1o5t0o\"yt(F9$\"1L:UK;X))pF97$$\"1_H5Dp#Q:*F9$\"10U=6*ez#zF97$$
\"1%[ZWq$)p0\"F*$\"1zlNN$p(3()F97$$\"1'f$RN$Qp<\"F*$\"14i\\mTNM#*F97$$
\"14(Rj'H*oH\"F*$\"1O(G'zSAF'*F97$$\"10heU3mm8F*$\"1$=Hl0mBz*F97$$\"1+
D$)=(GkV\"F*$\"1\\U**oC')4**F97$$\"1)pbpl78Z\"F*$\"1%Gtkzb0&**F97$$\"1
'*)y]f'>1:F*$\"1D9aG;9z**F97$$\"1%4-K`!3T:F*$\"1$\\PH=&e&***F97$$\"1#H
D8Zkfd\"F*$\"1=yq^k')****F97$$\"1k,UnE%og\"F*$\"1!ed+,/N***F97$$\"1P]^
j3sP;F*$\"1YG@*)Qhx**F97$$\"14*4'f!*fo;F*$\"1a?hQ7@_**F97$$\"1#y/dDx%*
p\"F*$\"1?\"=oF?t\"**F97$$\"1FX*ykL7w\"F*$\"1!\\iB\\:#>)*F97$$\"1rU3S+
*H#=F*$\"1k@2uNn$o*F97$$\"1:=FSJ]e>F*$\"1(eB8q%yd#*F97$$\"1f$f/C;S4#F*
$\"1qv$fsZ@m)F97$$\"1bX*yvcIA#F*$\"1gxFPy8ZzF97$$\"1](H`F(4_BF*$\"1(e:
M1!)**4(F97$$\"1b>o'R<%>CF*$\"1lyo7[=5mF97$$\"1fT.=vt'[#F*$\"1x`!\\_V/
4'F97$$\"1kjQRw0aDF*$\"1(>#[x26VbF97$$\"1o&Q2wx8i#F*$\"1BZ2Xhmq\\F97$$
\"1ot?Z')>$o#F*$\"13Ht/[3DWF97$$\"1ohnL&>]u#F*$\"11YTjpfiQF97$$\"1p\\9
?/%o!GF*$\"1E#f\\m^`G$F97$$\"1pPh18moGF*$\"13-iNVb&p#F97$$\"1Q?Q%[V`$H
F*$\"1Qf&f8+z/#F97$$\"12.:ic--IF*$\"18gs&HV6R\"F97$$\"1w&=*RyqoIF*$\"1
Wj?2,.#G(F37$$\"1Xoo<+RNJF*$\"1st+Jhf-iFdy7$$\"1KMqfSl/KF*$\"1e\"HWzh>
I'F37$$\"1=+s,\"=RF$F*$\"1)G:y:'R>8F97$$\"1/mtV@=VLF*$\"1B\"4M&)oA+#F9
7$$\"1\">`d=YCT$F*$\"1y$z))[Rbn#F97$$\"1wF%ztSFZ$F*$\"1#oz+^*G^KF97$$
\"1hB8!HNI`$F*$\"1NgWmKA:QF97$$\"1X>KU)HLf$F*$\"1o')fN7HlVF97$$\"1I:^%
RCOl$F*$\"1;\"H!)G%\\**[F97$$\"1;&*piTu=PF*$\"1QLr/1RcaF97$$\"1-v)3$R'
Qy$F*$\"1x&fI\"o:!*fF97$$\"1([v!*p$)*[QF*$\"1jPFK-`)\\'F97$$\"1tMEnM59
RF*$\"1\\bxBeNzpF97$$\"1cwG_Ol[SF*$\"1essH`'p(yF97$$\"1Q=JPQ?$=%F*$\"1
*fdO@(=K')F97$$\"1K*fM%\\$[J%F*$\"1X^32u5?#*F97$$\"1D!3'\\gYYWF*$\"1L#
zvl.&['*F97$$\"1`)z8)f95XF*$\"19ODx\\='z*F97$$\"1\"o^J\"f#Qd%F*$\"1#\\
1$z[:/**F97$$\"1&fP!zem0YF*$\"1WJ,J\\5V**F97$$\"13N#\\%e]PYF*$\"1\"3%G
zc(>(**F97$$\"1A%43\"eMpYF*$\"1fd!z&yt!***F97$$\"1O`pwd=,ZF*$\"1$G*pZC
P****F97$$\"1ogCI'Qlt%F*$\"1Dax8T3(***F97$$\"1,oz$[\"*=x%F*$\"1T&QH[-B
)**F97$$\"1MvMPVC2[F*$\"1&H0x-Y]&**F97$$\"1m#)*3>(fU[F*$\"1^mI5)[`\"**
F97$$\"1I(**z*GI8\\F*$\"1*4v;,Y))z*F97$$\"1&>,^g3S)\\F*$\"1M?`wiPL'*F9
7$$\"1d#*G]I26^F*$\"1_o$oVEd@*F97$$\"1=tZ&\\P\"Q_F*$\"1tu\\P`[\\')F97$
$\"1a$y?\"f!QP&F*$\"1&=o0%pA\"*yF97$$\"1!Rz'GVZ4bF*$\"15.8-f%z)pF97$$
\"1\"RXN?U4d&F*$\"1`W?S@LNlF97$$\"1\"R6%y+TKcF*$\"1vS%)3P.egF97$$\"1#R
xK&z(Qp&F*$\"1fU!oT`yb&F97$$\"1#RV\"GeMbdF*$\"1v&[,]!oO]F97$$\"1v_;^ka
AeF*$\"10%z&[Y?XWF97$$\"1er=uqu*)eF*$\"1c2;X@mLQF97$$\"1S!4spZp&fF*$\"
1%\\g+j8[?$F97$$\"1B4B?$[T-'F*$\"1w)om'y\\hDF97$$\"1+vUp)yt3'F*$\"1+vY
didX>F97$$\"1ySi=%41:'F*$\"1r7bE'y=K\"F97$$\"1c1#y'*RQ@'F*$\"1+%y=]x*G
pF37$$\"1Ls,<02xiF*$\"1D'>k0_Z6'Fdy7$$\"1fA)*fY;VjF*$\"1(R9BkRV*fF37$$
\"1&GZH!)e#4kF*$\"1w**R8yRd7F97$$\"16B\"f%HNvkF*$\"1a\"Q.!3()4>F97$$\"
1Pt())3Z9a'F*$\"193>wO+aDF97$$\"1!4NX>mgg'F*$\"11@t<*)*H<$F97$$\"1VG>+
`oqmF*$\"1KBD.&\\(yPF97$$\"1&f]eS/`t'F*$\"1Q\"yJ#psoVF97$$\"1[$3:^B**z
'F*$\"1\")e!z]o/%\\F97$$\"1\"*\\aEPdnoF*$\"12lDpR*o^&F97$$\"1L;eTRANpF
*$\"1)eW%323ogF97$$\"1u#=m:uG+(F*$\"1Fd\"G:2:f'F97$$\"1;\\lrV_qqF*$\"1
m:pF(yZ3(F97$$\"1ZIlmZ$3?(F*$\"16)Hl=a<%zF97$$\"1z6lh^9JtF*$\"1:jePI1k
')F97$$\"1YTohjSkuF*$\"1]S$)HXq]#*F97$$\"18rrhvm(f(F*$\"1,*oh\"*4Ln*F9
7$$\"1'RCnUYPm(F*$\"1y*G(pyf>)*F97$$\"1z;t\"HD)HxF*$\"1<I?p\\-B**F97$$
\"1?`BCZ'Gw(F*$\"1Sw'eD<&e**F97$$\"1i*Qn:/fz(F*$\"1%f3JxRJ)**F97$$\"1.
EC*eV*GyF*$\"1H+LXc'o***F97$$\"1Wiu@I)>'yF*$\"1xkn!*)z'****F97$$\"1sV(
)GGM#*yF*$\"12\")*e*Hk#***F97$$\"1+D+OEqAzF*$\"1a2u5kRw**F97$$\"1F18VC
1`zF*$\"1=C#)3^&4&**F97$$\"1b(e-DAM)zF*$\"1!3X!QDM;**F97$$\"15]^k=9W!)
F*$\"19n[:'R(>)*F97$$\"1m7xy9'[5)F*$\"1?$>ipVpo*F97$$\"15l0RX/W#)F*$\"
1W$RCU^)[#*F97$$\"1`<M*fFKQ)F*$\"1bdV[,)=j)F97$$\"1$=uwe9x])F*$\"1/qyU
7>QzF97$$\"18m+w:?K')F*$\"16m#=*Gk@rF97$$\"1x-0Bhc)p)F*$\"1(o'\\`,USmF
97$$\"1RR4q1$\\w)F*$\"1)Q[k0i*HhF97$$\"1,w8<_HJ))F*$\"1_%HB'f^#f&F97$$
\"1l7=k(fw*))F*$\"1W!R60[/.&F97$$\"1[ui.\"z6'*)F*$\"10L!)**)*prWF97$$
\"1KO2V%)pC!*F*$\"16ID<e\"\\*QF97$$\"19)>Dy<#)3*F*$\"1%>Ac<ACI$F97$$\"
1(*f'>7P<:*F*$\"1Xk>4(3mp#F97$$\"1)*p&pK(**>#*F*$\"1+!H8#Q_L?F97$$\"1)
*z%>`d#)G*F*$\"1Lb.)\\n4O\"F97$$\"1***Qpt<lN*F*$\"1\"4h-pA2#oF37$$\"1+
+$>%zxC%*F*$\"1%)G8?!QR)=!#B-%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!Fibn-%+AX
ESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;$!+aEfTJ!\"*$\"+izxC%*Ffcn%(DEFAULTG" 
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 
1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "The  coefficients in the
 complex Fourier series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "c[k] = 1/(2*L);" "6#/&%\"cG6#%\"kG*&\"\"\"F)*&\"\"#F
)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*exp(-i*k*Pi*x/
L),x = -L .. L);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*,%\"iGF
+%\"kGF+%#PiGF+F*F+%\"LG!\"\"F5F+/F*;,$F4F5F4" }{TEXT -1 2 "  " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&
\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(sin*x)*exp
(-2*k*i*x),x = -Pi/2 .. Pi/2);" "6#-%$IntG6$*&-%$absG6#*&%$sinG\"\"\"%
\"xGF,F,-%$expG6#,$**\"\"#F,%\"kGF,%\"iGF,F-F,!\"\"F,/F-;,$*&%#PiGF,F3
F6F6*&F;F,F3F6" }{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(sin*x*exp(-2*k*i*x),x = 0 .. Pi);" "6#-%$IntG6$*
(%$sinG\"\"\"%\"xGF(-%$expG6#,$**\"\"#F(%\"kGF(%\"iGF(F)F(!\"\"F(/F);
\"\"!%#PiG" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 8 "because " }
{XPPEDIT 18 0 "exp(2*k*i*x)" "6#-%$expG6#**\"\"#\"\"\"%\"kGF(%\"iGF(%
\"xGF(" }{TEXT -1 12 " has period " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "
" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c[k]= 1/Pi" "6#/&%\"cG6#%\"kG*&\"\"
\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``((exp(i*x)-exp(-
i*x))/2)*exp(-2*k*i*x),x = 0 .. Pi);" "6#-%$IntG6$*&-%!G6#*&,&-%$expG6
#*&%\"iG\"\"\"%\"xGF1F1-F-6#,$*&F0F1F2F1!\"\"F7F1\"\"#F7F1-F-6#,$**F8F
1%\"kGF1F0F1F2F1F7F1/F2;\"\"!%#PiG" }{TEXT -1 2 "  " }}{PARA 256 "" 0 
"" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi*i);" "6#/%!G*&\"\"\"F&*
(\"\"#F&%#PiGF&%\"iGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-
(2*k-1)*i*x)-exp(-(2*k+1)*i*x),x=0..Pi)" "6#-%$IntG6$,&-%$expG6#,$*(,&
*&\"\"#\"\"\"%\"kGF/F/F/!\"\"F/%\"iGF/%\"xGF/F1F/-F(6#,$*(,&*&F.F/F0F/
F/F/F/F/F2F/F3F/F1F1/F3;\"\"!%#PiG" }{TEXT -1 2 "  " }}{PARA 256 "" 0 
"" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi*i);" "6#/%!G*&\"\"\"F&*
(\"\"#F&%#PiGF&%\"iGF&!\"\"" }{XPPEDIT 18 0 "``(exp(-(2*k-1)*i*x)/(-(2
*k-1)*i)-exp(-(2*k+1)*i*x)/(-(2*k+1)*i));" "6#-%!G6#,&*&-%$expG6#,$*(,
&*&\"\"#\"\"\"%\"kGF0F0F0!\"\"F0%\"iGF0%\"xGF0F2F0,$*&,&*&F/F0F1F0F0F0
F2F0F3F0F2F2F0*&-F)6#,$*(,&*&F/F0F1F0F0F0F0F0F3F0F4F0F2F0,$*&,&*&F/F0F
1F0F0F0F0F0F3F0F2F2F2" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([Pi,
 ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!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&!\"\"" }{XPPEDIT 18 0 "``(exp(-(2*k-1)*i*x)/(2*k-1)-exp(-(2*
k+1)*i*x)/(2*k+1));" "6#-%!G6#,&*&-%$expG6#,$*(,&*&\"\"#\"\"\"%\"kGF0F
0F0!\"\"F0%\"iGF0%\"xGF0F2F0,&*&F/F0F1F0F0F0F2F2F0*&-F)6#,$*(,&*&F/F0F
1F0F0F0F0F0F3F0F4F0F2F0,&*&F/F0F1F0F0F0F0F2F2" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG
6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*P
i);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(exp(-(
2*k-1)*Pi*i)/(2*k-1)-exp(-(2*k+1)*Pi*i)/(2*k+1)-(1/(2*k-1)-1/(2*k+1)))
;" "6#-%!G6#,(*&-%$expG6#,$*(,&*&\"\"#\"\"\"%\"kGF0F0F0!\"\"F0%#PiGF0%
\"iGF0F2F0,&*&F/F0F1F0F0F0F2F2F0*&-F)6#,$*(,&*&F/F0F1F0F0F0F0F0F3F0F4F
0F2F0,&*&F/F0F1F0F0F0F0F2F2,&*&F0F0,&*&F/F0F1F0F0F0F2F2F0*&F0F0,&*&F/F
0F1F0F0F0F0F2F2F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%
!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(exp(-Pi*i)/(2*k-
1)-exp(-Pi*i)/(2*k+1)-(1/(2*k-1)-1/(2*k+1)));" "6#-%!G6#,(*&-%$expG6#,
$*&%#PiG\"\"\"%\"iGF.!\"\"F.,&*&\"\"#F.%\"kGF.F.F.F0F0F.*&-F)6#,$*&F-F
.F/F.F0F.,&*&F3F.F4F.F.F.F.F0F0,&*&F.F.,&*&F3F.F4F.F.F.F0F0F.*&F.F.,&*
&F3F.F4F.F.F.F.F0F0F0" }{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 1 " " }{XPPEDIT 18 0 
"``((-1)/(2*k-1)-(-1)/(2*k+1)-(1/(2*k-1)-1/(2*k+1)));" "6#-%!G6#,(*&,$
\"\"\"!\"\"F),&*&\"\"#F)%\"kGF)F)F)F*F*F)*&,$F)F*F),&*&F-F)F.F)F)F)F)F
*F*,&*&F)F),&*&F-F)F.F)F)F)F*F*F)*&F)F),&*&F-F)F.F)F)F)F)F*F*F*" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/Pi;" "6#/%!G,$*&\"\"\"F'%#PiG!
\"\"F)" }{XPPEDIT 18 0 "``(1/(2*k-1)-1/(2*k+1));" "6#-%!G6#,&*&\"\"\"F
(,&*&\"\"#F(%\"kGF(F(F(!\"\"F-F(*&F(F(,&*&F+F(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 "``=-2/((4*k^2-1)*Pi)" "6#/%!G,$*&\"\"#\"\"\"*&,&
*&\"\"%F(*$%\"kGF'F(F(F(!\"\"F(%#PiGF(F/F/" }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The constant coe
fficient does not have to be calculated separately here, but if we do \+
so, it provides a check for the case " }{XPPEDIT 18 0 "k = 0" "6#/%\"k
G\"\"!" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "c[0] = 1/(2*L);" "6#/&%\"cG6#\"\"!*&\"\"\"F)*&\"\"#F)%
\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = -L .. L);" "
6#-%$IntG6$-%\"fG6#%\"xG/F);,$%\"LG!\"\"F-" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"
F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(sin*x),x = -Pi/2
 .. Pi/2);" "6#-%$IntG6$-%$absG6#*&%$sinG\"\"\"%\"xGF+/F,;,$*&%#PiGF+
\"\"#!\"\"F3*&F1F+F2F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/Pi;" "6
#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*
x,x = 0 .. Pi/2);" "6#-%$IntG6$*&%$sinG\"\"\"%\"xGF(/F);\"\"!*&%#PiGF(
\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 
2 "  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "assume(k_,integer)
:\n1/Pi*Int(abs(sin(x))*exp(-I*2*k*x),x=-Pi/2..Pi/2);\nc[k]=subs(k_=k,
simplify(value(subs(k=k_,%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-
%$IntG6$*&-%$absG6#-%$sinG6#%\"xG\"\"\"-%$expG6#*(^#!\"#F/%\"kGF/F.F/F
//F.;,$*&\"\"#!\"\"%#PiGF/F<,$*&F;F<F=F/F/*$F=F<" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"cG6#%\"kG,$*(\"\"#\"\"\"%#PiG!\"\",&*&\"\"%F+)F'F*
F+F+F+F-F-F-" }}}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The \+
complex Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 24 " for all real values of " }{TEXT 278 1 "x" }{TEXT -1 10 "
, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = \+
Sum(c[k]*exp(2*k*i*x),k = -infinity .. infinity);" "6#/-%\"fG6#%\"xG-%
$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#**\"\"#F0F/F0%\"iGF0F'F0F0/F/;,$%
)infinityG!\"\"F:" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "wher
e  " }{XPPEDIT 18 0 "c[k] = -2/(Pi*(4*k^2-1));" "6#/&%\"cG6#%\"kG,$*&
\"\"#\"\"\"*&%#PiGF+,&*&\"\"%F+*$F'F*F+F+F+!\"\"F+F2F2" }{TEXT -1 2 " \+
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The \+
coefficients " }{XPPEDIT 18 0 "a[k]" "6#&%\"aG6#%\"kG" }{TEXT -1 5 " a
nd " }{XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" }{TEXT -1 47 " in the cor
responding trigonometric series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6
#%\"xG" }{TEXT -1 15 " are given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "a[k] = 2*Re(c[k]);" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"
-%#ReG6#&%\"cG6#F'F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -4/(Pi*(4*k
^2-1));" "6#/%!G,$*&\"\"%\"\"\"*&%#PiGF(,&*&F'F(*$%\"kG\"\"#F(F(F(!\"
\"F(F0F0" }{TEXT -1 8 "   and  " }{XPPEDIT 18 0 "b[k] = -2*Im(c[k])" "
6#/&%\"bG6#%\"kG,$*&\"\"#\"\"\"-%#ImG6#&%\"cG6#F'F+!\"\"" }{XPPEDIT 
18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The trigonometric series is " }
}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(x) = 2/Pi" "6#/-%\"fG6#%\"xG*&\"\"
#\"\"\"%#PiG!\"\"" }{TEXT -1 5 "  +  " }{XPPEDIT 18 0 "Sum(``(-4/(Pi*(
4*k^2-1)))*cos*2*k*x,k = 1 .. infinity);" "6#-%$SumG6$*,-%!G6#,$*&\"\"
%\"\"\"*&%#PiGF-,&*&F,F-*$%\"kG\"\"#F-F-F-!\"\"F-F5F5F-%$cosGF-F4F-F3F
-%\"xGF-/F3;F-%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "f := x -> abs(sin
(x));\nFS := (x,n) -> 2/Pi+Sum(-4/(Pi*(4*k^2-1))*cos(2*k*x),k=1..n);\n
plot([f(x),FS(x,2),FS(x,5)],x=-Pi..3*Pi,color=[black,red,blue],\n     \+
             linestyle=[3,1],ytickmarks=[0,0.5,1],numpoints=60);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(-%$absG6#-%$sinG6#9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FS
Gf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF),&*&\"\"#\"\"\"%#PiG!\"\"F0-%
$SumG6$,$**\"\"%F0F1F2,&*&F8F0)%\"kGF/F0F0F0F2F2-%$cosG6#,$*(F/F0F<F09
$F0F0F0F2/F<;F09%F0F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 724 200 200 
{PLOTDATA 2 "6(-%'CURVESG6%7[v7$$!3*****4tk#fTJ!#<$\"3=5KT_Kzzi!#E7$$!
3i!**Q;\">)e3$F*$\"3'Hx92$G>ob!#>7$$!3!3)z'f<r,.$F*$\"3)=[F!e2\">6\"!#
=7$$!3WrpHS/YuHF*$\"3CNb`.@bj;F97$$!31ifi/(\\(=HF*$\"3owY\"3kJ+@#F97$$
!30T'y\"yC!=#GF*$\"3G.lh2QnVJF97$$!3/?8t^_&[s#F*$\"3oVHaeCzZSF97$$!3E;
u>cz#eh#F*$\"3:)\\e-pX(=]F97$$!3[7Nmg1!o]#F*$\"3&39o\"4/5IfF97$$!3#zgU
9m]qR#F*$\"3k#4iKj]jx'F97$$!3O.<Ai1I(G#F*$\"3mn')z'=h5a(F97$$!3'>L*e&G
s!y@F*$\"3TYyD%*[07#)F97$$!3egp&*3R%)o?F*$\"31tsDw#p^y)F97$$!3o#H8;Avv
'>F*$\"39[SN5d5B#*F97$$!3yC'pU`1j'=F*$\"3#QP$\\,o`m&*F97$$!3;^lBu'[9w
\"F*$\"3!\\l&fX#3)=)*F97$$!3wxM?93fc;F*$\"3?Zr.*4>K'**F97$$!3%ou\"*pxz
%H;F*$\"3'pi7U?'y#)**F97$$!3$f,!yR(oBg\"F*$\"3SMjpFj,&***F97$$!3-&GoDq
d_d\"F*$\"3;6hz![+*****F97$$!35alNlm9[:F*$\"3wz&QQ2Nu***F97$$!3=B[9Gc.
@:F*$\"37@u!)=>i()**F97$$!3E#4L4fCR\\\"F*$\"3cQ;,G#o/(**F97$$!3Oh8s`N
\"oY\"F*$\"3EpZy3m)f%**F97$$!3VI'4l^-(R9F*$\"3Tf*=U0&>9**F97$$!3%3RB%
\\hgJ8F*$\"3e]&))G`,`r*F97$$!3-^rL#y4NA\"F*$\"3[w$yV\"p*HS*F97$$!3%\\
\\c?Q>B6\"F*$\"3/-@zahFn*)F97$$!3()Qex\")*G6+\"F*$\"37Uul)R/3U)F97$$!3
e/W'o;C>.*F9$\"3!f\"oZEN2`yF97$$!3]?/(f^eD0)F9$\"343x?I&z+@(F97$$!3)GK
(=bZ-]pF9$\"3n4^g3M(QS'F97$$!3EDUS%*4\\ZeF9$\"3l+xAh=!*>bF97$$!3cerBf0
'RH&F9$\"3WL1@=;7]]F97$$!3!>4qS7I/u%F9$\"3YCp'y&>([c%F97$$!3ADI!*)o**o
=%F9$\"3))yb)eGRc1%F97$$!3aeft`#pLj$F9$\"35PyRSG&Rb$F97$$!31Qqv7,%**4$
F9$\"3Y]yuJ$H00$F97$$!3g<\"y<(4^mDF9$\"3Y4D>-xUQDF97$$!38(>*zI=3L?F9$
\"3`A7EuZ5>?F97$$!3mw-#)*o_'*\\\"F9$\"3'R)Rd8z.%\\\"F97$$!38JkQEzC:5F9
$\"36agH]Z]85F97$$!3,ce_H;V3`F3$\"3\\hHPO)QfI&F37$$!3OGmN7yS')GF3$\"3+
Ar[Mq+')GF37$$!3,2S(=&*RQk%!#?$\"3QQT'4EBQk%Fdy7$$\"3'p#=)>#)Rw&>F3$\"
3aD%3\\y9v&>F37$$\"3ia5:ROmzVF3$\"3-lw%[jj#yVF37$$\"3@q(p-[qR,\"F9$\"3
Xz+wtQB75F97$$\"3'\\VCmfu**e\"F9$\"3yyyT:RG$e\"F97$$\"3s*4zHryf;#F9$\"
3'oTq\"yB3\\@F97$$\"3ZkPLHG)>u#F9$\"3je0kqDv2FF97$$\"3BBsg6\\)yr$F9$\"
3Rl#H%>J#Gj$F97$$\"3+#o!)Q*py$p%F9$\"3*o[83mAL_%F97$$\"3$Q$f&e6$**GeF9
$\"3\\^$4!fzY/bF97$$\"3m&=JyB*>kpF9$\"3\\.Q:8Pv9kF97$$\"39_T04!\\!pzF9
$\"3=u(Q.#G'>:(F97$$\"3i=rF!y)*Q(*)F9$\"3'yw[Ll<q\"yF97$$\"3s,I%R#yj25
F*$\"33sd<X8tb%)F97$$\"3e\"He)pd)y6\"F*$\"39Ry#>)Hx\"**)F97$$\"3$GxPd)
y'GA\"F*$\"3J7+o(H53S*F97$$\"33ash,+&yK\"F*$\"3;6SC1TL1(*F97$$\"3?$z.R
\"*=EQ\"F*$\"3#3L?p:oM#)*F97$$\"3MK.>EyQP9F*$\"3t,q+mF96**F97$$\"3!=gL
BGsZY\"F*$\"3.Hw?Tr%Q%**F97$$\"3YroZQn:#\\\"F*$\"39a1_2]4p**F97$$\"39T
,i%>T&>:F*$\"3_c7aKu'o)**F97$$\"3g5Mw]c#pa\"F*$\"3)G!3T*3^r***F97$$\"3
W-lC41A(f\"F*$\"3QCFi*)*3l***F97$$\"3I%fHxc:vk\"F*$\"35\"**f@A&eq**F97
$$\"37'o7i_5yp\"F*$\"3*o#z^\\`W>**F97$$\"3)zx&p%[0\"[<F*$\"3%R6upq=K%)
*F97$$\"3Ur&p$*H0m&=F*$\"3Ag4'***)QVf*F97$$\"3'[OVS60^'>F*$\"3X4dw\"GB
EB*F97$$\"3C[?<Gf!y2#F*$\"3*e2yyZ/?u)F97$$\"3gJ2IUn]!>#F*$\"3A\"\\wI\"
oYS\")F97$$\"3WBTIKJh)G#F*$\"3%f-5I,NC`(F97$$\"3E:vIA&>nQ#F*$\"3m]xl#o
i>&oF97$$\"3sfNA6rn#\\#F*$\"3\"y(f))3tAVgF97$$\"3=/'R,qM')f#F*$\"3qJ)[
4,3n;&F97$$\"3-4NzypO`EF*$\"3y+XWFgf!p%F97$$\"3(QTZuD*43FF*$\"3(>R#4(H
O/?%F97$$\"3s=85O:$Gw#F*$\"3'4F8oy'p(p$F97$$\"3bB_v9Qc<GF*$\"3!f8\"RQJ
)Q=$F97$$\"33JM\\r'36(GF*$\"3-$\\+H`y>n#F97$$\"3/R;BGNlCHF*$\"31pp)p0:
C:#F97$$\"3-Z)p\\Q)>yHF*$\"3XOB7q>oE;F97$$\"3aa!3<CV<.$F*$\"35cg9;iG'4
\"F97$$\"3a2v#Qk]N3$F*$\"3$\\L/teU4!eF37$$\"3bgp%f/e`8$F*$\"3[;#*p_scM
iFdy7$$\"3a8k1[a;(=$F*$\"3e1JoC@qbXF37$$\"3cme=]G(*QKF*$\"3*[G)GthjA(*
F37$$\"3;h#>t8'\\'H$F*$\"3CI/%QRZGa\"F97$$\"3AcEXC%>SN$F*$\"3U_#3hSF$3
@F97$$\"3F^ge6Fa6MF*$\"3wr\\1PI$om#F97$$\"3)eW>()*f1pMF*$\"3cY*45u;l@$
F97$$\"3#*e7gi/WsNF*$\"3KiOr=GTwTF97$$\"3%>2$[E\\\"en$F*$\"3!*e,t5$=<4
&F97$$\"3WBW^J#*='y$F*$\"3[Y/9^,w3gF97$$\"3!\\xXl`jl*QF*$\"3!>$Q_KYn_o
F97$$\"39r9vh!zl*RF*$\"3i\"***[o*>ca(F97$$\"3%p;dpe%f'4%F*$\"3=\\3ur&[
J;)F97$$\"3kV!pN(z$f?%F*$\"3\"pnSZX:Zu)F97$$\"3N?4=g8G:VF*$\"3/+gYa[$=
A*F97$$\"3H<Lll\\;=WF*$\"3u&z$oSYFq&*F97$$\"3B9d7r&[5_%F*$\"3H&=(*z/-v
\")*F97$$\"3+_nEx+#[d%F*$\"3!Rxw]7Bb!**F97$$\"3l!z2Me\"fGYF*$\"3]!o9<_
5\\'**F97$$\"3_4$ykLxal%F*$\"3;7If%p4Q)**F97$$\"3GH)[&*3jBo%F*$\"37%)R
0oD\\&***F97$$\"3<[$>E%)[#4ZF*$\"3y&e9xp]*****F97$$\"31n)*o&fMht%F*$\"
3g2zBh3=(***F97$$\"3IYXeL.UiZF*$\"3uj=(*Gp[()**F97$$\"3cD#z921()y%F*$
\"3EUakRE*3(**F97$$\"3#R!RP4=*\\\"[F*$\"3GGqDe%4u%**F97$$\"3<$eosax7%[
F*$\"3(z9U$4O0<**F97$$\"3ySz0B!\\Q*[F*$\"3lfHAMD\"e$)*F97$$\"3G*HZ))\\
?k%\\F*$\"3Lz$>M*QRF(*F97$$\"3%\\h3]U&\\c]F*$\"3,ttGVFx8%*F97$$\"3cI*p
6Nql;&F*$\"3?iuSQ[?')*)F97$$\"3))z&yxe&es_F*$\"3Wz%\\]f1:Z)F97$$\"3?Hs
QC3gy`F*$\"3)*[u#3G%ohyF97$$\"3gt%Rfr;q[&F*$\"3G\"))R'RY\"o9(F97$$\"3+
=<\\2EV&f&F*$\"35Lw+\\N-[jF97$$\"3R2Rsg2&Hq&F*$\"3AI'f16,A[&F97$$\"3x'
4cR\"*o/\"eF*$\"3'Re6ehkIb%F97$$\"3%Q0t.$pE4fF*$\"3e)QSSceEl$F97$$\"3
\"4,!zY\\13gF*$\"3JzlgGxi;FF97$$\"3QW@([y\"okgF*$\"3)zzh]q*on@F97$$\"3
&yFaHi)H@hF*$\"3z\\#*)430=h\"F97$$\"3I6k.ha\"z<'F*$\"335B#*\\_v]5F97$$
\"3zW&=\"*HKXB'F*$\"3o[Mpy:Qj[F37$$\"3M))fd'o^)fiF*$\"3P=&3*H.:LBF37$$
\"3!>VLS2r^G'F*$\"3KIlx[bw&)>Fdy7$$\"3Xv3\\h/\\5jF*$\"3td\"=Z9w,t#F37$
$\"36=$[*[)4eL'F*$\"3m(ym-fD+E&F37$$\"3K/K'Qi[kQ'F*$\"3*3#H-I*)zI5F97$
$\"3W\"4y()R(3PkF*$\"3/8c..E&H`\"F97$$\"3Y?!yY*)y5\\'F*$\"3O-]eBI*R1#F
97$$\"3[\\zd!Rq]a'F*$\"3W)\\Xu>=!*e#F97$$\"3]yyZ')=1*f'F*$\"3KlQ80!)\\
1JF97$$\"3_2yP#Q`Il'F*$\"319D=<V#\\h$F97$$\"3Yj&\\Q)oScnF*$\"3M@%o&e=c
dXF97$$\"3Q>8K&Qg(foF*$\"3K-[(**3f:X&F97$$\"3?<ECm6_upF*$\"3)G,\"Q**)o
cP'F97$$\"3-:R;Z>G*3(F*$\"3v\")H)*GI!f@(F97$$\"3K+90^jl)=(F*$\"3e(G@qC
gr'yF97$$\"3k&))Q\\vI!)G(F*$\"3(Q'4**H5zS%)F97$$\"3/)e.Af@3S(F*$\"3#\\
(=;&*4o!**)F97$$\"3W!Ho%HCh8vF*$\"3'4(fRjZJE%*F97$$\"3gV$eI.4kh(F*$\"3
$3.Mt6@\">(*F97$$\"3w'R[mj0#>xF*$\"33$=N$\\XJ4**F97$$\"3#QAZ*okJZxF*$
\"3nV[XGm;V**F97$$\"3w^gC,tUvxF*$\"39/O&o%>;p**F97$$\"3oz[aL\"QN!yF*$
\"3yMC*Q'*zs)**F97$$\"3t1P%e'*[;$yF*$\"3%4w9LO1v***F97$$\"3yLD9)zf(fyF
*$\"3C&peX1L)****F97$$\"3sh8WI1(y)yF*$\"3)*G?9H#eU***F97$$\"3l*=SFY\")
f\"zF*$\"3CT\\6iiy!)**F97$$\"3q;!R]H#4WzF*$\"3!4Fz&3yUf**F97$$\"3^ID$=
(=6T!)F*$\"3XA5=3<UD)*F97$$\"35Ygi[98Q\")F*$\"3A7MX<M+*f*F97$$\"3O%f8<
UD'[#)F*$\"3;(\\p>zN8B*F97$$\"3'39,[R>\"f$)F*$\"3K&R*3k\"y5v)F97$$\"3o
*=Qz6^eY)F*$\"3Ml+<'>cd=)F97$$\"3sO_2TGes&)F*$\"3MkgPfLFFvF97$$\"3=3'f
*Q[Cz')F*$\"3=I#Q[0OOy'F97$$\"3)y(R%o$o!fy)F*$\"3u#)yAVf*G'fF97$$\"3'o
'>[Wk<#*))F*$\"3m_^CYBwx]F97$$\"3%e&*>@0Y%)**)F*$\"3/AbGg$Q`8%F97$$\"3
,W-9wy`+\"*F*$\"3%QgsPf&)e=$F97$$\"3%R`g,qHE?*F*$\"3weu\\**fD.AF97$$\"
3y+Hggn;e#*F*$\"3G?^CXVTe;F97$$\"3%eEX5#Qq8$*F*$\"3=X#4.&)e%36F97$$\"3
oKw[\")3Cp$*F*$\"3c:%)oAM&3b&F37$$\"3^***H>%zxC%*F*$\"3)Q)G8?!QR)=!#D-
%'COLOURG6&%$RGBG\"\"!FfbnFfbn-%*LINESTYLEG6#\"\"$-F$6%7]w7$F($\"35k^t
W&RKF\"F97$F5$\"35G$p'[75h9F97$F@$\"30&em\"zoH.?F97$$!3K,BS\"4w-(GF*$
\"3Qya1E$*fNBF97$FE$\"3%R1iTT!)or#F97$$!3w!)\\&\\')GLx#F*$\"3%)H!3BA)G
RJF97$FJ$\"3Slog-xK%f$F97$$!3#zOkRgT.n#F*$\"3k0`F\"4%)R8%F97$FO$\"3'fH
Q]PR3p%F97$$!3gk/V3VJhDF*$\"3Q:i-?VI__F97$FT$\"3;7!3'p)ek!eF97$$!3UgI0
hc#>X#F*$\"3K7!=Do7fM'F97$FY$\"3-U>Yk?IdoF97$$!3Ub@$=mv@M#F*$\"3lPFK:)
pEL(F97$Fhn$\"3#Q+q#zr$fw(F97$F]o$\"3+e#*4Ulq)[)F97$Fbo$\"3'z))3#=xV?!
*F97$Fgo$\"3m9Hc$f;lN*F97$F\\p$\"3')fd\"*H:no&*F97$$!3(z3`UgxQ\"=F*$\"
3)4#Q%H:g.k*F97$Fap$\"31\"f/?o$y\"p*F97$$!3d9+AW(>!4<F*$\"3m!=[c(p#os*
F97$Ffp$\"3!)3:u!zr'[(*F97$F`q$\"3O)\\-gu/)f(*F97$Fjq$\"35aOpI1jg(*F97
$Fdr$\"3a?7x^[D^(*F97$F^s$\"3]+`lc?^I(*F97$Fcs$\"3SP4@];#[k*F97$Fhs$\"
3+07k(*ftt%*F97$F]t$\"3+1Nx)**>%o\"*F97$Fbt$\"3)=()QBT7Ip)F97$Fgt$\"3#
3t2yfwJ6)F97$F\\u$\"3)=94Fvg3Q(F97$$!3pr)ybj\"H,vF9$\"3c;t%)foY2pF97$F
au$\"3i!*y3V!3qR'F97$$!33udzuyv)R'F9$\"3k)Hgcaot&eF97$Ffu$\"3M?-'o'>=)
H&F97$F[v$\"3=y=P)fL#GZF97$F`v$\"3+6J[g\\>iTF97$Fev$\"3.(fN1,:Kh$F97$F
jv$\"3Gp?^a%=Z4$F97$F_w$\"3YlBgbeCOEF97$Fdw$\"3p&H/,pc&HAF97$Fiw$\"3Qi
S9h,%[)=F97$F^x$\"3,Q&3%=M!4h\"F97$Fcx$\"3au&z+Tm%H9F97$Fhx$\"3RK'3&=.
?;8F97$F]y$\"3mX>Uk1'fG\"F97$Fby$\"3)p3]>.pNF\"F97$Fhy$\"3u'e!z/K4z7F9
7$F]z$\"3<[tugH]-8F97$Fgz$\"3/\\2F@8)>l\"F97$Fa[l$\"3'3,*3]1(pN#F97$$
\"3&Q\\q/(Q$*HKF9$\"326*)R\"*\\hVFF97$Ff[l$\"3qa`R\\7QrJF97$$\"3h_Ru_f
$e?%F9$\"3C7%=&R%o:j$F97$F[\\l$\"3#e\"*fET!4:TF97$$\"3!zIo[0!Rh_F9$\"3
SK3]sHp%p%F97$F`\\l$\"3w2gH&R1#z_F97$$\"3ufN%o<'f'R'F9$\"3MW\\2Xr?beF9
7$Fe\\l$\"3nZuh3k`5kF97$$\"3!*oEWBTimuF9$\"37c<tf\\QwoF97$Fj\\l$\"3kCj
'[t)y6tF97$$\"3QNcm%*QZr%)F9$\"3R4PAWa27xF97$F_]l$\"3#Q7A+lcR2)F97$Fd]
l$\"3L<(*Rb42E()F97$Fi]l$\"3E]ihj3Y(=*F97$F^^l$\"3!*HD(=)**Qs%*F97$Fc^
l$\"3ICI(Hu;0k*F97$Fh^l$\"3%RU\")fFyPp*F97$F]_l$\"3X[ZYX=MH(*F97$Fg_l$
\"3'3hq&pvw](*F97$Fa`l$\"3#3^oz\"Q`g(*F97$Ff`l$\"3g&Ro$pZJg(*F97$F[al$
\"3%[)*QFK'H^(*F97$F`al$\"3Z.%*HeT^K(*F97$Feal$\"3=I3Qkg6-(*F97$Fjal$
\"3='=*f6'eOe*F97$F_bl$\"3;6Zl^./j$*F97$Fdbl$\"3EFQ!3.dP)*)F97$Fibl$\"
3\\gJ&*QD8;%)F97$F^cl$\"3iWm'>Y,hv(F97$Fccl$\"3k-')p15w\\pF97$$\"3GPbw
;$)pRCF*$\"3O)o&zZ.gikF97$Fhcl$\"3%R%4l>_ZZfF97$$\"3=#e\"o0flXDF*$\"3r
CMY'3FFT&F97$F]dl$\"3)y;>\"zb+o[F97$Fbdl$\"3vEyPDU81VF97$Fgdl$\"3c7IS9
?gdPF97$F\\el$\"3u;uk%zj`B$F97$Fael$\"3ORxzA^H_FF97$F[fl$\"3IDR)o$z^m>
F97$Fefl$\"3Y<DjXo)eX\"F97$$\"3E\"ynF%pkdIF*$\"3/a'4Qn>.Q\"F97$Fjfl$\"
3&Ge(3!\\xXK\"F97$$\"3#QB()[Ma%4JF*$\"3#z%z\\xz+*G\"F97$F_gl$\"3'GT8<U
LQF\"F97$$\"3G(o1quh7;$F*$\"3O!>WSv[\"z7F97$Fdgl$\"38-)*>:2#\\I\"F97$$
\"3$)Rh7\\\"pI@$F*$\"3%RY(H[\"))4N\"F97$Figl$\"3w&>X\"=M1<9F97$Fchl$\"
3\"ot6_8v*Q>F97$F]il$\"3MHDo:EI\"y#F97$$\"3<_.mIKv?NF*$\"3(RHn*f)[!RKF
97$Fbil$\"3_tntM$\\;t$F97$$\"3ml@a%pFTi$F*$\"3%\\uxHGg#[UF97$Fgil$\"35
B?5\"*p&zx%F97$$\"3pZ()**y?+JPF*$\"3'4]>H;XfM&F97$F\\jl$\"3@%4G^V$\\/f
F97$$\"3<*4ISQw8%QF*$\"3]UBLrEcUkF97$Fajl$\"3AUnWn&H1&pF97$Ffjl$\"3m)y
1aWI6x(F97$F[[m$\"3-;!\\(3!f#R%)F97$F`[m$\"3;FU'GGyg)*)F97$Fe[m$\"3'[A
NO$Gkb$*F97$Fj[m$\"3_B?DJjqq&*F97$F_\\m$\"3qLsr:FA\"p*F97$Fd\\m$\"3-p$
*['Gvrs*F97$Fi\\m$\"3]@MTeoF\\(*F97$Fc]m$\"3UVnH%fn*f(*F97$F]^m$\"3s*3
l$pRag(*F97$Fg^m$\"3Cs*Q-E09v*F97$Fa_m$\"3iG?%z!QgJ(*F97$Ff_m$\"3A7,Ip
:,*p*F97$F[`m$\"3[+zLowc]'*F97$F``m$\"3#*HW^n-M![*F97$Fe`m$\"3m&>UGS^J
=*F97$Fj`m$\"3C<(Gr'=*3u)F97$F_am$\"3+`$H[L0D7)F97$$\"3S^L;q(3GV&F*$\"
3poKCbi^OxF97$Fdam$\"3Ys7VW;l0tF97$$\"3!ef:<mC7a&F*$\"3/8)yQ:)*R$oF97$
Fiam$\"3p9LP(*\\^FjF97$$\"3E7y5%o\">\\cF*$\"3UPxIX#\\%)z&F97$F^bm$\"3Q
=$GLha=D&F97$$\"3_-+MP)4nv&F*$\"3EtH*)Qr:)p%F97$Fcbm$\"3%>!=xt[y[TF97$
$\"3IvX;Az')feF*$\"31IJT'yE\"eOF97$Fhbm$\"3aSAb@l&3>$F97$$\"3QK:eQfmef
F*$\"3ah<AaOTcFF97$F]cm$\"3')4w/%GXQO#F97$Fgcm$\"3yjB-<()el;F97$Fadm$
\"398UP$))R$48F97$Ffdm$\"3Sf^1EZb\"G\"F97$F[em$\"3=UzF$z*Ht7F97$F`em$
\"31P\\wLWi%G\"F97$Feem$\"3I-\"*4_4Y:8F97$Fjem$\"3Hmp)z,=[V\"F97$F_fm$
\"3bCdjcX`G;F97$Fdfm$\"3fVDx\\h!=\">F97$Fifm$\"3_/*pgLuoE#F97$F^gm$\"3
!H#)zi9hWo#F97$Fcgm$\"3-'*RE<V(Q:$F97$$\"3/&o8J8IZq'F*$\"35F<\"zP)*3k$
F97$Fhgm$\"3M]ZyQ.*Q:%F97$$\"3'=W&eMO33oF*$\"3y5D!\\q?>o%F97$F]hm$\"3v
h0\\eWE9_F97$$\"3Ho>yv29<pF*$\"3=;H&ROm!)z&F97$Fbhm$\"3c4gncO#>O'F97$$
\"36mKqc:!>.(F*$\"3aKe*>v;[*oF97$Fghm$\"3'z')ymTgxQ(F97$F\\im$\"3U**R^
k-VG\")F97$Faim$\"3THGu*)3(>r)F97$Ffim$\"3!e())y$e;m=*F97$F[jm$\"3mXZf
sg'z[*F97$F`jm$\"3u4B5oRkY'*F97$Fejm$\"3d2+s?(Q'G(*F97$F_[n$\"3))=^zW8
z](*F97$Fi[n$\"3SCHWF\\lg(*F97$Fc\\n$\"3I(fQ'z_af(*F97$F]]n$\"3%pSVOJ5
tu*F97$$\"3gtdV$3-E*zF*$\"3wj\"y?p9ms*F97$Fb]n$\"3[CPxa8h%p*F97$$\"3U(
GH-m@'*3)F*$\"3/VB0okz['*F97$Fg]n$\"3QZ1'=^Rhe*F97$F\\^n$\"3QrKI3*f@O*
F97$Fa^n$\"3mm]gHr^\"**)F97$Ff^n$\"3i$Hx<M)=i%)F97$F[_n$\"3;[G6p/@]xF9
7$$\"3d@u,SQ\"fi)F*$\"35SL<3=3GtF97$F`_n$\"3a1Bdl&Gi'oF97$$\"3/$z,z$ed
K()F*$\"37*=.d:C-P'F97$Fe_n$\"3CVxHQ2KZeF97$$\"3EtHmS;/R))F*$\"3#H\\a
\\@0$3`F97$Fj_n$\"3cG>$z(fDhZF97$$\"3[g4I[7JX*)F*$\"3&z%\\bf+7<UF97$F_
`n$\"3O#4gqU&\\(o$F97$$\"3%**4IT'>\\\\!*F*$\"3GP5DJSR.KF97$Fd`n$\"3q!Q
9QDoSv#F97$$\"3')*Q]\")y$e^\"*F*$\"3IT>,>W]\\BF97$Fi`n$\"35gE$)4e#*)*>
F97$Fcan$\"3'zD/t-V*f9F97$F]bn$\"3ao^tW&RKF\"F9-Fcbn6&Febn$\"*++++\"!
\")$FfbnFfbnF^]q-Fhbn6#\"\"\"-F$6%7gv7$F($\"3)Q%*ogZ_uy&F37$F/$\"3#)*o
yP&3w`oF37$F5$\"3f1@9Y'p$>**F37$F;$\"3SI>TUuEh9F97$F@$\"3*4K147m,/#F97
$Fhcn$\"3](p*)=b=ue#F97$FE$\"3w^yhv9,QJF97$F`dn$\"3+uPRIknkOF97$FJ$\"3
Wm=$yo84:%F97$FO$\"3u(>@!\\A*e3&F97$FT$\"3OsXq:3e\"*eF97$FY$\"3t(HTAy!
4.nF97$Fhn$\"3L.dY/PRHvF97$F]o$\"3?G*ob*\\Am#)F97$Fbo$\"3tGh-ERYH))F97
$Fgo$\"3^2gKz>!z?*F97$F\\p$\"3R]B))y;C8&*F97$Fap$\"3%GA*3vH#3z*F97$Ffp
$\"3ex&R@S$3$***F97$F[q$\"3uqF!3`!R-5F*7$F`q$\"3vk\\+wVO/5F*7$Feq$\"3)
e[Tl5j^+\"F*7$Fjq$\"3b)R,5<fZ+\"F*7$Fdr$\"3Z+xqS&R/+\"F*7$F^s$\"3K)4&)
\\?7(>**F97$Fcs$\"3e%RBt/2ym*F97$Fhs$\"3?]Cjg5^i$*F97$F]t$\"3cUdB$)eW!
**)F97$Fbt$\"3a*RL%o_O\"[)F97$Fgt$\"3cuNe]V%p(yF97$F\\u$\"3s&)zoNZVjrF
97$Fau$\"3dCeM[V?KjF97$Ffu$\"3t_%\\Jn2\"GbF97$F[v$\"3XRe%QctR6&F97$F`v
$\"3I.&fE22sm%F97$Fev$\"3))p__)fU&pTF97$Fjv$\"3Alf6$G2Gh$F97$F_w$\"3J$
pKN\"Q$z-$F97$Fdw$\"38+!4)QE+?CF97$Fiw$\"3=@J=c1QG=F97$F^x$\"3B-k&eh&G
-8F97$Fcx$\"3#[#=C!R'eU#*F37$Fhx$\"3#p^hI`Slv'F37$F]y$\"3GI@&GkFf2'F37
$Fby$\"3GQwC\"RS\\z&F37$$\"3GJ@(R8ziY(Fdy$\"3*[q\\X;1o!eF37$Fhy$\"3Iq&
3hNa.#fF37$$\"3!3Wm0t^'oJF3$\"3nU)G?*z*[8'F37$F]z$\"3;+%))\\1U\"\\kF37
$Fbz$\"3*fh,$\\1?M#*F37$Fgz$\"31d58cc_%Q\"F97$F\\[l$\"3#f&yi3Gmr>F97$F
a[l$\"3lb![\"*[s.i#F97$Fb^o$\"3!\\*[v(R%ztJF97$Ff[l$\"3!RLN_:A:q$F97$F
j^o$\"3%><%[#HPv=%F97$F[\\l$\"3X9cc4wVFYF97$F`\\l$\"38yu!*o:d9bF97$Fe
\\l$\"3E/mlJpqUjF97$Fj\\l$\"3S)))y4L8-5(F97$F_]l$\"309.[1L)p$yF97$Fd]l
$\"3Rhbj!H3j^)F97$Fi]l$\"3N*>'erZd6!*F97$F^^l$\"3wYV\"R#Heg$*F97$Fc^l$
\"3af#*e43$yl*F97$F]_l$\"3B'***QJhJ:**F97$Fa`l$\"3Oz1iTFr/5F*7$$\"3kc
\\+IJ2s:F*$\"3g]?41\"y^+\"F*7$Ff`l$\"3k'Q-#Qyg/5F*7$$\"3E[!)[)3oBi\"F*
$\"3Rg_P4*=I+\"F*7$F[al$\"3Hx-H**yX+5F*7$F`al$\"3*ym\"*z`>t#**F97$Feal
$\"3#*H5nx'GA#)*F97$Fjal$\"3h6bZOAcS&*F97$F_bl$\"39rw?`S&f@*F97$Fdbl$
\"3#ppR'fp7!z)F97$Fibl$\"3ml`1*zM->)F97$F^cl$\"3>#e(**GUx>vF97$Fccl$\"
3aCMpD#f7y'F97$Fhcl$\"3=fXjDA:%*fF97$F]dl$\"3W&yd#e#[w@&F97$Fbdl$\"3[Y
IcV^\"fy%F97$Fgdl$\"3X,8sXtL3VF97$F\\el$\"3=jv_\"=^Px$F97$Fael$\"39(ou
(>rL&=$F97$Ffel$\"3=T&*GD!**yd#F97$F[fl$\"3U!eJ;G)Qv>F97$F`fl$\"3C5r.#
3-dU\"F97$Fefl$\"3@oU,6eL3)*F37$Fjfl$\"3z9DZap\"Q%pF37$F_gl$\"37uO>\"G
[4!eF37$$\"3qBoZ'*)4$[JF*$\"3\\\\*>?I=J!eF37$Fjgo$\"3Fs#z#\\Uh@fF37$$
\"3S]l`(f8U<$F*$\"3ia0YzhhbhF37$Fdgl$\"3:WN35*3N]'F37$Fbho$\"3!GP.7=`1
`(F37$Figl$\"379%y8Zo^(*)F37$F^hl$\"3K\"\\,-S[oM\"F97$Fchl$\"36x)[`E/j
#>F97$Fhhl$\"3I\"zP%\\ZzrDF97$F]il$\"3OMDN:ekBKF97$F`io$\"3)GoP?=iyx$F
97$Fbil$\"3+s'R#Hy%QG%F97$Fhio$\"3gW!3%)*enQZF97$Fgil$\"3u`k&z?x5:&F97
$F\\jl$\"3+s4PwFvifF97$Fajl$\"3E'3gSv**>y'F97$Ffjl$\"3\"fH%R<#yW`(F97$
F[[m$\"3Lp!zIP>W@)F97$F`[m$\"35FCbY?h#z)F97$Fe[m$\"3d<&QkEFo?*F97$Fj[m
$\"3'y8Ou]zo^*F97$F_\\m$\"3#37CShq\"*y*F97$Fd\\m$\"3b2OD$zss!**F97$Fi
\\m$\"3OHNE@<r&***F97$F^]m$\"3vaJ6&HaD+\"F*7$Fc]m$\"36AOl#*>W/5F*7$Fh]
m$\"3Ls(*>]8<05F*7$F]^m$\"3(*Q$R\"3wr/5F*7$Fa_m$\"3RJ7m#)o%Q#**F97$F[`
m$\"3G]i8KqT\"o*F97$F``m$\"3!4;dx#)Q?P*F97$Fe`m$\"3!\\Un&47y1!*F97$Fj`
m$\"3A9R([l%)>`)F97$F_am$\"3-XQf#fsk)yF97$Fdam$\"3KQq,`Fj%4(F97$Fiam$
\"3kroHkLfyiF97$F^bm$\"3_xut5t/&\\&F97$Fcbm$\"3!)H46oO$fl%F97$Ff`p$\"3
whlNU6R8UF97$Fhbm$\"3FD>v1$zOs$F97$F^ap$\"3_K.#=1,3>$F97$F]cm$\"3ELgk%
z<4j#F97$Fbcm$\"3m8[R%)Q\\#*>F97$Fgcm$\"3k&QNS0-:T\"F97$F\\dm$\"3il][T
GV#\\*F37$Fadm$\"3q.VVM<w-mF37$$\"3cms%G*>>ZiF*$\"3ZEo-)G;`B'F37$Ffdm$
\"31cu,3)fh(fF37$$\"375ZI!Q6DF'F*$\"3+eOyk(**p#eF37$F[em$\"3C_OtY<#))y
&F37$$\"3m`@wn2$yH'F*$\"3uX1C]$y='eF37$F`em$\"3enSbIjoXgF37$$\"3B(f>_:
]JK'F*$\"3aq*>=0K!RjF37$Feem$\"30c6Fj^)*RnF37$Fjem$\"3'3G-]f4wN*F37$F_
fm$\"3dWD'3_KxL\"F97$Fdfm$\"3WnR2g/Sx=F97$Fifm$\"31zQ\")[nmzCF97$F^gm$
\"31w+(eFYT4$F97$Fcgm$\"3'pVMZLu9o$F97$Fjcp$\"3_>j$3eWm>%F97$Fhgm$\"3#
GR[p:I-m%F97$Fbdp$\"3a#Qcw2r\"z]F97$F]hm$\"3kcEIHV=oaF97$Fbhm$\"3u&\\U
.9o]I'F97$Fghm$\"3g#G<F_\"zprF97$F\\im$\"3GSF,g(GD*yF97$Faim$\"3q6/`u/
S,&)F97$Ffim$\"3P3/e(4N1,*F97$F[jm$\"3%y`K6p%=$Q*F97$F`jm$\"3Opn_Si3s'
*F97$Fejm$\"3/j'p\\G#p7**F97$F_[n$\"3pz\">=yM-+\"F*7$Fi[n$\"3mug^H3x/5
F*7$F^\\n$\"3:EdVS?:05F*7$Fc\\n$\"3/>b\\S4C/5F*7$Fh\\n$\"3(Q]m'o72-5F*
7$F]]n$\"3%H<B;Q;s)**F97$Fb]n$\"3[#))QeKK#*z*F97$Fg]n$\"3q=h]Ku?X&*F97
$F\\^n$\"3X#G97Vk[@*F97$Fa^n$\"3%H=N$yoV)z)F97$Ff^n$\"3Lt[b*fE%Q#)F97$
F[_n$\"3(*)e*Q9#>S^(F97$F`_n$\"3g1@0t3e5nF97$Fe_n$\"3+**=O%ov6#fF97$Fj
_n$\"3\"QbwL(QkQ^F97$F_`n$\"38#pRnQH<C%F97$Fd[q$\"3:w0M]L'yt$F97$Fd`n$
\"3!R1+U\\!p(=$F97$F\\\\q$\"3X))*4_L*))3EF97$Fi`n$\"3<LFMKI^K?F97$F^an
$\"3=V]4g+Gc9F97$Fcan$\"3%ey$ynXs%*)*F37$Fhan$\"3[+v%>i$>ZoF37$F]bn$\"
3!\\0pgZ_uy&F3-Fcbn6&FebnF^]qF^]qF[]qFgbn-%*AXESTICKSG6$%(DEFAULTG7%Ff
bn$\"\"&!\"\"Fa]q-%+AXESLABELSG6$Q\"x6\"Q!Fhbs-%%VIEWG6$;$!+aEfTJ!\"*$
\"+izxC%*F`csF_bs" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The following graph plots " }
{XPPEDIT 18 0 "abs(c[k]);" "6#-%$absG6#&%\"cG6#%\"kG" }{TEXT -1 46 " a
gainst the the associated angular frequency " }{XPPEDIT 18 0 "2*k;" "6
#*&\"\"#\"\"\"%\"kGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "c := k -> -2/(Pi*(4*k^2
-1)):\nplot([[seq([2*k,evalf(c(k))],k=-5..5)]$3],style=point,\n       \+
symbol=[circle,diamond,cross],color=[COLOR(RGB,.4,0,.9)$3]);" }}{PARA 
13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%7-7$$!#5\"
\"!$!3K+++\\F]Ik!#?7$$!\")F*$!3%******\\d20,\"!#>7$$!\"'F*$!3$******\\
j8*==F37$$!\"%F*$!3M+++9=8WUF37$$!\"#F*$!3.+++2f1A@!#=7$$F*F*$\"3!****
**>s(>mjFC7$$\"\"#F*FA7$$\"\"%F*F<7$$\"\"'F*F77$$\"\")F*F17$$\"#5F*F+-
%&COLORG6&%$RGBG$FM!\"\"F*$\"\"*Ffn-%'SYMBOLG6#%'CIRCLEG-F$6%F&FW-Fjn6
#%(DIAMONDG-F$6%F&FW-Fjn6#%&CROSSG-%&STYLEG6#%&POINTG-%+AXESLABELSG6$Q
!6\"F^p-%%VIEWG6$%(DEFAULTGFcp" 1 5 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "We find the \+
complex Fourier series " }{XPPEDIT 18 0 "Sum(c[k]*exp(k*i*x),k=-infini
ty..infinity)" "6#-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#*(F*F+%\"iGF+
%\"xGF+F+/F*;,$%)infinityG!\"\"F5" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "
f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " where" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([exp(-i*x), -Pi <= x a
nd x < 0],[exp(i*x), 0 <= x and x < Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWI
SEG6$7$-%$expG6#,$*&%\"iG\"\"\"F'F2!\"\"31,$%#PiGF3F'2F'\"\"!7$-F-6#*&
F1F2F'F231F9F'2F'F7" }{TEXT -1 2 ", " }}{PARA 0 "" 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 10 "Note that " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = cos*x+i*abs(sin*x);" "6#/-%\"f
G6#%\"xG,&*&%$cosG\"\"\"F'F+F+*&%\"iGF+-%$absG6#*&%$sinGF+F'F+F+F+" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 13 "The function " }{XPPEDIT 18 0 "exp(i*x) = cos*x+i*sin*x;
" "6#/-%$expG6#*&%\"iG\"\"\"%\"xGF),&*&%$cosGF)F*F)F)*(F(F)%$sinGF)F*F
)F)" }{TEXT -1 147 " can be thought of as \"wrapping\"  the real line \+
around the unit circle with its centre at the origin in the complex pl
ane. Similarly, the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 132 " described above can be thought of as  wrapping the r
eal line back and forth across the upper semi-circular part of the uni
t circle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 119 "plot([cos(x),sin(x),x=0..Pi],color=coral,thickness=2
,\n  labels=[`real`,`imag`],xtickmarks=3,ytickmarks=[0=`0`,1=`i`]);\n
" }}{PARA 13 "" 1 "" {GLPLOT2D 344 179 179 {PLOTDATA 2 "6(-%'CURVESG6#
7S7$$\"\"\"\"\"!F*7$$\"1)\\W\"HJcw**!#;$\"1q+d@uUUo!#<7$$\"1r#\\#p_6=*
*F.$\"1[%)4&*>5x7F.7$$\"1zC-I#[.\")*F.$\"1*>O?m:$Q>F.7$$\"1)HHf&*)Rd'*
F.$\"1)>yf`>^f#F.7$$\"1'*R4Kvch%*F.$\"1@(eB<)3PKF.7$$\"17SL)R3>C*F.$\"
1,zuO+J>QF.7$$\"1&)o(eEjn(*)F.$\"1U\"zKNalS%F.7$$\"1u\"z-'HOj')F.$\"1y
p0=8h%*\\F.7$$\"1\\q#Q(Ho7$)F.$\"1VQ$)RSrebF.7$$\"1*fHS#zr8zF.$\"1uV=o
;N8hF.7$$\"1eXk6bkJvF.$\"1>@'=m@$ylF.7$$\"1Q.n/k&*oqF.$\"1smy)eyJ2(F.7
$$\"1Q[!\\UE<d'F.$\"1R\"oZ'3SPvF.7$$\"1dPuw?vjgF.$\"1!Gv*Qmy^zF.7$$\"1
`,,Xd#)zbF.$\"1)\\(pbm_)H)F.7$$\"1V))>3uzy\\F.$\"1]$RB!3Ys')F.7$$\"1)e
b@\"f+]WF.$\"1&e+@v-`&*)F.7$$\"1b-rmU*[\"QF.$\"10V7C?tV#*F.7$$\"1@>94^
7PKF.$\"1IP#[*[bh%*F.7$$\"14B(*3`5*e#F.$\"1Ij+\"38!f'*F.7$$\"1Czu,%[4'
>F.$\"1B7VW$\\e!)*F.7$$\"1/xP3T(oH\"F.$\"12`:\"G\\b\"**F.7$$\"13BW%>!*
z\"oF1$\"1nE@J/tw**F.7$$\"1&*=b#ec1b\"!#=$\"0f8t(z)*****!#:7$$!16SKN^;
mnF1$\"14'*zXK3x**F.7$$!1@vEKjew7F.$\"1qAta;==**F.7$$!17_fjQH>>F.$\"1w
$4\"yt39)*F.7$$!1r=G7zpuDF.$\"1OkT\"[jGm*F.7$$!1S>V-7j/KF.$\"1L5WG*4EZ
*F.7$$!1`xMXL%4!QF.$\"17Y_lsZ\\#*F.7$$!1'*HrKB*[W%F.$\"1\"y7g$>%y&*)F.
7$$!1#4]_i`Y+&F.$\"13)Q%)flvl)F.7$$!1k#eVXs*zbF.$\"1@)z#3!G%)H)F.7$$!1
WZ0b**>zgF.$\"1exilG)*RzF.7$$!1k1Y5)[')f'F.$\"1j=%o\"G%Q^(F.7$$!13'\\U
0]-1(F.$\"14j1X!p=3(F.7$$!1IyW%*Qc7vF.$\"1FrLF[5+mF.7$$!1EdsL13BzF.$\"
1kGuUC@,hF.7$$!1/esK4R<$)F.$\"1\"Q*\\Vrm^bF.7$$!1Rlozn?h')F.$\"1.i^R%
\\$)*\\F.7$$!1!3Z7B%yu*)F.$\"1<$R^j$e5WF.7$$!1HHo[=VY#*F.$\"1#o>S%eM3Q
F.7$$!1R1516\\g%*F.$\"1bj`7FBSKF.7$$!1c)4yk-Hm*F.$\"14`(f\"4buDF.7$$!1
aNi=gL/)*F.$\"1is\"zv+&o>F.7$$!1JA3TjH8**F.$\"1)f2vk%)RJ\"F.7$$!1`8!pw
6n(**F.$\"1(R(G%y?2#oF17$$!\"\"F*$!1BmIq&o?5%!#D-%*THICKNESSG6#\"\"#-%
'COLOURG6&%$RGBG$\"*++++\"!\")$\")AR!)\\Fd[lF*-%+AXESLABELSG6$%%realG%
%imagG-%*AXESTICKSG6$\"\"$7$/F*%\"0G/F)%\"iG-%%VIEWG6$%(DEFAULTGFh\\l
" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 10 "Note that " }{XPPEDIT 18 0 "f(-2*Pi)=f(0)" "6#/-%\"fG6#,$*&\"\"
#\"\"\"%#PiGF*!\"\"-F%6#\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "f(2*P
i) = 1" "6#/-%\"fG6#*&\"\"#\"\"\"%#PiGF)F)" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "f(-3*Pi)=f(-Pi)" "6#/-%\"fG6#,$*&\"\"$\"\"\"%#PiGF*!\"
\"-F%6#,$F+F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "f(Pi) = f(3*Pi)" "6#/
-%\"fG6#%#PiG-F%6#*&\"\"$\"\"\"F'F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 
"-1" "6#,$\"\"\"!\"\"" }{TEXT -1 5 " and " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(-5*Pi/2) = f
(-3*Pi/2)" "6#/-%\"fG6#,$*(\"\"&\"\"\"%#PiGF*\"\"#!\"\"F--F%6#,$*(\"\"
$F*F+F*F,F-F-" }{XPPEDIT 18 0 "`` = f(-Pi/2);" "6#/%!G-%\"fG6#,$*&%#Pi
G\"\"\"\"\"#!\"\"F-" }{XPPEDIT 18 0 " ``= f(Pi/2)" "6#/%!G-%\"fG6#*&%#
PiG\"\"\"\"\"#!\"\"" }{XPPEDIT 18 0 "`` = f(3*Pi/2);" "6#/%!G-%\"fG6#*
(\"\"$\"\"\"%#PiGF*\"\"#!\"\"" }{XPPEDIT 18 0 " ``= f(5*Pi/2)" "6#/%!G
-%\"fG6#*(\"\"&\"\"\"%#PiGF*\"\"#!\"\"" }{XPPEDIT 18 0 "`` =i" "6#/%!G
%\"iG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 46 "The coefficients in the complex Fourier series" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 19 "First s
uppose that " }{XPPEDIT 18 0 "k<> 1" "6#0%\"kG\"\"\"" }{TEXT -1 5 " an
d " }{XPPEDIT 18 0 "k<>(-1)" "6#0%\"kG,$\"\"\"!\"\"" }{TEXT -1 2 ". " 
}}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "c[k] = 1/(2*Pi)" "6#/&%\"cG6#%\"kG*&\"\"\"F)*&\"\"#F
)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*exp(-k*i*x),x=
-Pi..Pi)" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*(%\"kGF+%\"iGF+
F*F+!\"\"F+/F*;,$%#PiGF3F7" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#P
iGF&!\"\"" }{XPPEDIT 18 0 "``(Int(exp(-i*x)*exp(-k*i*x),x = -Pi .. 0)+
Int(exp(i*x)*exp(-k*i*x),x = 0 .. Pi));" "6#-%!G6#,&-%$IntG6$*&-%$expG
6#,$*&%\"iG\"\"\"%\"xGF1!\"\"F1-F,6#,$*(%\"kGF1F0F1F2F1F3F1/F2;,$%#PiG
F3\"\"!F1-F(6$*&-F,6#*&F0F1F2F1F1-F,6#,$*(F8F1F0F1F2F1F3F1/F2;F=F<F1" 
}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"
#F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(Int(exp(-(k+1)*i*x),x = -Pi .. 0)+
Int(exp(-(k-1)*i*x),x = 0 .. Pi));" "6#-%!G6#,&-%$IntG6$-%$expG6#,$*(,
&%\"kG\"\"\"F1F1F1%\"iGF1%\"xGF1!\"\"/F3;,$%#PiGF4\"\"!F1-F(6$-F+6#,$*
(,&F0F1F1F4F1F2F1F3F1F4/F3;F9F8F1" }{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 1 " " }
{TEXT 266 2 "[ " }{XPPEDIT 18 0 "exp(-(k+1)*i*x)/(-(k+1)*i)" "6#*&-%$e
xpG6#,$*(,&%\"kG\"\"\"F+F+F+%\"iGF+%\"xGF+!\"\"F+,$*&,&F*F+F+F+F+F,F+F
.F." }{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([0, ``],[``, ``],[-Pi,
 ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$%#PiG!\"\"F(" }{TEXT -1 
3 " + " }{XPPEDIT 18 0 "exp(-(k-1)*i*x)/(-(k-1)*i)" "6#*&-%$expG6#,$*(
,&%\"kG\"\"\"F+!\"\"F+%\"iGF+%\"xGF+F,F+,$*&,&F*F+F+F,F+F-F+F,F," }
{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" 
"6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT 267 1 "]" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF
&!\"\"" }{XPPEDIT 18 0 "``((1-exp(-(k+1)*i*Pi))*i/(k+1)+(exp(-(k-1)*i*
Pi)-1)*i/(k-1));" "6#-%!G6#,&*(,&\"\"\"F)-%$expG6#,$*(,&%\"kGF)F)F)F)%
\"iGF)%#PiGF)!\"\"F3F)F1F),&F0F)F)F)F3F)*(,&-F+6#,$*(,&F0F)F)F3F)F1F)F
2F)F3F)F)F3F)F1F),&F0F)F)F3F3F)" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+
i/(2*Pi);" "6#/%!G*&%\"iG\"\"\"*&\"\"#F'%#PiGF'!\"\"" }{XPPEDIT 18 0 "
``((1-(-1)^(k+1))/(k+1)+((-1)^(k-1)-1)/(k-1));" "6#-%!G6#,&*&,&\"\"\"F
)),$F)!\"\",&%\"kGF)F)F)F,F),&F.F)F)F)F,F)*&,&),$F)F,,&F.F)F)F,F)F)F,F
),&F.F)F)F,F,F)" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = i/(2*Pi);" "6#/%
!G*&%\"iG\"\"\"*&\"\"#F'%#PiGF'!\"\"" }{XPPEDIT 18 0 "``((1+(-1)^k)/(k
+1)-(1+(-1)^k)/(k-1));" "6#-%!G6#,&*&,&\"\"\"F)),$F)!\"\"%\"kGF)F),&F-
F)F)F)F,F)*&,&F)F)),$F)F,F-F)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+(-1)^k)*i/(2*Pi);" "6#/%!G*(,&\"\"\"F'),$F'!\"
\"%\"kGF'F'%\"iGF'*&\"\"#F'%#PiGF'F*" }{XPPEDIT 18 0 "``(1/(k+1)-1/(k-
1));" "6#-%!G6#,&*&\"\"\"F(,&%\"kGF(F(F(!\"\"F(*&F(F(,&F*F(F(F+F+F+" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -(1+(-1)^k)*i/(Pi*(k^2-1));" "6#/%
!G,$*(,&\"\"\"F(),$F(!\"\"%\"kGF(F(%\"iGF(*&%#PiGF(,&*$F,\"\"#F(F(F+F(
F+F+" }{TEXT -1 4 "    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([-2*i/(Pi*(k^2-1)
), ` if k is even`],[0, ` if k is odd`]);" "6#/%!G-%*PIECEWISEG6$7$,$*
(\"\"#\"\"\"%\"iGF,*&%#PiGF,,&*$%\"kGF+F,F,!\"\"F,F3F3%.~if~k~is~evenG
7$\"\"!%-~if~k~is~oddG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 173 "f := x -> piecewise(x<0,exp(-I*x),exp(I*x)):\n'f(x)'
=f(x);\nassume(k_,integer);\n1/(2*Pi)*Int('f(x)'*exp(-k*I*x),x=-Pi..Pi
);\nsimplify(subs(k_=k,simplify(value(subs(k=k_,%)))));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$-%$expG6#*&^#!\"
\"\"\"\"F'F22F'\"\"!7$-F-6#*&F'F2^#F2F2%*otherwiseG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&,$-%$IntG6$*&-%\"fG6#%\"xGF&-%$expG
6#*(^#!\"\"F&%\"kGF&F0F&F&/F0;,$%#PiGF6F;*$F;F6F&F&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#**^#!\"\"\"\"\",&F&F&)F%%\"kGF&F&%#PiGF%,&*$)F)\"\"#F
&F&F&F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We need to conside
r the cases " }{XPPEDIT 18 0 "k=``" "6#/%\"kG%!G" }{TEXT 283 1 "+" }
{TEXT -1 4 " 1. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c
[1] = 1/(2*Pi);" "6#/&%\"cG6#\"\"\"*&F'F'*&\"\"#F'%#PiGF'!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*exp(-i*x),x = -Pi .. Pi);" "6#
-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*&%\"iGF+F*F+!\"\"F+/F*;,$%#P
iGF2F6" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "``(Int(exp(-i*x)*exp(-i*x),x = -Pi .. 0)+Int(
exp(i*x)*exp(-i*x),x = 0 .. Pi));" "6#-%!G6#,&-%$IntG6$*&-%$expG6#,$*&
%\"iG\"\"\"%\"xGF1!\"\"F1-F,6#,$*&F0F1F2F1F3F1/F2;,$%#PiGF3\"\"!F1-F(6
$*&-F,6#*&F0F1F2F1F1-F,6#,$*&F0F1F2F1F3F1/F2;F<F;F1" }{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&!\"\"
" }{XPPEDIT 18 0 "``(Int(exp(-2*i*x),x = -Pi .. 0)+Int(1,x = 0 .. Pi))
;" "6#-%!G6#,&-%$IntG6$-%$expG6#,$*(\"\"#\"\"\"%\"iGF0%\"xGF0!\"\"/F2;
,$%#PiGF3\"\"!F0-F(6$F0/F2;F8F7F0" }{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 1 " " }
{TEXT 268 2 "[ " }{XPPEDIT 18 0 "exp(-2*i*x)/(-2*i);" "6#*&-%$expG6#,$
*(\"\"#\"\"\"%\"iGF*%\"xGF*!\"\"F*,$*&F)F*F+F*F-F-" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "PIECEWISE([0, ``],[-Pi, ``]);" "6#-%*PIECEWISEG6$7$\"\"
!%!G7$,$%#PiG!\"\"F(" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Pi;" "6#%#PiG
" }{TEXT -1 1 " " }{TEXT 269 1 "]" }{TEXT -1 1 " " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"
\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 
"" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[-1] = 1/(2*Pi);" "6#/&%\"cG6#,$\"
\"\"!\"\"*&F(F(*&\"\"#F(%#PiGF(F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int
(f(x)*exp(i*x),x = -Pi .. Pi);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$ex
pG6#*&%\"iGF+F*F+F+/F*;,$%#PiG!\"\"F4" }{TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*
&\"\"#F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(Int(exp(-i*x)*exp(i*x),x = -P
i .. 0)+Int(exp(i*x)*exp(i*x),x = 0 .. Pi));" "6#-%!G6#,&-%$IntG6$*&-%
$expG6#,$*&%\"iG\"\"\"%\"xGF1!\"\"F1-F,6#*&F0F1F2F1F1/F2;,$%#PiGF3\"\"
!F1-F(6$*&-F,6#*&F0F1F2F1F1-F,6#*&F0F1F2F1F1/F2;F;F:F1" }{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&!\"\"
" }{XPPEDIT 18 0 "``(Int(1,x = -Pi .. 0)+Int(exp(2*i*x),x = 0 .. Pi));
" "6#-%!G6#,&-%$IntG6$\"\"\"/%\"xG;,$%#PiG!\"\"\"\"!F*-F(6$-%$expG6#*(
\"\"#F*%\"iGF*F,F*/F,;F1F/F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*P
i);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{TEXT 
270 1 "[" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 2 "
 +" }{TEXT 272 1 " " }{XPPEDIT 18 0 "exp(2*i*x)/(2*i);" "6#*&-%$expG6#
*(\"\"#\"\"\"%\"iGF)%\"xGF)F)*&F(F)F*F)!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([0, ``],[ -Pi,`` ])" "6#-%*PIECEWISEG6$7$\"\"
!%!G7$,$%#PiG!\"\"F(" }{TEXT -1 1 " " }{TEXT 271 1 "]" }{TEXT -1 1 " \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!
G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "f := x -> piecewise(x<0
,exp(-I*x),exp(I*x)):\n1/(2*Pi)*Int('f(x)'*exp(-I*x),x=-Pi..Pi);\n``=v
alue(%);\nf := x -> piecewise(x<0,exp(-I*x),exp(I*x)):\n1/(2*Pi)*Int('
f(x)'*exp(I*x),x=-Pi..Pi);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&#\"\"\"\"\"#F&,$-%$IntG6$*&-%\"fG6#%\"xGF&-%$expG6#*&^#!\"\"
F&F0F&F&/F0;,$%#PiGF6F:*$F:F6F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
%!G#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&,
$-%$IntG6$*&-%\"fG6#%\"xGF&-%$expG6#*&F0F&^#F&F&F&/F0;,$%#PiG!\"\"F9*$
F9F:F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"\"\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Alternatively, we can use the f
ormula " }{XPPEDIT 18 0 "f(x)= cos(x)+i*abs(sin(x))" "6#/-%\"fG6#%\"xG
,&-%$cosG6#F'\"\"\"*&%\"iGF,-%$absG6#-%$sinG6#F'F,F," }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 137 "f := x -> cos(x)+I*abs(sin(x));\nassume(k_,integer);\n1/(2*Pi)*
Int('f(x)'*exp(-k*I*x),x=-Pi..Pi);\nsubs(k_=k,simplify(value(subs(k=k_
,%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)opera
torG%&arrowGF(,&-%$cosG6#9$\"\"\"*&-%$absG6#-%$sinGF/F1^#F1F1F1F(F(F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#PiG!\"\"-%$IntG6$*&-%\"fG6#%
\"xG\"\"\"-%$expG6#*(^#F&F/%\"kGF/F.F/F//F.;,$F%F&F%F/#F/\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#**^#!\"\"\"\"\",&F&F&)F%%\"kGF&F&%#PiG
F%,&*$)F)\"\"#F&F&F&F%F%" }}}{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 "" }}{PARA 0 "" 0 "" 
{TEXT -1 30 "The complex Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#
-%\"fG6#%\"xG" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "f(x)" "6#
-%\"fG6#%\"xG" }{TEXT -1 24 " for all real values of " }{TEXT 279 1 "x
" }{TEXT -1 10 ", that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "f(x) = Sum(c[k]*exp(k*i*x),k = -infinity .. infinity);
" "6#/-%\"fG6#%\"xG-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#*(F/F0%\"iGF
0F'F0F0/F/;,$%)infinityG!\"\"F9" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" 
{TEXT -1 6 "where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c[k] = PIECEWISE([-2*i/(Pi*(k^2-1)),
 ` if k is even `],[0, ` if k is odd and`*abs(k) <> 1],[1/2, `if`*k = \+
1 or k = -1]);" "6#/&%\"cG6#%\"kG-%*PIECEWISEG6%7$,$*(\"\"#\"\"\"%\"iG
F/*&%#PiGF/,&*$F'F.F/F/!\"\"F/F5F5%/~if~k~is~even~G7$\"\"!0*&%1~if~k~i
s~odd~andGF/-%$absG6#F'F/F/7$*&F/F/F.F55/*&%#ifGF/F'F/F//F',$F/F5" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Since " }{XPPEDIT 18 0 "c[-k]=c[k]" "6#/&%\"cG6#,$%\"kG!\"
\"&F%6#F(" }{TEXT -1 39 " , we can write this series in the form" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = Sum(c[k]*exp(-
k*i*x),k = 1 .. infinity)+c[0]+Sum(c[k]*exp(k*i*x),k = 1 .. infinity);
" "6#/-%\"fG6#%\"xG,(-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#,$*(F0F1%
\"iGF1F'F1!\"\"F1/F0;F1%)infinityGF1&F.6#\"\"!F1-F*6$*&&F.6#F0F1-F36#*
(F0F1F7F1F'F1F1/F0;F1F;F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 12 "Then, since " }{XPPEDIT 18 0 "exp(k*i*x)+exp(-k*i*x) = 2*cos*k*
x;" "6#/,&-%$expG6#*(%\"kG\"\"\"%\"iGF*%\"xGF*F*-F&6#,$*(F)F*F+F*F,F*!
\"\"F***\"\"#F*%$cosGF*F)F*F,F*" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x) = c[0]+``;" "6#/-%\"fG6#%\"xG,&&%
\"cG6#\"\"!\"\"\"%!GF-" }{XPPEDIT 18 0 "Sum(2*c[k]*cos*k*x,k = 1 .. in
finity);" "6#-%$SumG6$*,\"\"#\"\"\"&%\"cG6#%\"kGF(%$cosGF(F,F(%\"xGF(/
F,;F(%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = c[0]+cos*x+``;" "6#/%!G,(&%\"cG6#\"\"!\"\"\"*&%$co
sGF*%\"xGF*F*F$F*" }{XPPEDIT 18 0 "Sum(2*c[2*k]*cos*2*k*x,k = 1 .. inf
inity);" "6#-%$SumG6$*.\"\"#\"\"\"&%\"cG6#*&F'F(%\"kGF(F(%$cosGF(F'F(F
-F(%\"xGF(/F-;F(%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT 
-1 6 "where " }{XPPEDIT 18 0 "c[2*k] = -2*i/(4*k^2-1);" "6#/&%\"cG6#*&
\"\"#\"\"\"%\"kGF),$*(F(F)%\"iGF),&*&\"\"%F)*$F*F(F)F)F)!\"\"F2F2" }
{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = 2*i/Pi+cos*x;" "6#/-%\"fG6#
%\"xG,&*(\"\"#\"\"\"%\"iGF+%#PiG!\"\"F+*&%$cosGF+F'F+F+" }{TEXT -1 4 "
  + " }{XPPEDIT 18 0 "Sum(``(-4*i/(4*k^2-1))*cos*2*k*x,k = 1 .. infini
ty);" "6#-%$SumG6$*,-%!G6#,$*(\"\"%\"\"\"%\"iGF-,&*&F,F-*$%\"kG\"\"#F-
F-F-!\"\"F4F4F-%$cosGF-F3F-F2F-%\"xGF-/F2;F-%)infinityG" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 36 "The imaginary part of this series i
s" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2/Pi;" "6#*&\"\"
#\"\"\"%#PiG!\"\"" }{TEXT -1 4 "  + " }{XPPEDIT 18 0 "Sum(``(-4/(4*k^2
-1))*cos*2*k*x,k = 1 .. infinity);" "6#-%$SumG6$*,-%!G6#,$*&\"\"%\"\"
\",&*&F,F-*$%\"kG\"\"#F-F-F-!\"\"F3F3F-%$cosGF-F2F-F1F-%\"xGF-/F1;F-%)
infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 53 "and gives the Fourier series for the imaginary pa
rt  " }{XPPEDIT 18 0 "abs(sin*x);" "6#-%$absG6#*&%$sinG\"\"\"%\"xGF(" 
}{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 
35 ", as shown in the previous example." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 31 "The real part of the series is " }
{XPPEDIT 18 0 "cos*x;" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "C
onsider the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 
-1 12 " defined by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "f(x) = PIECEWISE([-exp(i*x/2), -Pi <= x and x < 0],[exp(i*x/2), 0 <
= x and x < Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,$-%$expG6#*(%\"i
G\"\"\"F'F2\"\"#!\"\"F431,$%#PiGF4F'2F'\"\"!7$-F.6#*(F1F2F'F2F3F431F:F
'2F'F8" }{TEXT -1 3 " , " }}{PARA 0 "" 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 10 "Note that " }{XPPEDIT 18 0 "f(x)" "6#-%\"
fG6#%\"xG" }{TEXT -1 18 " is also given by " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=cos(x/2)+i*abs(sin(x/2))" "6#/-%\"
fG6#%\"xG,&-%$cosG6#*&F'\"\"\"\"\"#!\"\"F-*&%\"iGF--%$absG6#-%$sinG6#*
&F'F-F.F/F-F-" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG
" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ",
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 "and
 " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic \+
with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The function  " }{XPPEDIT 18 0 "f(
x)" "6#-%\"fG6#%\"xG" }{TEXT -1 43 " can be thought of as mapping the \+
interval " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }
{TEXT -1 137 " onto the upper semi-circular part of the unit circle wi
th its centre at the origin in the complex plane. There are a disconti
nuities at " }{XPPEDIT 18 0 "-2*Pi,0,2*Pi" "6%,$*&\"\"#\"\"\"%#PiGF&!
\"\"\"\"!*&F%F&F'F&" }{TEXT -1 6 ", etc." }}{PARA 0 "" 0 "" {TEXT -1 
13 "For example, " }{XPPEDIT 18 0 "Limit(f(x),x=0,left)=-1" "6#/-%&Lim
itG6%-%\"fG6#%\"xG/F*\"\"!%%leftG,$\"\"\"!\"\"" }{TEXT -1 7 " while " 
}{XPPEDIT 18 0 "Limit(f(x),x = 0,right) = 1;" "6#/-%&LimitG6%-%\"fG6#%
\"xG/F*\"\"!%&rightG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "plot([[-cos(x/2),
-sin(x/2),x=-Pi..0],[cos(x/2),sin(x/2),x=0..Pi]],\n   color=[green,cor
al],thickness=2,labels=[`real`,`imag`],xtickmarks=3,\n    ytickmarks=[
-1=`-i`,0=`0`,1=`i`]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 321 178 178 
{PLOTDATA 2 "6(-%'CURVESG6$7S7$$\"3_6L:&GM50#!#F$\"\"\"\"\"!7$$!3u1GQ<
+ABM!#>$\"3KT#G[1RT***!#=7$$!3+UK;E@i)R'F1$\"3kCK(4#y]z**F47$$!3!p[ZkJ
eyt*F1$\"3+5;DATZ_**F47$$!3vJiL^#=)38F4$\"3W#y3_wzR\"**F47$$!33u?385yS
;F4$\"3%Gs1>^tW')*F47$$!3MlfOf)4p%>F4$\"3wmo\"yGY'3)*F47$$!3uDf(HC)*=E
#F4$\"3$o=L`UK3u*F47$$!3!)[dT;d=&e#F4$\"3%HjW#yG1g'*F47$$!3i\\siC+e/HF
4$\"3%=ZI'4u()o&*F47$$!3i/JZ)Rp(HKF4$\"3#[QvYMoSY*F47$$!3n$H*)R)e38NF4
$\"3QD?Mmsfi$*F47$$!37rI[;M@GQF4$\"3^V1clRAQ#*F47$$!3a#\\eT]8-9%F4$\"3
$eu0MqrE5*F47$$!3(p[qSJajV%F4$\"3cY%GJWt?'*)F47$$!3/\"QXXPc6q%F4$\"3'f
6#R3$[g#))F47$$!3=)>W\"z+f5]F4$\"3%3<-M@JTl)F47$$!3wot*)oS#yE&F4$\"3(3
c\\*R<++&)F47$$!3gn^j;E2hbF4$\"3ZOlp-`56$)F47$$!32HI2\\G,:eF4$\"3Q'z[o
%[XN\")F47$$!3K)zOAdQs3'F4$\"3F=WA(o@Q$zF47$$!3&>Teg-t*RjF4$\"3%3c!otD
NLxF47$$!3'HxXttPmf'F4$\"3Q::ai%3c^(F47$$!3S=1e)Qgd#oF4$\"38K9s0e93tF4
7$$!3))e5^%G$elqF4$\"3!)eab)3[l2(F47$$!3%GM/`tsjI(F4$\"3U]g1q#ew#oF47$
$!3Cx$y(zo&)3vF4$\"36H)Qa%>K/mF47$$!3qO5^`\\()>xF4$\"3#[7A!orPcjF47$$!
3pAK`=0GHzF4$\"3Q3Z$>v_J4'F47$$!33J**[KMYD\")F4$\"3kcalYf'*GeF47$$!3A.
0^qx!pI)F4$\"3'[MbQxStc&F47$$!3Yx*f5h(\\)\\)F4$\"3%zM[BJ]-F&F47$$!3;bt
`=tfh')F4$\"37\"o$\\ZEn(*\\F47$$!3Fd8:i)*3E))F4$\"38-l#)f$y5q%F47$$!3O
O:]U;Qm*)F4$\"3ecvN#[SwU%F47$$!3(fK`**yi+6*F4$\"3s]Q5D[#R7%F47$$!3)yK=
jfneB*F4$\"3M'*R#H&[*Q$QF47$$!39X:y02]d$*F4$\"3o'prHqQm_$F47$$!3/\"exc
CTlY*F4$\"3C:%Hk09DA$F47$$!3YN***zM2,d*F4$\"3j5WV7]_+HF47$$!3A'RYp%[]f
'*F4$\"3%RM'Q6+F(e#F47$$!3Q/P!))[C.u*F4$\"3a8/s]X3kAF47$$!3f5$*[V\"*z4
)*F4$\"3%pi*zsE4T>F47$$!3wk(\\Pq+U')*F4$\"3\"y-R/J?Ck\"F47$$!38s/$*GvO
:**F4$\"3mLBQ!)HE)H\"F47$$!3I&>&*oxj4&**F4$\"3GE#e_c05*)*F17$$!3EA:\"=
a+$y**F4$\"3A6`@Q1@%e'F17$$!3qq(\\eCwT***F4$\"3[?E+!oZBT$F17$$!\"\"F-$
F-F--%'COLOURG6&%$RGBGFhz$\"*++++\"!\")Fhz-F$6$7S7$F+Fhz7$$\"3gG!eZ1RT
***F4$\"3%*Q6)y.?KU$F17$$\"3G')>%3#y]z**F4$\"3Kf:jY@i)R'F17$$\"3)>)=0A
TZ_**F4$\"3of.'oLeyt*F17$$\"3GQ.%\\wzR\"**F4$\"3)=ipLD=)38F47$$\"39$>q
:^tW')*F4$\"307`5:5yS;F47$$\"3S[vT(GY'3)*F4$\"3Y_xPh)4p%>F47$$\"3fk#p[
UK3u*F4$\"3q.Q(\\C)*=E#F47$$\"3k-WrxG1g'*F4$\"37hqR=d=&e#F47$$\"3%GtM!
4u()o&*F4$\"3Dg)*eE+e/HF47$$\"31[H,W$oSY*F4$\"3*o@9/Sp(HKF47$$\"3Uy9il
sfi$*F4$\"3#ff4f)e38NF47$$\"3\"QYvZ'RAQ#*F4$\"3bhyP=M@GQF47$$\"3(GdcDq
rE5*F4$\"3)=[Dg]8-9%F47$$\"3UM&=AWt?'*)F4$\"3$*R'3fJajV%F47$$\"3\"G*yU
2$[g#))F4$\"35McNwj:,ZF47$$\"3!=[uB@JTl)F4$\"3n\">>43!f5]F47$$\"3#H6p)
Q<++&)F4$\"3aZ2kqS#yE&F47$$\"32Tfb,`56$)F4$\"3//)R$=E2hbF47$$\"35<hlX[
XN\")F4$\"3wQ;u]G,:eF47$$\"3w/f(fo@Q$zF4$\"3?_S'QdQs3'F47$$\"3!*4-QsDN
LxF4$\"3t[XkFI(*RjF47$$\"3AB&)=h%3c^(F4$\"3!)[s))Qxj'f'F47$$\"3XY9K/e9
3tF4$\"3$Qaz+Rgd#oF47$$\"3e$G1r3[l2(F4$\"3U$[ifG$elqF47$$\"39*[n&o#ew#
oF4$\"3@=ZqOFP1tF47$$\"3sP()*Q%>K/mF4$\"3#f%H8\")o&)3vF47$$\"3]_(Qk;xj
N'F4$\"3h^Z\"[&\\()>xF47$$\"3)fQ3.v_J4'F4$\"3d[Hy>0GHzF47$$\"3`(*)))\\
%f'*GeF4$\"3UraoLMYD\")F47$$\"3)*p::s2MnbF4$\"39$Q_;x2pI)F47$$\"3#yF01
J]-F&F4$\"3!R#497w\\)\\)F47$$\"3IerrXEn(*\\F4$\"3?&Ri&>tfh')F47$$\"3uS
i,e$y5q%F4$\"3([c:J')*3E))F47$$\"3W?&=0[SwU%F4$\"3#*f'4Mk\"Qm*)F47$$\"
3$fMNK#[#R7%F4$\"3:d\"*z!zi+6*F47$$\"3I)oH5&[*Q$QF4$\"3?sY5(fneB*F47$$
\"3#3W_5qQm_$F4$\"3)4([]12]d$*F47$$\"3/&z([aS^AKF4$\"3+H&QjCTlY*F47$$
\"34\\:Z5]_+HF4$\"3-W[f[t5q&*F47$$\"3]Y^S4+F(e#F4$\"3'R0xu%[]f'*F47$$
\"3kREs[X3kAF4$\"3Ew!o#*[C.u*F47$$\"3M.wyqE4T>F4$\"3YNu)Q9*z4)*F47$$\"
3YYeT3.UU;F4$\"3)>j'3/2?k)*F47$$\"3md'[$yHE)H\"F4$\"3C]n>HvO:**F47$$\"
3'zX[[a05*)*F1$\"37k!)4xP'4&**F47$$\"3Cv%\\xh5Ue'F1$\"3Yml%>a+$y**F47$
$\"3OAU]fwM7MF1$\"3;f(>fCwT***F47$$!3_6L:&GM50#F*F+-Fjz6&F\\[lF][l$\")
AR!)\\F_[lFhz-%*THICKNESSG6#\"\"#-%*AXESTICKSG6$\"\"$7%/Fgz%#-iG/F-%\"
0G/F,%\"iG-%+AXESLABELSG6$%%realG%%imagG-%%VIEWG6$%(DEFAULTGF]\\m" 1 
2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve
 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "W
e can plot separate graphs for the real and imaginary parts of " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "plot([co
s((x-2*Pi*floor(x/(2*Pi)))/2),abs(sin(x/2))],x=-Pi..4*Pi,\n   color=[C
OLOR(RGB,.4,0,.9),COLOR(RGB,.6,.4,0)],ytickmarks=3,\n   title=`real an
d imag parts of f(x)`);" }}{PARA 13 "" 1 "" {GLPLOT2D 618 165 165 
{PLOTDATA 2 "6(-%'CURVESG6$7go7$$!1++JZEfTJ!#:$!1D>u.l*)RJ!#C7$$!1f%**
4v.#*z#F*$!1)355h%f.<!#;7$$!1QJ1.IH,DF*$!1hUJlx3ZJF37$$!1mi\\47Em@F*$!
1^6<x*\\co%F37$$!1aXs2'3!H=F*$!1#QBrqL=5'F37$$!1]))p3*eL\\\"F*$!1.oGNC
[RtF37$$!1I)\\:no@=\"F*$!1C6a&3pLI)F37$$!1\"=S#e?\\*f)F3$!1t='oOj(*3*F
37$$!1VnIMP4n_F3$!1v$e?<@_l*F37$$!1^Q=%4Qig$F3$!1mflZ!yy$)*F37$$!1g41a
CQX>F3$!14vNv3t_**F37$$!1>Pdo`=\"4\"F3$!1$o'yA,7&)**F37$$!1*yk3$G))pB!
#<$!1l`8lzH****F37$$!1Qd+uk8-8Fco$!1fS#e0)y****F37$$!1yoYr6!RM#!#=$!0
\"*fE8$******F*7$$\"1GO7(RiNL)F^p$\"1=yz*=8*****F37$$\"189d'f-6!>Fco$
\"19j^H#[&****F37$$\"1:&*G5`fOSFco$\"1=nU1L'z***F37$$\"1;w+C!)3shFco$
\"1!o$3W&Q_***F37$$\"1#QW^M2V/\"F3$\"1x67u3P')**F37$$\"1-!))y)eSr9F3$
\"1$\\CME\\H(**F37$$\"1d1e./;wHF3$\"1Z],F][*))*F37$$\"16LF>\\\"4[%F3$
\"1;8fLc1](*F37$$\"1Db!Q57\\<'F3$\"1$>U]%[:F&*F37$$\"1SxL)G4*oyF3$\"13
m'*ok$fB*F37$$\"1M$z(3:3F6F*$\"1D\"H4/!p`%)F37$$\"1h%Q$o\"=\\X\"F*$\"1
O;=YXmouF37$$\"1?9`i_i_<F*$\"1`$[j/)))*R'F37$$\"1!\\tBzFm5#F*$\"1eD'\\
qPp%\\F37$$\"13!Q(pV^1CF*$\"1:hlS,?$f$F37$$\"1)4p4v``v#F*$\"1dM;bG@>>F
37$$\"1Oq\">rNT1$F*$\"1+xf7!z=(QFco7$$\"1L]Qi%=HS$F*$!130G[U\"HI\"F37$
$\"1,A(fv>bs$F*$!1$4+bLK$yGF37$$\"1lBti.7iSF*$!1rpc!>X=W%F37$$\"1!ev]z
C7P%F*$!1eo$*RG3odF37$$\"1vg\"RoNYq%F*$!1D6i/FgVqF37$$\"1D\")*R*e&40&F
*$!1fG<,!)Hh\")F37$$\"1#GX\\lGCN&F*$!1O#oe^=l$*)F37$$\"1\"Q%)e\\F!ycF*
$!1R0)Q%>rX&*F37$$\"1\"=krA:i%eF*$!1f\\f%QpAw*F37$$\"1\")RWeHS9gF*$!1&
[Ee*3$)4**F37$$\"1Ijy*Rsm4'F*$!1rV9uwac**F37$$\"1!oG6%=%*yhF*$!1(Q)e*p
>k)**F37$$\"1a)*zhl2?iF*$!1\"=!)3-A]***F37$$\"1H5Z#G67E'F*$!1m#pQV'R**
**F37$$\"1B)QE'\\\\riF*$!1H?Hq\"H)****F37$$\"1;m!Gky<G'F*$!0G>m_(*****
*F*7$$\"14W(HKi?H'F*$\"1K**R)\\,*****F37$$\"1/A9.gM-jF*$\"1IG<)3T&****
F37$$\"1#zxMO8HK'F*$\"1>(*zmr-)***F37$$\"1zL\"Qs![VjF*$\"1&*=gAfX&***F
37$$\"1UDC`0o-lF*$\"16f;=x$)R**F37$$\"11<n#Q!)=m'F*$\"1Iz&Q\\s7#)*F37$
$\"1<\\U]YkQoF*$\"18H;!z.oh*F37$$\"1G\"y\"=*3a,(F*$\"1bQ<dKEP$*F37$$\"
1tu9J+2LtF*$\"1Cx3\"HGNl)F37$$\"1+#\\E2TAn(F*$\"1^m0K&fNo(F37$$\"1@%yp
W!ezzF*$\"1%Qn].iLh'F37$$\"1yp3iNe:$)F*$\"1*eu+<4gE&F37$$\"1g!p!3jtJ')
F*$\"1#=&\\2Y6iQF37$$\"1\\L*G-2A'*)F*$\"1ryQF&*G#H#F37$$\"1\"H\"=^DI&G
*F*$\"1Iz%*3#>\"opFco7$$\"1QOOEObB'*F*$!1i[>1*GC#**Fco7$$\"1Q\"eQhH$\\
**F*$!1Pg)>?#z#f#F37$$\"1rRRh#[#G5!#9$!1@x!HKt#eTF37$$\"1Gv*Qp(Gh5F^`l
$!1z2VsL@(f&F37$$\"1yb-,vk\"4\"F^`l$!11e'e1`fy'F37$$\"1,o4mKWE6F^`l$!1
bO*Q8q\\&zF37$$\"1H)zJ,lv:\"F^`l$!1&o6tEvzz)F37$$\"1w:q'GZ2>\"F^`l$!1%
Q#)[)y?i%*F37$$\"1#3j:7Fm?\"F^`l$!1+c%pC,!*o*F37$$\"1(eCk&p]A7F^`l$!1Q
$)=5]ua)*F37$$\"1$Hsw+s&R7F^`l$!1b\"\\A2?O'**F37$$\"1++#*eqjc7F^`l$!\"
\"\"\"!-%&COLORG6&%$RGBG$\"\"%F]clF^cl$\"\"*F]cl-F$6$7^r7$F($\"\"\"F^c
l7$$!1lBBBa*f0$F*$\"1H+h!zU3***F37$$!1HZ:*>)RqHF*$\"1j7-OzQj**F37$$!1%
4x](4![)GF*$\"1tn>=do<**F37$F/$\"1?*z!Q)>Q&)*F37$$!1*HJqP[-l#F*$\"1.va
W2u*p*F37$F5$\"1d!3Tx#)=\\*F37$F:$\"1RE(RD!HM))F37$$!15/h3\\j(*>F*$\"1
/lGWYJ3%)F37$F?$\"1)\\?o=-E#zF37$$!1-<@eP=h;F*$\"1Mp\"p'oI$Q(F37$FD$\"
1A-=.S0#z'F37$$!1SV7!zjxL\"F*$\"1PqyT?4,iF37$FI$\"1U@hVrhsbF37$$!1CpoQ
*e5-\"F*$\"1y_&*)p#R')[F37$FN$\"1Zk6srZoTF37$$!1iMF'*GHLpF3$\"1%)><Vwi
(R$F37$FS$\"1l)=&[.@.EF37$FX$\"1KL$>IkLz\"F37$Fgn$\"1-LdZ8e6(*Fco7$F\\
o$\"1h_%pX?KX&Fco7$Fao$\"13kH'o8\\=\"Fco7$Faq$\"1s-P%>ab3$Fco7$F[r$\"1
?=]tWR]tFco7$F`r$\"1P@=9Vf#[\"F37$Fer$\"1\"\\7[fg<A#F37$Fjr$\"18sjWzjQ
IF37$F_s$\"1qyBM*GP$QF37$$\"1Tb1)=i)p&*F3$\"13^`u/U/YF37$Fds$\"1EO%=%y
#=M&F37$$\"1)*)e&Q)**4H\"F*$\"1$e:v[yf,'F37$Fis$\"1;.n)eS(\\mF37$$\"1T
\\V:<x.;F*$\"1g&=A/(o'=(F37$F^t$\"1(*psKB%Qo(F37$$\"1bCXFliH>F*$\"1?3#
HrD%>#)F37$Fct$\"1(3wRet1p)F37$$\"1\\d0\"3rlD#F*$\"1j\"oep$zO!*F37$Fht
$\"1&G.'fS9K$*F37$$\"1`NNgS$4e#F*$\"1cJDKQk4'*F37$F]u$\"15;>?K59)*F37$
$\"1$317C\\D$GF*$\"1f!e!oD&3))*F37$$\"1nIWJZu4HF*$\"1`PB)Q$)G$**F37$$
\"1_+o@-%p)HF*$\"1`ybv\"=,(**F37$Fbu$\"1'ea]Y,D***F37$$\"1ObsbN[1JF*$
\"1ZoiG#f%)***F37$$\"1OS`*RJ)[JF*$\"1Yzw+X$*****F37$$\"1NDMV#z6>$F*$\"
10l3?m#p***F37$$\"1M5:(3FNB$F*$\"1L-2NpV*)**F37$$\"1L!oZxA#=LF*$\"1o8&
3fF5'**F37$Fgu$\"1cx53uv9**F37$$\"1<'y\"4\">Uc$F*$\"1&=%GhJcx(*F37$F\\
v$\"1TGrna!od*F37$Fav$\"1M?)fG`$f*)F37$$\"1tR!*yDn;UF*$\"1P]%\\y3(*e)F
37$Ffv$\"1&)*oj_&zo\")F37$$\"1Ge\\R-$z`%F*$\"1n<9-g?gwF37$F[w$\"1qlR_m
U)4(F37$$\"1+r&*)y&zx[F*$\"1N4J6TwikF37$F`w$\"1TP&QE'oydF37$$\"1/<ZuAp
,_F*$\"1xj!*=9wZ^F37$Few$\"1uQ)QD4w[%F37$$\"1J[Tv!G_^&F*$\"12I;QK7YPF3
7$Fjw$\"1\"o>)Q%H)zHF37$F_x$\"17JCw#4v;#F37$Fdx$\"1*)*4(***p)R8F37$Fix
$\"13*=n3M@J*Fco7$F^y$\"1@jbfO\")4_Fco7$Fcy$\"1[/ehO\"\\:$Fco7$Fhy$\"1
u/rI\"z')4\"Fco7$F]z$\"1a;/')*Q^%eF^p7$Fbz$\"1`G)\\G@L.(!#>7$Fgz$\"1tU
1CL\\QWF^p7$F\\[l$\"1WPpH9?!e*F^p7$Fa[l$\"1;Z9&=si)>Fco7$Ff[l$\"18!)ys
TJ9IFco7$F[\\l$\"1X(orXu_4\"F37$F`\\l$\"1B1ET8=#)=F37$Fe\\l$\"1(3DyEI<
u#F37$Fj\\l$\"1[2$ePw)zNF37$$\"1,Gmu%RU<(F*$\"1jH\"*=dL4VF37$F_]l$\"1R
.'GGJ;,&F37$$\"1P$)*=bbE](F*$\"1FntNL]EdF37$Fd]l$\"1L9invA+kF37$$\"15Q
\")f2\"f#yF*$\"1()*yr2H6(pF37$Fi]l$\"1n*R$>h*3](F37$$\"1*pKX+#eZ\")F*$
\"1\\hg.DKH!)F37$F^^l$\"1O3DWi7,&)F37$$\"1=!y]$*fOZ)F*$\"1^w>Y#p.*))F3
7$Fc^l$\"1UM]2F5C#*F37$Fh^l$\"1W:.E`sL(*F37$$\"1@t.(yaP7*F*$\"1ZQiz\\%
p))*F37$F]_l$\"1MZ6;Jpv**F37$$\"1&3zb$ReF$*F*$\"1OEgvR>))**F37$$\"1yo(
*>`')p$*F*$\"1tE8*)4B'***F37$$\"1qYP/n97%*F*$\"1t_'[c+)****F37$$\"1jCx
)3GWX*F*$\"1>M^26!*)***F37$$\"1]!ov&3**Q&*F*$\"1Hhb)p)p$)**F37$Fb_l$\"
1`QzN4l]**F37$$\"1*)36?;W'y*F*$\"1!*\\8.V%p$)*F37$Fg_l$\"12-V_S-e'*F37
$F\\`l$\"1XB$[#eV%4*F37$Fb`l$\"1cPMV*3oG)F37$$\"1`:Y(fnk2\"F^`l$\"1&ez
%3L`QyF37$Fg`l$\"1t\\k>O7XtF37$$\"1!>hNQX!46F^`l$\"18hW@%)yFnF37$F\\al
$\"1dKa\"3v&fgF37$$\"1:$Q'RT+U6F^`l$\"1%p@aHOHU&F37$Faal$\"1\"\\b\\@'[
`ZF37$$\"1-2%*\\h:u6F^`l$\"1$o=BVM\"3SF37$Ffal$\"1Whl.f@NKF37$F[bl$\"1
#f-G8@XZ#F37$F`bl$\"1Md53YB)p\"F37$Febl$\"1mLuzs<A&)Fco7$Fjbl$\"1B\"[1
lezD\"!#B-F`cl6&Fbcl$\"\"'F]clFcclF^cl-%&TITLEG6#%<real~and~imag~parts
~of~f(x)G-%+AXESLABELSG6$Q\"x6\"%!G-%*AXESTICKSG6$%(DEFAULTG\"\"$-%%VI
EWG6$;$!+aEfTJ!\"*$\"+iqjc7!\")F]`n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
46.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "The coeffic
ients in the complex Fourier series " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"c[k] = 1/(2*Pi)" "6#/&%\"cG6#%\"kG*&\"\"\"F)*&\"\"#F)%#PiGF)!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*exp(-k*i*x),x=-Pi..Pi)" "6#-%$
IntG6$*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*(%\"kGF+%\"iGF+F*F+!\"\"F+/F*;,
$%#PiGF3F7" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"
" }{XPPEDIT 18 0 "``(Int((-exp(i*x/2))*exp(-k*i*x),x = -Pi .. 0)+Int(e
xp(i*x/2)*exp(-k*i*x),x = 0 .. Pi));" "6#-%!G6#,&-%$IntG6$*&,$-%$expG6
#*(%\"iG\"\"\"%\"xGF1\"\"#!\"\"F4F1-F-6#,$*(%\"kGF1F0F1F2F1F4F1/F2;,$%
#PiGF4\"\"!F1-F(6$*&-F-6#*(F0F1F2F1F3F4F1-F-6#,$*(F9F1F0F1F2F1F4F1/F2;
F>F=F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F
&*&\"\"#F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(Int(-exp(-(2*k-1)*i*x/2),x \+
= -Pi .. 0)+Int(exp(-(2*k-1)*i*x/2),x = 0 .. Pi));" "6#-%!G6#,&-%$IntG
6$,$-%$expG6#,$**,&*&\"\"#\"\"\"%\"kGF3F3F3!\"\"F3%\"iGF3%\"xGF3F2F5F5
F5/F7;,$%#PiGF5\"\"!F3-F(6$-F,6#,$**,&*&F2F3F4F3F3F3F5F3F6F3F7F3F2F5F5
/F7;F<F;F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"
\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{TEXT 280 2 "[ " }
{XPPEDIT 18 0 "-2*exp(-(2*k-1)*i*x/2)/(-(2*k-1)*i);" "6#,$*(\"\"#\"\"
\"-%$expG6#,$**,&*&F%F&%\"kGF&F&F&!\"\"F&%\"iGF&%\"xGF&F%F/F/F&,$*&,&*
&F%F&F.F&F&F&F/F&F0F&F/F/F/" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWIS
E([0, ``],[``, ``],[-Pi, ``]);" "6#-%*PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$
%#PiG!\"\"F(" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "2*exp(-(2*k-1)*i*x/2)/
(-(2*k-1)*i);" "6#*(\"\"#\"\"\"-%$expG6#,$**,&*&F$F%%\"kGF%F%F%!\"\"F%
%\"iGF%%\"xGF%F$F.F.F%,$*&,&*&F$F%F-F%F%F%F.F%F/F%F.F." }{TEXT -1 2 " \+
 " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEW
ISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT 281 1 "]" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = i/Pi;" "6#/%!G*&%\"iG\"\"\"%#PiG!\"\"" }{XPPEDIT 
18 0 "``((exp((2*k-1)*i*Pi/2)-1)/(2*k-1)+(exp(-(2*k-1)*i*Pi/2)-1)/(2*k
-1));" "6#-%!G6#,&*&,&-%$expG6#**,&*&\"\"#\"\"\"%\"kGF0F0F0!\"\"F0%\"i
GF0%#PiGF0F/F2F0F0F2F0,&*&F/F0F1F0F0F0F2F2F0*&,&-F*6#,$**,&*&F/F0F1F0F
0F0F2F0F3F0F4F0F/F2F2F0F0F2F0,&*&F/F0F1F0F0F0F2F2F0" }{TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = i/(Pi*(2*k-1));" "6#/%!G*&%\"iG\"\"\"*&%#PiGF',&*&
\"\"#F'%\"kGF'F'F'!\"\"F'F." }{XPPEDIT 18 0 "``(exp((2*k-1)*i*Pi/2)+ex
p(-(2*k-1)*i*Pi/2)-2);" "6#-%!G6#,(-%$expG6#**,&*&\"\"#\"\"\"%\"kGF.F.
F.!\"\"F.%\"iGF.%#PiGF.F-F0F.-F(6#,$**,&*&F-F.F/F.F.F.F0F.F1F.F2F.F-F0
F0F.F-F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = i/(Pi*(2*k-1));" "6#/%!G*&
%\"iG\"\"\"*&%#PiGF',&*&\"\"#F'%\"kGF'F'F'!\"\"F'F." }{XPPEDIT 18 0 "`
`(2*cos((2*k-1)*Pi/2)-2);" "6#-%!G6#,&*&\"\"#\"\"\"-%$cosG6#*(,&*&F(F)
%\"kGF)F)F)!\"\"F)%#PiGF)F(F1F)F)F(F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``
 = -2*i/(Pi*(2*k-1));" "6#/%!G,$*(\"\"#\"\"\"%\"iGF(*&%#PiGF(,&*&F'F(%
\"kGF(F(F(!\"\"F(F/F/" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "f := x -> piecewise(x<
0,-exp(I*x/2),exp(I*x/2)):\n'f(x)'=f(x);\nassume(k_,integer);\n1/(2*Pi
)*Int('f(x)'*exp(-k*I*x),x=-Pi..Pi);\nsubs(k_=k,simplify(value(subs(k=
k_,%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWIS
EG6$7$,$-%$expG6#*&^##\"\"\"\"\"#F3F'F3!\"\"2F'\"\"!7$F-%*otherwiseG" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&,$-%$IntG6$*&-%\"f
G6#%\"xGF&-%$expG6#*(^#!\"\"F&%\"kGF&F0F&F&/F0;,$%#PiGF6F;*$F;F6F&F&" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(^#!\"#\"\"\"%#PiG!\"\",&*&\"\"#F&%
\"kGF&F&F&F(F(" }}}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Th
e complex Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" 
}{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F
(x) = Sum(c[k]*exp(k*i*x),k = -infinity .. infinity);" "6#/-%\"FG6#%\"
xG-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#*(F/F0%\"iGF0F'F0F0/F/;,$%)in
finityG!\"\"F9" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " 
}{XPPEDIT 18 0 "c[k]=(-2*i)/(Pi*(2*k-1))" "6#/&%\"cG6#%\"kG*&,$*&\"\"#
\"\"\"%\"iGF,!\"\"F,*&%#PiGF,,&*&F+F,F'F,F,F,F.F,F." }{TEXT -1 11 ", f
or each " }{TEXT 282 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x = 0" "
6#/%\"xG\"\"!" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "Sum(c[k]*exp(k*i*x),k
 = -infinity .. infinity) = -i/Pi" "6#/-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-
%$expG6#*(F+F,%\"iGF,%\"xGF,F,/F+;,$%)infinityG!\"\"F6,$*&F1F,%#PiGF7F
7" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/``(k-1/2),k = -infinity .. in
finity);" "6#-%$SumG6$*&\"\"\"F'-%!G6#,&%\"kGF'*&F'F'\"\"#!\"\"F/F//F,
;,$%)infinityGF/F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "We \+
can show that this sum is 0." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Sum(1/``(k-1/2),k = -n .. n) = ``(Sum(1/``(k-1/2),k = -
n .. 0));" "6#/-%$SumG6$*&\"\"\"F(-%!G6#,&%\"kGF(*&F(F(\"\"#!\"\"F0F0/
F-;,$%\"nGF0F4-F*6#-F%6$*&F(F(-F*6#,&F-F(*&F(F(F/F0F0F0/F-;,$F4F0\"\"!
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``+Sum(1/``(k-1/2),k = 1 .. n+1)-1/`
`(n+1/2);" "6#,(%!G\"\"\"-%$SumG6$*&F%F%-F$6#,&%\"kGF%*&F%F%\"\"#!\"\"
F0F0/F-;F%,&%\"nGF%F%F%F%*&F%F%-F$6#,&F4F%*&F%F%F/F0F%F0F0" }{TEXT -1 
2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "`` = ``(Sum(1/``(-k-1/2),k = 0 .. n))+``(Sum(1/``(
k-1/2),k = 1 .. n+1))-1/``(n+1/2)" "6#/%!G,(-F$6#-%$SumG6$*&\"\"\"F,-F
$6#,&%\"kG!\"\"*&F,F,\"\"#F1F1F1/F0;\"\"!%\"nGF,-F$6#-F)6$*&F,F,-F$6#,
&F0F,*&F,F,F3F1F1F1/F0;F,,&F7F,F,F,F,*&F,F,-F$6#,&F7F,*&F,F,F3F1F,F1F1
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(Sum(1/``(-k+1/2),k = 1 .. n+1))+`
`(Sum(1/``(k-1/2),k = 1 .. n+1))-1/``(n+1/2)" "6#/%!G,(-F$6#-%$SumG6$*
&\"\"\"F,-F$6#,&%\"kG!\"\"*&F,F,\"\"#F1F,F1/F0;F,,&%\"nGF,F,F,F,-F$6#-
F)6$*&F,F,-F$6#,&F0F,*&F,F,F3F1F1F1/F0;F,,&F7F,F,F,F,*&F,F,-F$6#,&F7F,
*&F,F,F3F1F,F1F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-1/``(n+1/2)" "6#/
%!G,$*&\"\"\"F'-F$6#,&%\"nGF'*&F'F'\"\"#!\"\"F'F.F." }{TEXT -1 2 ", " 
}}{PARA 0 "" 0 "" {TEXT -1 38 "because the first two sums cancel out.
" }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "-i/Pi" "6#,$*&%\"iG\"\"\"%#PiG!\"\"F(" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "Sum(1/``(k-1/2),k = -infinity .. infinity) = -i/
Pi;" "6#/-%$SumG6$*&\"\"\"F(-%!G6#,&%\"kGF(*&F(F(\"\"#!\"\"F0F0/F-;,$%
)infinityGF0F4,$*&%\"iGF(%#PiGF0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Li
mit(``(Sum(1/``(k-1/2),k = -n .. n)),n = infinity) = -i/Pi;" "6#/-%&Li
mitG6$-%!G6#-%$SumG6$*&\"\"\"F.-F(6#,&%\"kGF.*&F.F.\"\"#!\"\"F5F5/F2;,
$%\"nGF5F9/F9%)infinityG,$*&%\"iGF.%#PiGF5F5" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Limit(``(-1/``(n+1/2)),n = infinity) = 0;" "6#/-%&Limit
G6$-%!G6#,$*&\"\"\"F,-F(6#,&%\"nGF,*&F,F,\"\"#!\"\"F,F3F3/F0%)infinity
G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 44 "Thus the Fourier series converges to 0 when " }
{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 25 ", which is the val
ue of  " }{XPPEDIT 18 0 "(Limit(f(x),x = 0,left)+Limit(f(x),x = 0,righ
t))/2;" "6#*&,&-%&LimitG6%-%\"fG6#%\"xG/F+\"\"!%%leftG\"\"\"-F&6%-F)6#
F+/F+F-%&rightGF/F/\"\"#!\"\"" }{TEXT -1 1 "." }}{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 0 "" 0 "" {TEXT -1 16 "code for pi
cture" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 
0 "" 0 "" {TEXT -1 18 "Example 2 picture " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "p1 := plot([[[-1,0],
[-1,1]],[[1,0],[1,1]],[[-1,1],[1,1]],\n  [[3,0],[3,1]],[[5,0],[5,1]],[
[3,1],[5,1]]],\n  color=COLOR(RGB,.4,0,.9)):\nt1 := plots[textplot]([[
6.35,-.05,`x`],[-.1,1.45,`y`]],\n   color=black):\nplots[display]([p1,
t1],view=[-2..6.5,-.05..1.5],ytickmarks=[0,1],\nxtickmarks=[-2=`-L`,-1
=`-a`,1=`a`,2=`L`,3=`2L-a`,4=`2L`,5=`2L+a`]);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }