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{TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 "Section 1: A first example:
 The square wave " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "setopti
ons(thickness=2): #set the tickness of the lines in the plots" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=signum(sin(x));\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%'signumG6#-%$sinG6#%\"xG" }}}
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{XPPEDIT 18 0 "a[0]" "6#&%\"aG6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "a[0]:=1/2/Pi*Int('f',x=-Pi..Pi)=1/2/Pi*int(f,x=-Pi..P
i);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"!/,$*&#\"\"\"\"
\"#F,,$-%$IntG6$%\"fG/%\"xG;,$%#PiG!\"\"F7*$F7F8F,F,F'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "T
he coefficient " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "a[n]:=1/Pi*Int('f'*cos(n*x),x=-Pi..
Pi)=1/Pi*int(f*cos(n*x),x=-Pi..Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>&%\"aG6#%#n|irG/,$-%$IntG6$*&%\"fG\"\"\"-%$cosG6#*&F'F/%\"xGF/F//F
4;,$%#PiG!\"\"F8*$F8F9\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Th
e coefficient " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "b[n]:=1/Pi*Int('f'*sin(n*x),x=-Pi..
Pi)=1/Pi*int(f*sin(n*x),x=-Pi..Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>&%\"bG6#%#n|irG/,$-%$IntG6$*&%\"fG\"\"\"-%$sinG6#*&F'F/%\"xGF/F//F
4;,$%#PiG!\"\"F8*$F8F9,$**\"\"#F/F8F9,&)F9F'F/F/F9F/F'F9F9" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B[k]:=-2*((-1)^k-1)/k/Pi;\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#%\"kG,$**\"\"#\"\"\",&)!\"\"F
'F+F+F.F+F'F.%#PiGF.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Let us \+
look at some coefficients " }{XPPEDIT 18 0 "b[k]" "6#&%\"bG6#%\"kG" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eval(seq(([i,subs(k=i,B[k])
]),i=1..10));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,7$\"\"\",$*&\"\"%F$
%#PiG!\"\"F$7$\"\"#\"\"!7$\"\"$,$*(F'F$F.F)F(F)F$7$F'F,7$\"\"&,$*(F'F$
F3F)F(F)F$7$\"\"'F,7$\"\"(,$*(F'F$F9F)F(F)F$7$\"\")F,7$\"\"*,$*(F'F$F?
F)F(F)F$7$\"#5F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "All even coef
ficients are 0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Let call S(n,x
), the Fourier series approximations with n coefficients:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "S:=(m,x)->sum(B[k]*sin(k*x),k=1..m)
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atorG%&arrowGF)-%$sumG6$*&&%\"BG6#%\"kG\"\"\"-%$sinG6#*&F4F59%F5F5/F4;
F59$F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Now, we look at dif
ferent approximations of S as m increases:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 66 "plot(\{f,S(1,x)\},x=-2*Pi..2*Pi,title=\"First appro
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" {TEXT -1 74 "The approximation gets better., Let's try some higher o
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "The slow divergence of the series
 at the points where the function is discontinuous is known as the " }
{TEXT 256 18 "Gibbs phenomenon. " }{TEXT -1 99 "Can you argue why the \+
convergence is slower? To what value does the series converge at these
 points" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "Section 2: A Second \+
example: Homework #3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "f:=
Heaviside(signum(sin(x)));g:=signum(sin(x)):assume(n,integer):\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%*HeavisideG6#-%'signumG6#-%$si
nG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot( f,x=-4*P
i..4*Pi,-0.2..1.2,title=\"Problem 3\");\n" }}{PARA 13 "" 1 "" 
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*eqjc7F*F\\r-%'COLOURG6&%$RGBG$\"*++++\"!\")F\\rF\\r-%*THICKNESSG6#\"
\"#-%&TITLEG6#Q*Problem~36\"-%+AXESLABELSG6$%\"xGQ!Fi`n-%%VIEWG6$;$!+i
qjc7F``n$\"+iqjc7F``n;$!\"#!\"\"$\"#7Fjan" 1 2 0 1 10 2 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 16 "The coefficient " }{XPPEDIT 18 0 "a[0]" "6#&%\"aG6#\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a[0]:=1/2/Pi*Int('f',x=
-Pi..Pi)=1/2/Pi*int(g,x=0..Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%\"aG6#\"\"!/,$*&#\"\"\"\"\"#F,,$-%$IntG6$%\"fG/%\"xG;,$%#PiG!\"\"F7*
$F7F8F,F,F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 16 "The coefficient " }{XPPEDIT 18 0 "a[n];" "6#&%
\"aG6#%\"nG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a[n]:=1/Pi*I
nt('f'*cos(n*x),x=-Pi..Pi)=1/Pi*int(g*cos(n*x),x=0..Pi);\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#%#n|irG/,$-%$IntG6$*&%\"fG\"\"\"-%$
cosG6#*&F'F/%\"xGF/F//F4;,$%#PiG!\"\"F8*$F8F9\"\"!" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 16 "The coefficient " }{XPPEDIT 18 0 "b[n];" "6#&%\"b
G6#%\"nG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "b[n]:=1/Pi*Int(
'f'*sin(n*x),x=-Pi..Pi)=1/Pi*int(g*sin(n*x),x=0..Pi);\n" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"bG6#%#n|irG/,$-%$IntG6$*&%\"fG\"\"\"-%$sinG
6#*&F'F/%\"xGF/F//F4;,$%#PiG!\"\"F8*$F8F9,$*(F8F9,&F/F9)F9F'F/F/F'F9F9
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "B[k]:=subs(n=k,1/Pi*int
(g*sin(n*x),x=0..Pi));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#%
\"kG,$*(,&)!\"\"F'\"\"\"F-F,F-F'F,%#PiGF,F," }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 33 "Let us look at some coefficients " }{XPPEDIT 18 0 "b[k]
" "6#&%\"bG6#%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eval(
seq(([i,subs(n=i,B)]),i=1..10));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,
7$\"\"\"%\"BG7$\"\"#F%7$\"\"$F%7$\"\"%F%7$\"\"&F%7$\"\"'F%7$\"\"(F%7$
\"\")F%7$\"\"*F%7$\"#5F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "All e
ven coefficients are 0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Let ca
ll S(n,x), the Fourier series approximations with n coefficients:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "S:=(m,x)->1/2+sum(B[k]*sin(k
*x),k=1..m);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SGf*6$%\"mG%\"xG
6\"6$%)operatorG%&arrowGF),&#\"\"\"\"\"#F/-%$sumG6$*&&%\"BG6#%\"kGF/-%
$sinG6#*&F8F/9%F/F//F8;F/9$F/F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 61 "Now, we look at different approximations of S as m increases:" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Let us see n=,7,9,11,..." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "plot([f,S(7,x),S(9,x)],x=-P
i-0.01..Pi+0.01,title=\"Higher approximations: n=7 (Green),n=9 (Blue)
\",color=[red,green,blue],numpoints=100);\n" }}{PARA 13 "" 1 "" 
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\\l-Fj]m6&F\\^mF@F@F]^m-%*THICKNESSG6#\"\"#-%&TITLEG6#QDHigher~approxi
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J!\"*$\"+aEf^JFih]l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 11 " References" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "E. \+
Kreyszig : " }{TEXT 257 47 "Advanced Engineering Mathematics (8th Edit
ion) " }{TEXT -1 27 "John Wiley  New York (1999)" }}}{EXCHG {PARA 257 
"" 0 "" {TEXT 258 11 "Disclaimer:" }{TEXT -1 240 " While every effort \+
has been made to validate the solutions in this worksheet, Waterloo Ma
ple Inc. and the contributors are not responsible for any errors conta
ined and are not liable for any damages resulting from the use of this
 material." }}}}}{MARK "7" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }