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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 55 "Piecewise Periodic Functi
ons and Finite Fourier Series'" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 
"Piecewise periodic functions" }}{PARA 0 "" 0 "" {TEXT -1 78 "Suppose \+
I want to generate a function that looks like the one pictured below. \+
" }}{PARA 4 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 254 176 176 {PLOTDATA 2 
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e 1" "Curve 2" "Curve 3" "Curve 4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
182 "Notice that the fundamental period is 2.  We can start anywhere. \+
 Working from -1 to 1 or 0 to 2 is easiest. Let's first plot a section
 over the interval x = -1 to x = 1. We use the " }{TEXT 256 9 "piecewi
se" }{TEXT -1 22 " command as follows.  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "g := x -> piecewise(x<0,x+1,x>0,0);\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecew
iseG6&29$\"\"!,&F0\"\"\"F3F32F1F0F1F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 30 "Now lets plot this one piece. " }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 45 "plot(g(x),x=-1..1,discont=true,thickness=3);\n" }}
{PARA 13 "" 1 "" {GLPLOT2D 392 392 392 {PLOTDATA 2 "6&-%'CURVESG6&7S7$
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F(F\\dl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
99 "Now we want g(x+2) = g(x) and g(x-2) = g(x) as well.  We define an
other piecewise function called f" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "f := x -> piecewise(x<-1,g(x+2),x>1,g(x-2),g(x));\n" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arr
owGF(-%*piecewiseG6'29$!\"\"-%\"gG6#,&F0\"\"\"\"\"#F62F6F0-F36#,&F0F6F
7F1-F36#F0F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "and plot it f
rom -3 to 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(f(x),x
=-3..3,discont=true,thickness=3);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 392 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "There you have it.  If you want t
o plot this over a broader range of x values. You would have to length
en your definition of g(x).  We will be happy with just 3 periods.  " 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
56 "Finite Fourier series representation of such functions. " }}{PARA 
0 "" 0 "" {TEXT -1 96 "Now we want to find a Fourier series representa
tion of the this function defined above as f(x). " }}{PARA 0 "" 0 "" 
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Sls[#FavFa`l7$$!1]rfJBr+@FavFf`l7$$!1;Eq\"*)ywl\"FavF[al7$$!1p4i,XUh7F
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F]u-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;FeblFdt%(DEFAULTG" 1 2 0 1 10 
3 2 9 1 4 2 1.000000 70.000000 -148.000000 0 0 "Curve 1" "Curve 2" "Cu
rve 3" "Curve 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "L := 1:" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Now for the coefficients. I cre
ate a function a(m) and b(m) by " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "a := m -> 1/L*int(f(x)*cos(m*Pi*x/L),x=-L..L);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"mG6\"6$%)operatorG%&arrow
GF(,$-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**9$F5%#PiGF5F4F5%\"LG!\"
\"F5/F4;,$F<F=F<*$F<F=F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "b := m -> 1/L*int(f(x)*sin(m*Pi*x/L),x=-L..L);\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"bGf*6#%\"mG6\"6$%)operatorG%&arrowGF(,$-%$intG
6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#**9$F5%#PiGF5F4F5%\"LG!\"\"F5/F4;,$F<F
=F<*$F<F=F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Now we will d
efine a function F that sums the first n terms of the series at x.  Th
is is defined as " }{XPPEDIT 18 0 "F[n](x);" "6#-&%\"FG6#%\"nG6#%\"xG
" }{TEXT -1 179 " above only in Maple we need to make F a function of \+
two variables: n and x, where n is the number of terms to use and x is
 the value at which to evaluate F. We do this using the " }{TEXT 260 
3 "sum" }{TEXT -1 13 " command by  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "F := (x,n) -> a(0)/2 + Sum(a(m)*cos(m*Pi*x/L)+b(m)*si
n(m*Pi*x/L),m=1..n);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6$%
\"xG%\"nG6\"6$%)operatorG%&arrowGF),&*&#\"\"\"\"\"#F0-%\"aG6#\"\"!F0F0
-%$SumG6$,&*&-F36#%\"mGF0-%$cosG6#**F=F0%#PiGF09$F0%\"LG!\"\"F0F0*&-%
\"bGF<F0-%$sinGF@F0F0/F=;F09%F0F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 21 "Notice the format of " }{TEXT 261 3 "sum" }{TEXT -1 134 "
 and be sure to use it correctly.  To plot this function we must decid
e on the number of terms to use.  Here we use the first 4 terms." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(F(x,4),x=-3..3);\n" }}
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}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 339 "It looks like things are improv
ing except near the points of discontinuity.  This is known as the Gib
bs phenomenon. Here's how we show the two graphs together.  When you g
ive a graph a name such as 'graphf' below, be sure to end the command \+
with a colon. Otherwise you will get a very messy output. We then comb
ine the two graphs with the " }{TEXT 257 7 "display" }{TEXT -1 22 " co
mmand found in the " }{TEXT 258 5 "plots" }{TEXT -1 10 " library. " }
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "graphf := \+
plot(f(x),x=-3..3,discont=true,thickness=3):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "graphF := plot(F(x,10),x=-3..3):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" 
{TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display(\{graphf,graphF\});
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{PARA 0 "" 0 "" {TEXT -1 308 "This looks pretty good, except near the \+
points of discontinuity of  f(x).  If we want, we can graph the error \+
in order to visualize the improvements in using more terms. We define \+
a function error which compares the two for a given number of terms. W
e use absolute value so that error is always postitive.   " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "errorfunction := (x,n) -> abs(f(x) \+
- F(x,n)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(errorfun
ction(x,4),x=-3..3);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 392 392 392 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "This displays what was clear from
 the two graphs.  Specifically, the error on the continuous region is \+
pretty small be we get larger errors near the points of discontinuitie
s in f. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 87 "See i
f increasing the number of terms in the finite Fourier series reduces \+
this error. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# end of se
ction" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 194 "Assignment: Do probl
ems 21 and 25 from Section 10.2 in the text. As usual, presentation co
unts.  This will count as a 20 point question on the next exam.  You m
ust work by yourself on this lab. " }}}}{MARK "0 1" 0 }{VIEWOPTS 1 1 
0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }