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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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{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. L
et's first plot a section over the interval x = -1 to x = 1. We use th
e " }{TEXT 256 9 "piecewise" }{TEXT -1 22 " command as follows.  " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g := x -> piecewise(x<0,x+1,
x>0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operat
orG%&arrowGF(-%*piecewiseG6&29$\"\"!,&F0\"\"\"F3F32F1F0F1F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Now lets plot this one piece. " }}
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1/Y2d<.Y&)F.F*7$$\"1*fqqJnjv)F.F*7$$\"1825_Qk\\*)F.F*7$$\"185\\s0;r\"*
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\"F*F*FizF`[l-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fhdl%(DEFAULTG" 1 
2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 99 "Now we want g(x+2) = g(x) and g(x-2) = g(x) as wel
l.  We define another piecewise function called f" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 49 "f := x -> piecewise(x<-1,g(x+2),x>1,g(x-2),g
(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operator
G%&arrowGF(-%*piecewiseG6'29$!\"\"-%\"gG6#,&F0\"\"\"\"\"#F62F6F0-F36#,
&F0F6!\"#F6-F36#F0F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "and p
lot it from -3 to 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plo
t(f(x),x=-3..3,discont=true,thickness=3);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%7S7$$\"1+++1+++?!#:\"
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1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
171 "There you have it.  If you want to plot this over a broader range
 of x values. You would have to lengthen 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 repres
entation of such functions. " }}{PARA 0 "" 0 "" {TEXT -1 96 "Now we wa
nt to find a Fourier series representation of the this function define
d above as f(x). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{GLPLOT2D 254 176 
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7$$!1;Eq\"*)ywl\"FavF[al7$$!1p4i,XUh7FavF`al7$$!10EWTnM*Q)FjuFeal7$$!1
mwh2+eXVFjuFjal7$$!1+++++++SFfuF_blFftF]u-%+AXESLABELSG6$Q\"x6\"%!G-%%
VIEWG6$;FeblFdt%(DEFAULTG" 1 2 0 1 3 2 9 1 4 2 1.000000 70.000000 
-148.000000 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Recall that the f
inite Fourier series has the form" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "F[
n](x) = a[0]/2+sum(a[m]*cos(m*Pi*x/L)+b[m]*sin(m*Pi*x/L),m = 1 .. n);
" "6#/-&%\"FG6#%\"nG6#%\"xG,&*&&%\"aG6#\"\"!\"\"\"\"\"#!\"\"F1-%$sumG6
$,&*&&F.6#%\"mGF1-%$cosG6#**F;F1%#PiGF1F*F1%\"LGF3F1F1*&&%\"bG6#F;F1-%
$sinG6#**F;F1F@F1F*F1FAF3F1F1/F;;\"\"\"F(F1" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 12 "Recall that " }{XPPEDIT 18 0 "a[m];" "6#&%\"aG6
#%\"mG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[m];" "6#&%\"bG6#%\"mG" }
{TEXT -1 63 "  have specific values given by the definite integrals be
low.  " }}{PARA 0 "" 0 "" {TEXT -1 79 "First though, lets define L whi
ch is one half the fundamental period of  f(x). " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 7 "L := 1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
63 "Now for the coefficients. I create a function a(m) and b(m) by " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "a := m -> 1/L*int(f(x)*cos
(m*Pi*x/L),x=-L..L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGR6#%\"mG
6\"6$%)operatorG%&arrowGF(*&-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#*&*
(9$F5%#PiGF5F4F5\"\"\"%\"LG!\"\"F5/F4;,$F>!\"\"F>F=F>F?F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "b := m -> 1/L*int(f(x)*sin(m
*Pi*x/L),x=-L..L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGR6#%\"mG6
\"6$%)operatorG%&arrowGF(*&-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#*&*(
9$F5%#PiGF5F4F5\"\"\"%\"LG!\"\"F5/F4;,$F>!\"\"F>F=F>F?F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Now we will define a function F t
hat sums the first n terms of the series at x.  This is defined as " }
{XPPEDIT 18 0 "F[n](x);" "6#-&%\"FG6#%\"nG6#%\"xG" }{TEXT -1 179 " abo
ve only in Maple we need to make F a function of two variables: n and \+
x, where n is the number of terms to use and x is the value at which t
o evaluate F. We do this using the " }{TEXT 260 3 "sum" }{TEXT -1 13 "
 command by  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "F := (x,n)
 -> a(0)/2 + Sum(a(m)*cos(m*Pi*x/L)+b(m)*sin(m*Pi*x/L),m=1..n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6$%\"xG%\"nG6\"6$%)operatorG%&a
rrowGF),&-%\"aG6#\"\"!#\"\"\"\"\"#-%$SumG6$,&*&-F/6#%\"mGF3-%$cosG6#*&
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F3F)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 decide on the number of terms to use.  \+
Here we use the first 4 terms." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "plot(F(x,4),x=-3..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
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{PARA 0 "" 0 "" {TEXT -1 28 "How about the first 10 terms" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(F(x,10),x=-3..3);" }}{PARA 13 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 339 "It looks like things are improvin
g except near the points of discontinuity.  This is known as the Gibbs
 phenomenon. Here's how we show the two graphs together.  When you giv
e a graph a name such as 'graphf' below, be sure to end the command wi
th a colon. Otherwise you will get a very messy output. We then combin
e the two graphs with the " }{TEXT 257 7 "display" }{TEXT -1 22 " comm
and 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):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "display(\{graphf,graphF\});" }}{PARA 13 "" 1 "
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{TEXT -1 308 "This looks pretty good, except near the points of discon
tinuity of  f(x).  If we want, we can graph the error in order to visu
alize the improvements in using more terms. We define a function error
 which compares the two for a given number of terms. We use absolute v
alue so that error is always postitive.   " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "error := (x,n) -> abs(f(x) - F(x,n)):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 126 "Notice 'error(x,n)' is a function of two
 variables: x is where to evaluate f and F, and n is the number of ter
ms to use in F. " }}{PARA 0 "" 0 "" {TEXT -1 81 "Now let's plot the er
ror over x = -3 ..3 using a four term finite Fourier series." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(error(x,4),x=-3..3);" }
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0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "This displays what was clea
r from the two graphs.  Specifically, the error on the continuous regi
on is pretty small be we get larger errors near the points of disconti
nuities in f. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 87 
"See if increasing the number of terms in the finite Fourier series re
duces this error. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# end
 of section" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 197 "Assignment: Do
 problems 21 and 25 from Section 10.2 in the text. As usual, presentat
ion counts.  This will count as a 20 point question on the next exam. \+
 You may work in groups of no more than 4. " }}}}{MARK "1 0 0" 0 }
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