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{PSTYLE "" 0 398 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 
1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 399 1 {CSTYLE "" -1 
-1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 
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0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 17 "" 0 "" {TEXT -1 39 "     Paradigms in Physics: Oscil
lations" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 "" {TEXT -1 
77 "© 2000 Oregon State University                       version Oct 1
6, 2000 DMc" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 "" 
{TEXT -1 6 "      " }}{PARA 18 "" 0 "" {TEXT -1 29 "Fourier Series App
roximations" }}{PARA 18 "" 0 "" {TEXT -1 16 "for Square Waves" }}
{PARA 19 "" 0 "" {TEXT 304 75 "by Jason Janesky, Catherine Meyer, Cori
nne A. Manogue and Philip J. Siemens" }}{PARA 19 "" 0 "" {TEXT -1 43 "
Physics Department, Oregon State University" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 256 "" 0 "" 
{TEXT 258 22 "Worksheet Introduction" }}{PARA 265 "" 0 "" {TEXT -1 78 
"This worksheet is designed as a first experience in the use of Fourie
r series." }}{PARA 266 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" 
{TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 53 "A Fourier series can be
 used to represent a function " }{XPPEDIT 18 0 "g(t) " "6#-%\"gG6#%\"t
G" }{TEXT -1 48 "which is defined on a finite interval of length " }
{XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT 303 3 ",  " }{TEXT -1 22 "or a peri
odic function" }{TEXT 306 8 "  g(t) =" }{TEXT -1 3 " g(" }{TEXT 302 3 
"t+T" }{TEXT -1 61 ").   The fundamental angular frequency of such a f
unction is " }{XPPEDIT 18 0 "omega = 2*Pi/T" "6#/%&omegaG*(\"\"#\"\"\"
%#PiGF'%\"TG!\"\"" }{TEXT -1 176 " ,   Fourier series analysis involve
s the approximation of such a function as an infinite sum of terms who
se frequencies are integer multiples ofmega the fundamental frequency \+
" }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 1 "." }}{PARA 269 "" 
0 "" {TEXT -1 1 " " }}{PARA 270 "" 0 "" {TEXT -1 0 "" }}{PARA 271 "" 
0 "" {TEXT -1 286 "There are two types of Fourier series.  The Fourier
 Exponential Series is especially convenient for representing complex \+
functions.  The Fourier Sine and Cosine Series are often more convenie
nt for representing real functions.  In this worksheet we will use the
  Sine and Cosine Series:" }}{PARA 272 "" 0 "" {XPPEDIT 18 0 "g(t)=a[0
]/2 +a[1]*cos(omega t)+a[2]*cos(2omega t)+a[3]*cos(3omega t)" "6#/-%\"
gG6#%\"tG,**&&%\"aG6#\"\"!\"\"\"\"\"#!\"\"F.*&&F+6#F.F.-%$cosG6#*&%&om
egaGF.F'F.F.F.*&&F+6#F/F.-F56#*(F/F.F8F.F'F.F.F.*&&F+6#\"\"$F.-F56#*(F
BF.F8F.F'F.F.F." }{TEXT -1 3 " + " }{TEXT 257 3 "..." }}{PARA 272 "" 
0 "" {TEXT -1 5 "...+ " }{XPPEDIT 18 0 "+ b[1]*sin(omega t)+b[2]*sin(2
omega t)+b[3]*sin(3omega t) " "6#,(*&&%\"bG6#\"\"\"F(-%$sinG6#*&%&omeg
aGF(%\"tGF(F(F(*&&F&6#\"\"#F(-F*6#*(F2F(F-F(F.F(F(F(*&&F&6#\"\"$F(-F*6
#*(F9F(F-F(F.F(F(F(" }{TEXT -1 6 " + ..." }}{PARA 273 "" 0 "" {TEXT 
-1 0 "" }}{PARA 274 "" 0 "" {TEXT -1 37 "The coefficients in the serie
s Ð the " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 7 "'s and \+
" }{XPPEDIT 18 0 "b[n]" "6#&%\"bG6#%\"nG" }{TEXT -1 79 "'s Ð  can be c
hosen to provide the best approximation to the original function " }
{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" }{TEXT -1 3 ".  " }}{PARA 275 "
" 0 "" {TEXT -1 0 "" }}{PARA 276 "" 0 "" {TEXT -1 196 "It might not se
em much of an advantage to approximate a function with an infinite sum
.  But we will see that we can often obtain a useful approximation by \+
keeping only a few terms of the series.  " }}{PARA 277 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 366 "This worksheet presents the F
ourier series approximation of a square wave by sines and cosines.  Al
l coefficients are provided.  In a following worksheet you will guess \+
the most useful coefficients for a Fourier series approximation of a s
awtooth waveform.  Finally, for homework, you will use a systematic me
thod to find the best possible values of the coefficients" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 11 "Square Wave" }}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 0 
20 "restart:with(plots):" }}}{PARA 279 "" 0 "" {TEXT -1 0 "" }}{PARA 
280 "" 0 "" {TEXT -1 187 "We will choose an example of a periodic func
tion.  Of course, we cannot show the whole function on a graph.  So we
 will just show a representative piece out of the its range, say betwe
en " }{XPPEDIT 18 0 "-Pi" "6#,$%#PiG!\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 50 ".  In this interval, we can
 picture a square wave " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }
{TEXT -1 15 " using Maple's " }{HYPERLNK 17 "" 1 ":[?GetError?]" "" }
{HYPERLNK 17 "piecewise" 2 "" "" }{TEXT -1 14 " function:    " }}
{PARA 281 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 
1 0 54 "f(t):=piecewise(t<=-Pi, 2, t<0, 1, t<=Pi, 2, t>Pi, 1);" }}}
{EXCHG {PARA 283 "> " 0 "" {MPLTEXT 1 0 45 "plot(f(t), t=-2*Pi..2*Pi,0
..2.5,axes=normal);" }}}{PARA 284 "" 0 "" {TEXT -1 0 "" }}{PARA 285 "
" 0 "" {TEXT -1 31 "What is the fundamental period " }{XPPEDIT 18 0 "T
" "6#%\"TG" }{TEXT -1 4 " ?  " }}{PARA 286 "" 0 "" {TEXT -1 0 "" }}
{PARA 287 "" 0 "" {TEXT -1 34 "What is the fundamental frequency " }
{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 2 " ?" }}{PARA 288 "" 0 "
" {TEXT -1 0 "" }}{PARA 289 "" 0 "" {TEXT -1 53 "What would you call t
he amplitude of the square wave?" }}{PARA 290 "" 0 "" {TEXT -1 2 "  " 
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Plotting the Fourier Series, g(
t)" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 291 "" 0 "" {TEXT -1 203 "
In theory, a Fourier series can have an infinite number of terms.  In \+
practice, we need to truncate the series to a finite set of terms.  Us
ually we choose the terms which have the \"smoothest\" behavior. " }}
{PARA 292 "" 0 "" {TEXT -1 0 "" }}{PARA 293 "" 0 "" {TEXT -1 41 "A gen
eric term in the series looks like  " }{XPPEDIT 18 0 "a[n]*cos(n omega
 t)" "6#*&&%\"aG6#%\"nG\"\"\"-%$cosG6#*(F'F(%&omegaGF(%\"tGF(F(" }
{TEXT -1 4 " or " }{XPPEDIT 18 0 "b[n]*sin(n omega t)" "6#*&&%\"bG6#%
\"nG\"\"\"-%$sinG6#*(F'F(%&omegaGF(%\"tGF(F(" }{TEXT -1 28 " .  Each o
f these terms has " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 12 " maxima
 and " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 33 " minima in an interv
al of length " }{XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT -1 65 " .  The smoo
thest terms are those with fewest maxima and minima, " }{TEXT 305 4 "i
.e." }{TEXT -1 29 " with the smallest values of " }{XPPEDIT 18 0 "n" "
6#%\"nG" }{TEXT -1 3 " . " }{MPLTEXT 1 0 0 "" }}{PARA 294 "" 0 "" 
{TEXT -1 0 "" }}{PARA 295 "" 0 "" {TEXT -1 109 "We can find a smooth  \+
approximation to the square wave by using the first few terms of the s
eries, say up to " }{XPPEDIT 18 0 "n=Nsum" "6#/%\"nG%%NsumG" }{TEXT 
-1 2 " :" }}{PARA 295 "" 0 "" {TEXT -1 0 "" }}{PARA 295 "" 0 "" 
{XPPEDIT 18 0 "g[Nsum]=a[0]/2+Sum(a[n]*cos(n omega t) + b[n]*sin(n ome
ga t),n=1..Nsum)" "6#/&%\"gG6#%%NsumG,&*&&%\"aG6#\"\"!\"\"\"\"\"#!\"\"
F.-%$SumG6$,&*&&F+6#%\"nGF.-%$cosG6#*(F8F.%&omegaGF.%\"tGF.F.F.*&&%\"b
G6#F8F.-%$sinG6#*(F8F.F=F.F>F.F.F./F8;F.F'F." }}{PARA 295 "" 0 "" 
{TEXT -1 0 "" }}{PARA 295 "" 0 "" {TEXT -1 36 "We will start with a sm
all value of " }{XPPEDIT 18 0 "Nsum" "6#%%NsumG" }{TEXT -1 52 ".  Then
 we can choose increasingly larger values of " }{XPPEDIT 18 0 "nmax" "
6#%%nmaxG" }{TEXT -1 79 " to  see how the approximation gets closer an
d closer to the exact square wave " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%
\"tG" }{TEXT -1 1 "." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 296 "
" 0 "" {TEXT -1 23 "We will start with the " }{XPPEDIT 18 0 "a[0]" "6#
&%\"aG6#\"\"!" }{TEXT -1 6 " term:" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 
285 1 " " }{TEXT 299 10 "Constant (" }{XPPEDIT 300 0 "n=0" "6#/%\"nG\"
\"!" }{TEXT 275 6 ") term" }}{PARA 5 "" 0 "" {TEXT 272 3 "   " }{TEXT 
301 36 "Let's start with our function again:" }}{PARA 377 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 378 "> " 0 "" {MPLTEXT 1 0 45 "plot(f(t),
 t=-2*Pi..2*Pi,0..2.5,axes=normal);" }}}{EXCHG {PARA 379 "" 0 "" 
{TEXT -1 0 "" }}{PARA 380 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "a[
0]" "6#&%\"aG6#\"\"!" }{TEXT -1 161 " term is just a constant.  We're \+
not going to make a great approximation with this one term.  The best \+
choice for this term is the average value of the function:" }}{PARA 
381 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 382 "> " 0 "" {MPLTEXT 1 0 
9 "a[0] :=3;" }}}{EXCHG {PARA 383 "> " 0 "" {MPLTEXT 1 0 15 "g(t) := a
[0]/2;" }}}{EXCHG {PARA 384 "" 0 "" {TEXT -1 0 "" }}{PARA 385 "" 0 "" 
{TEXT -1 50 "Plotting this against our original function gives:" }}
{PARA 386 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 387 "> " 0 "" 
{MPLTEXT 1 0 69 "plot([f(t),g(t)],t=-2*Pi..2*Pi,0..2.5,axes=normal, co
lor=[red,blue]);" }}}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 
"" {TEXT 273 125 "Subtracting the original square wave from our guess \+
so far gives us an idea on what we should use for our next coefficient
s; " }{XPPEDIT 18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "b[1]" "6#&%\"bG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 388 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 389 "> " 0 "" {MPLTEXT 1 0 70 "plo
t(f(t)-g(t),t=-2*Pi..2*Pi,-1.5..1.5,axes=normal, color=[red,blue]);" }
}}}{SECT 1 {PARA 258 "" 0 "" {TEXT 266 1 "F" }{TEXT 276 12 "undamental
 (" }{XPPEDIT 18 0 "n=1" "6#/%\"nG\"\"\"" }{TEXT 277 6 ") term" }}
{PARA 297 "" 0 "" {TEXT -1 121 "Now it looks like we should have a zer
o coefficient for the cosine term and some positive coefficient for th
e sine.  WHY?" }}{PARA 298 "" 0 "" {TEXT -1 29 "The correct coefficien
ts are:" }}{PARA 299 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 300 "> " 0 "
" {MPLTEXT 1 0 8 "a[1]:=0;" }}{PARA 301 "> " 0 "" {MPLTEXT 1 0 11 "b[1
]:=2/Pi;" }}}{EXCHG {PARA 302 "" 0 "" {TEXT -1 78 "Maybe you were thin
king that the coefficient for the sine term should be .5.  " }}{PARA 
303 "" 0 "" {TEXT -1 19 "WHY IS THE CHOICE  " }{XPPEDIT 18 0 "b[1]=2/P
i" "6#/&%\"bG6#\"\"\"*&\"\"#F'%#PiG!\"\"" }{TEXT -1 10 "  BETTER ?" }}
{PARA 304 "" 0 "" {TEXT -1 0 "" }}{PARA 305 "" 0 "" {TEXT -1 26 "Our F
ourier series is now:" }}{PARA 306 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 307 "> " 0 "" {MPLTEXT 1 0 39 "g(t):=1/2*a[0]+a[1]*cos(t)+b[1]*s
in(t);" }}}{EXCHG {PARA 308 "" 0 "" {TEXT -1 11 "This gives:" }}{PARA 
309 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "> " 0 "" {MPLTEXT 1 0 68 "plot(
[f(t),g(t)],t=-2*Pi..2*Pi,0..2.5,axes=normal,color=[red,blue]);" }}
{PARA 311 "" 0 "" {TEXT -1 0 "" }}{PARA 312 "" 0 "" {TEXT -1 65 "It sh
ould now be evident why .5 wasn't a good enough coefficient." }}{PARA 
313 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 314 "" 0 "" {TEXT -1 106 " \+
Again subtracting the original from our approximation so far gives us \+
a clue about the next terms to use." }}{PARA 315 "" 0 "" {TEXT -1 0 "
" }}{PARA 316 "> " 0 "" {MPLTEXT 1 0 48 "plot(g(t)-f(t),t=-2*Pi..2*Pi,
-1..1,axes=normal);" }}{PARA 317 "" 0 "" {TEXT -1 0 "" }}{PARA 318 "" 
0 "" {TEXT -1 128 "This doesn't look much like a sine or cosine, but n
otice that it is still an oscillating function with more maxima and mi
nima.  " }}}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 
267 7 "Second " }{TEXT 278 8 "harmonic" }{TEXT -1 2 " (" }{XPPEDIT 18 
0 "n=2" "6#/%\"nG\"\"#" }{TEXT 279 6 ") term" }}{PARA 319 "" 0 "" 
{TEXT -1 208 "The second harmonic terms are sines and cosines oscillat
ing at twice the fundamental frequency.  From the graph just produced \+
it should be evident why a Fourier component of this frequency will no
t be needed." }}{PARA 320 "" 0 "" {TEXT -1 0 "" }}{PARA 321 "" 0 "" 
{TEXT -1 8 "WHY NOT?" }}{PARA 322 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 323 "> " 0 "" {MPLTEXT 1 0 16 "a[2]:=0;b[2]:=0;" }}}{PARA 324 "
" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 325 "" 0 "" {TEXT 268 14 "Third \+
harmonic" }{TEXT 271 1 " " }{TEXT -1 1 "(" }{XPPEDIT 18 0 "n=3" "6#/%
\"nG\"\"$" }{TEXT 280 6 ") term" }}{PARA 326 "" 0 "" {TEXT -1 236 "How
ever the subtracted graph does show oscillations with 3 maxima and min
ima.  The best way to represent this remainder is with components of 3
 times the fundamental frequency.  Thus we will have nonzero coefficie
nts for the terms with " }{XPPEDIT 18 0 "n=3" "6#/%\"nG\"\"$" }{TEXT 
-1 59 ".  Looking at the subtracted graph, the cosine coefficient " }
{XPPEDIT 18 0 "a[3]" "6#&%\"aG6#\"\"$" }{TEXT -1 52 " will again be ze
ro (WHY?) and the sine coefficent  " }{XPPEDIT 18 0 "b[3]" "6#&%\"bG6#
\"\"$" }{TEXT -1 18 " will be positive." }}{PARA 327 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 328 "> " 0 "" {MPLTEXT 1 0 8 "a[3]:=0;" }}{PARA 
329 "> " 0 "" {MPLTEXT 1 0 15 "b[3]:=2/(3*Pi);" }}}{EXCHG {PARA 330 "
" 0 "" {TEXT -1 50 "Adding these terms to the Fourier expansion gives:
" }}{PARA 331 "" 0 "" {TEXT -1 0 "" }}{PARA 332 "> " 0 "" {MPLTEXT 1 
0 96 "g(t):=1/2*a[0]+a[1]*cos(t)+a[2]*cos(2*t)+a[3]*cos(3*t)+b[1]*sin(
t)+ b[2]*sin(2*t)+b[3]*sin(3*t);" }}{PARA 333 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 334 "" 0 "" {TEXT -1 19 "And plotting gives:" }}{PARA 
335 "" 0 "" {TEXT -1 0 "" }}{PARA 336 "> " 0 "" {MPLTEXT 1 0 68 "plot(
[f(t),g(t)],t=-2*Pi..2*Pi,0..2.5,axes=normal,color=[red,blue]);" }}}
{PARA 337 "" 0 "" {TEXT -1 0 "" }}{PARA 338 "" 0 "" {TEXT -1 86 "Isn't
 it amazing how they are starting to match up?  We must be doing somet
hing right!" }}{PARA 339 "" 0 "" {TEXT -1 76 "How does the graph chang
e as you add additional terms of the Fourier Series?" }}{PARA 340 "" 
0 "" {TEXT 274 71 "Subtracting the latest approximation gives us a clu
e to the next terms." }}{PARA 341 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 342 "> " 0 "" {MPLTEXT 1 0 48 "plot(g(t)-f(t),t=-2*Pi..2*Pi,-1..
1,axes=normal);" }}{PARA 343 "" 0 "" {TEXT -1 0 "" }}{PARA 344 "" 0 "
" {TEXT -1 57 "Again, what frequency would the next term most likely b
e?" }}}}{SECT 1 {PARA 345 "" 0 "" {TEXT 269 7 "Fourth " }{TEXT 281 10 
"harmonic (" }{XPPEDIT 18 0 "n=4" "6#/%\"nG\"\"%" }{TEXT 282 6 ") term
" }}{PARA 346 "" 0 "" {TEXT -1 166 "The fourth harmonic terms have a f
requency of 4 times the fundamental frequency.  Looking at the subtrac
ted graph, this does not match up with what we need.  WHY NOT?" }}
{PARA 347 "" 0 "" {TEXT -1 32 "So  these coefficients are zero:" }}
{EXCHG {PARA 348 "> " 0 "" {MPLTEXT 1 0 8 "a[4]:=0;" }}{PARA 349 "> " 
0 "" {MPLTEXT 1 0 8 "b[4]:=0;" }}}}{SECT 1 {PARA 350 "" 0 "" {TEXT 
270 6 "Fifth " }{TEXT 283 10 "harmonic (" }{XPPEDIT 18 0 "n=5" "6#/%\"
nG\"\"&" }{TEXT 284 6 ") term" }}{PARA 351 "" 0 "" {TEXT -1 110 "We do
 need terms with  five maxima and minima, since they look most like th
e remainder.  The coefficients are:" }}{EXCHG {PARA 352 "> " 0 "" 
{MPLTEXT 1 0 8 "a[5]:=0;" }}{PARA 353 "> " 0 "" {MPLTEXT 1 0 15 "b[5]:
=2/(5*Pi);" }}}{EXCHG {PARA 354 "" 0 "" {TEXT -1 37 "giving us a Fouri
er series so far of:" }}{PARA 355 "" 0 "" {TEXT -1 0 "" }}{PARA 356 ">
 " 0 "" {MPLTEXT 1 0 138 "g(t):=1/2*a[0]+a[1]*cos(t)+a[2]*cos(2*t)+a[3
]*cos(3*t)+a[4]*cos(4*t)+a[5]*sin(5*t)+b[1]*sin(t)+ b[2]*sin(2*t)+b[3]
*sin(3*t)+b[5]*sin(5*t);" }}}{EXCHG {PARA 357 "" 0 "" {TEXT -1 24 "and
 producing a plot of:" }}{PARA 358 "" 0 "" {TEXT -1 0 "" }}{PARA 359 "
> " 0 "" {MPLTEXT 1 0 68 "plot([f(t),g(t)],t=-2*Pi..2*Pi,0..2.5,axes=n
ormal,color=[red,blue]);" }}}{PARA 360 "" 0 "" {TEXT -1 0 "" }}}{SECT 
1 {PARA 260 "" 0 "" {TEXT 265 21 "Looking at many terms" }}{PARA 361 "
" 0 "" {TEXT -1 76 "Surely by now you have guessed some characteristic
s of this Fourier series; " }}{PARA 362 "" 0 "" {TEXT -1 39 "all even \+
terms have zero coefficients, " }}{PARA 363 "" 0 "" {TEXT -1 26 "all c
osine terms are zero " }}{PARA 364 "" 0 "" {TEXT -1 38 "the odd sine c
oefficients are positive" }}{PARA 365 "" 0 "" {TEXT -1 32 "the coeffic
ients get smaller as " }{XPPEDIT 18 0 "n " "6#%\"nG" }{TEXT -1 11 " in
creases." }}{PARA 366 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 
-1 196 "After some math we can write a general form for the Fourier Se
ries Expansion out to any number of terms.  The calculation is given i
n the class notes on Fourier Series, and also in every textbook." }}
{PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 95 "In th
e formula below, there are Nsum+1 terms and you need to assign Nsum a \+
value at the prompt." }}{PARA 367 "" 0 "" {TEXT -1 0 "" }}{PARA 368 "
" 0 "" {TEXT 286 8 "WARNING:" }{TEXT -1 36 "  The formula written belo
w is only " }{TEXT 287 34 "valid for Nsum values larger than " }
{XPPEDIT 288 0 "1" "6#\"\"\"" }{TEXT -1 31 " and it is recommended tha
t you" }{TEXT 289 17 " keep Nsum below " }{XPPEDIT 290 0 "20" "6#\"#?
" }{TEXT -1 22 " for the sake of time." }}{EXCHG {PARA 369 "> " 0 "" 
{MPLTEXT 1 0 9 "Nsum:=10;" }}}{EXCHG {PARA 370 "> " 0 "" {MPLTEXT 1 0 
58 "g(t):=a[0]/2+2/Pi*Sum(1/(2*m+1)*sin((2*m+1)*t),m=0..Nsum);" }}}
{PARA 371 "" 0 "" {TEXT -1 201 "If you want to see the terms of the ab
ove sum written out, change the \"S\" in \"Sum\" written above to an \+
\"s\" and execute the statement again.  (If you are using a large Nsum
 this will speed things up.  )" }}{EXCHG {PARA 372 "> " 0 "" {MPLTEXT 
1 0 69 "plot([f(t),g(t)],t=-2*Pi..2*Pi,0..2.5,axes=normal, color=[red,
blue]);" }}}{PARA 373 "" 0 "" {TEXT -1 150 "Is there any correlation b
etween the number of terms in the series and the number of times the t
wo curves cross each other in each interval of length " }{XPPEDIT 18 
0 "Pi" "6#%#PiG" }{TEXT -1 32 " (half period) along the x-axis?" }}
{PARA 374 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 375 "" 0 "" {TEXT 
291 16 "Gibbs phenomenon" }}{PARA 376 "" 0 "" {TEXT -1 65 "Each time w
e add a term to the Fourier series, it makes a better " }{TEXT 292 7 "
overall" }{TEXT -1 53 " approximation to the square wave.  Specificall
y, the" }{TEXT 293 8 " average" }{TEXT -1 97 " difference between the \+
approximate and exact functions gets smaller and smaller.    However, \+
the" }{TEXT 294 8 " largest" }{TEXT -1 153 " deviation between the two
 does not get smaller!  There is always an overshoot of the same amoun
t close to the discontinuity in the square wave.  But the" }{TEXT 295 
7 " region" }{TEXT -1 73 " of this overshoot gets less as we add terms
 to the series approximation." }}}}}{SECT 1 {PARA 257 "" 0 "" {TEXT 
-1 43 "Comparison between Fourier and Power Series" }}{PARA 390 "" 0 "
" {TEXT -1 34 "In Math 253 you learned how to use" }{TEXT 263 1 " " }
{TEXT -1 12 "power series" }{TEXT 264 1 " " }{TEXT -1 262 "to approxim
ate complicated functions, but now we want to determine an approximati
on which will effectively represent periodic functions.  It turns out \+
that, for many purposes, a Fourier Series is more effective than a Pow
er Series.  Some reasons for this follow:" }}{PARA 391 "" 0 "" {TEXT 
-1 0 "" }}{PARA 392 "" 0 "" {TEXT 259 12 "Power Series" }{TEXT 260 1 "
:" }{TEXT -1 2 "  " }}{PARA 393 "" 0 "" {TEXT -1 228 "If you recall, w
hen you used Power Series to approximate functions you expanded about \+
a point on the curve.  No matter how many terms you had in the series,
 the approximation perfectly matched the original function at that poi
nt" }{TEXT 297 1 "." }{TEXT -1 419 "  The more terms in your Power Ser
ies, the further you could move along the curve (from the point of exp
ansion) and still obtain an accurate approximation of the original fun
ction.  Far from the expansion point, the Power Series was not an accu
rate approximation of the original function.  In fact, while the extra
 terms increased the range of validity, they made the approximate poly
nomial blow up wildly farther away." }}{PARA 394 "" 0 "" {TEXT -1 0 "
" }}{PARA 395 "" 0 "" {TEXT 261 15 "Fourier Series:" }}{PARA 396 "" 0 
"" {TEXT -1 244 "A Fourier Series Expansion also becomes a better and \+
better approximation of the original function as we use more and more \+
terms in the series.  But, unlike the Power Series, a Fourier Series d
oes not favor any particular point in the interval." }{TEXT 262 1 " " 
}{TEXT -1 304 " Due to its periodic nature,  the Fourier Series is uni
quely adapted to approximating periodic functions.   Another advantage
 is that the Fourier Series can - with difficulty - represent function
s with discontinuities or sharp corners while the Power Series cannot \+
even begin to represent such functions." }}{PARA 397 "" 0 "" {TEXT -1 
0 "" }}{PARA 398 "" 0 "" {TEXT 296 23 "Oversimplified Summary:" }}
{PARA 399 "" 0 "" {TEXT -1 86 "The Taylor series is a good tool for th
e neighborhood of a single point on the curve. " }}{PARA 400 "" 0 "" 
{TEXT -1 74 "The Fourier series is a good tool for the overall behavio
r in an interval." }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "6 0" 16 
}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }