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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa
tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 39 "Section \+
2.4: Calculus on Fourier Series" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 0 {PARA 3 "" 0 "" {TEXT 261 30 "Maple Packages for Section 2.4" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 298 "In the previous Section 2.3, we examined
 the nature of the convergence of series. Since we are going to be usi
ng representations of functions by their associated Fourier series to \+
solve partial differential equations, we should pause a bit to ask if \+
the derivative of a Fourier series representing " }{TEXT 263 2 "f " }
{TEXT -1 51 "is a series which will represent the derivative of " }
{TEXT 264 1 "f" }{TEXT -1 325 ". To see that this is a non-trivial que
stion, recall that differentiation is a limiting process, as is integr
ation. Also, forming an infinite series from the finite sums is a limi
ting process. So, to ask whether the derivative on an infinite sum be \+
the sum of the derivatives is to ask whether the limits can be interch
anged." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 8 
"Example:" }{TEXT -1 57 " We show that the derivative of the series co
nverging to " }{TEXT 289 1 "f" }{TEXT -1 62 " is not the same as the s
eries representing the derivative of " }{TEXT 290 1 "f" }{TEXT -1 33 "
. We consider this example: take " }{TEXT 291 1 "f" }{TEXT -1 1 "(" }
{TEXT 292 1 "x" }{TEXT -1 4 ") = " }{TEXT 293 1 "x" }{TEXT -1 95 " on \+
[-1, 1]. The Fourier Trigonometric Series must be a sine series since \+
the function is odd. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "L:=
1;\nf:=x->x;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "We compute coeffi
cients for 20 terms." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "for
 n from 1 to 20 do\n   a[n]:=int(f(x)*sin(n*Pi*x/L),x=-L..L)/int(sin(n
*Pi*x/L)^2,x=-L..L):\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Next, we de
fine the approximation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "a
pprox:=x->add(a[n]*sin(n*Pi*x/L),n=1..20): approx(x);" }}}{PARA 0 "" 
0 "" {TEXT -1 65 "It is a good idea to check the approximation by comp
aring graphs." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot([f(x),
approx(x)],x=-1..1);" }}}{PARA 0 "" 0 "" {TEXT -1 118 "Maybe we can lo
ok at the coefficients and guess at the general formula. If not, we le
t Maple find the general formula." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 98 "n:='n':  assume(n,integer):\na:=n->int(x*sin(n*Pi*x/L
),x=-L..L)/int(sin(n*Pi*x/L)^2,x=-L..L):\na(n);" }}}{PARA 0 "" 0 "" 
{TEXT -1 54 "Hence, the series could be written rather compactly as" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "        \+
  " }{XPPEDIT 18 0 "2/Pi;" "6#*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 4 "  \+
  " }{XPPEDIT 18 0 "sum((-1)^n/n*sin(n*Pi*x),n);" "6#-%$sumG6$*(),$\"
\"\"!\"\"%\"nGF)F+F*-%$sinG6#*(F+F)%#PiGF)%\"xGF)F)F+" }{TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Now, we
 differentiate the series term by term." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "Sum(a(n)*diff(sin(n*Pi*x/L),x),n=1..20);" }}}{PARA 0 
"" 0 "" {TEXT -1 79 "This series will not converge in norm, for the co
efficients of this last are 2 " }{XPPEDIT 18 0 "(-1)^(n+1);" "6#),$\"
\"\"!\"\",&%\"nGF%F%F%" }{TEXT -1 41 ". That is not a square summable \+
sequence." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 213 "The question, then,
 is this: what additional hypothesis must there be so that the derivat
ive of the series term by term converges the derivative of the functio
n to which the series converged?  Here is the theorem." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 8 "THEOREM." }{TEXT -1 
4 " If " }{TEXT 265 1 "f" }{TEXT -1 1 "(" }{TEXT 266 1 "x" }{TEXT -1 
93 ") is periodic, continuous, and sectionally smooth, then the differ
entiated Fourier series of " }{TEXT 267 1 "f" }{TEXT -1 1 "(" }{TEXT 
268 1 "x" }{TEXT -1 15 ") converges to " }{TEXT 269 1 "f" }{TEXT -1 3 
" '(" }{TEXT 270 1 "x" }{TEXT -1 17 ") at every point " }{TEXT 271 1 "
x" }{TEXT -1 7 " where " }{TEXT 272 1 "f" }{TEXT -1 4 " ''(" }{TEXT 
273 1 "x" }{TEXT -1 9 ") exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 38 "The function in the previous example: " }
{TEXT 294 1 "f" }{TEXT -1 1 "(" }{TEXT 295 1 "x" }{TEXT -1 4 ") = " }
{TEXT 296 1 "x" }{TEXT -1 127 " on the interval [-1, 1] did not have a
 continuous periodic extension. Thus, it did not satisfy the hypothesi
s of this Theorem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 259 8 "Example:" }{TEXT -1 89 "  An example that has a periodic \+
extension that is continuous and sectionally smooth is  " }{TEXT 274 
1 "f" }{TEXT -1 1 "(" }{TEXT 275 1 "x" }{TEXT -1 5 ") = |" }{TEXT 276 
1 "x" }{TEXT -1 170 "| on [-1, 1]. The periodic extension has period 2
 and is continuous. The extension is periodic, continuous, and section
ally smooth. We expect that the Fourier Series for " }{TEXT 277 1 "f" 
}{TEXT -1 136 " to converge uniformly and that the differentiated seri
es will converge at every point except possibly -1, 0, and 1. Look at \+
the graphs." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=x->abs(x):
 f(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "for n from 0 to \+
10 do\n   a[n]:=int(abs(x)*cos(n*Pi*x),x=-1..1)/int(cos(n*Pi*x)^2,x=-1
..1):\nend do:\nn:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "
approx:=x->add(a[n]*cos(n*Pi*x),n=0..10): approx(x);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 68 "plot([[x,f(x),x=-1..1],[x,approx(x),x=-3.
.3]], scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 35 "We now pl
ot both the derivative of " }{TEXT 278 1 "f" }{TEXT -1 45 " and the de
rivative of the approximation for " }{TEXT 279 1 "f" }{TEXT -1 1 "." }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot([[x,diff(f(x),x),x=-1.
.1],[x,diff(approx(x),x),x=-3..3]]);" }}}{PARA 0 "" 0 "" {TEXT -1 292 
"Perhaps these two examples give experimental evidence for some unders
tanding for the nature of the results of the previous theorem and, sec
ond, make it believable. In both examples, do remember that outside th
e interval [-1, 1], we have that the series converges to the periodic \+
extension of " }{TEXT 280 1 "f" }{TEXT -1 1 "(" }{TEXT 281 1 "x" }
{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 23 "What about integration?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 257 8 "THEOREM:" }{TEXT -1 4 " If " }{TEXT 
287 1 "f" }{TEXT -1 1 "(" }{TEXT 288 1 "x" }{TEXT -1 70 ") is periodic
 and sectionally continuous, then the Fourier series for " }{TEXT 283 
1 "f" }{TEXT -1 1 "(" }{TEXT 284 1 "x" }{TEXT -1 81 ") may be integrat
ed term-by-term to provide an approximation for the integral of " }
{TEXT 285 1 "f" }{TEXT -1 1 "(" }{TEXT 286 1 "x" }{TEXT -1 3 "). " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 8 "Example:
" }{TEXT -1 36 " We integrate over the interval [0, " }{TEXT 297 1 "x
" }{TEXT -1 2 "]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(t);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "int(f(t),t=-1..x);\nint(su
m(a[n]*cos(n*Pi*t),n=0..5),t=-1..x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "plot([int(f(t)+0.01,t=-1..x),\n          int(sum(a[n]
*cos(n*Pi*t),n=0..5),t=-1..x)],x=-1..1);" }}}{EXCHG }{EXCHG }{EXCHG }
{EXCHG }{PARA 0 "" 0 "" {TEXT -1 75 "These two graphs are nearly the s
ame! In fact, I have off-set the graph of " }{TEXT 282 1 "f" }{TEXT 
-1 131 " by a little so that the graphs can be distinguished. This gra
ph gives visual understanding to the results of the previous theorem.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 262 "This
 section has considered the issues involved with differentiating and i
ntegrating series term-by-term. We see that continuity of the periodic
 functions is an important condition for differentiability of Fourier \+
Series. Not so much is need for the integration." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "EMAIL: herod@math.gatech.
edu   or    jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "URL: http:/
/www.math.gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
257 "" 0 "" {TEXT -1 36 "Copyright \251  2003  by James V. Herod" }}
{PARA 257 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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