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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 297 23 "Section \+
2.2: Extensions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 
0 "" {TEXT 298 30 "Maple Packages for Section 2.2" }}{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 -1 188 "Before examining the character of convergence for Fourie
r Trigonometric Series, there is one more idea we need to have clear i
n our understanding of functions on intervals of the form [0, " }
{TEXT 258 1 "a" }{TEXT -1 7 "] or [-" }{TEXT 259 1 "a" }{TEXT -1 2 ", \+
" }{TEXT 260 1 "a" }{TEXT -1 232 "]. We need to consider how such func
tions might be extended to the entire real line. After all, as we expl
ain next, with whatever character the Fourier Trigonometric Series con
verges to a function, it will converge on the real line." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Most often, Fourie
r Trigonometric Series will come in one of three types:" }}{PARA 0 "" 
0 "" {TEXT -1 32 "                                " }{XPPEDIT 18 0 "su
m(a[p]*cos(p*Pi*x/L),p = 0 .. infinity);" "6#-%$sumG6$*&&%\"aG6#%\"pG
\"\"\"-%$cosG6#**F*F+%#PiGF+%\"xGF+%\"LG!\"\"F+/F*;\"\"!%)infinityG" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 30 "        or             \+
       " }{XPPEDIT 18 0 "sum(b[p]*sin(p*Pi*x/L),p = 1 .. infinity);" "
6#-%$sumG6$*&&%\"bG6#%\"pG\"\"\"-%$sinG6#**F*F+%#PiGF+%\"xGF+%\"LG!\"
\"F+/F*;F+%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 21 
"        or           " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }
{TEXT -1 4 "  + " }{XPPEDIT 18 0 "sum(a[p]*cos(p*Pi*x/L)+b[p]*sin(p*Pi
*x/L),p = 1 .. infinity);" "6#-%$sumG6$,&*&&%\"aG6#%\"pG\"\"\"-%$cosG6
#**F+F,%#PiGF,%\"xGF,%\"LG!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2
F,F3F4F,F,/F+;F,%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "If the first of these, if the c
osine series, converges to a function " }{TEXT 262 1 "f" }{TEXT -1 7 "
, then " }{TEXT 263 1 "f" }{TEXT -1 147 " will be even and will be per
iodic. To see this, simply recall elementary properties of the cosine \+
function. If the second converges to a function " }{TEXT 264 1 "f" }
{TEXT -1 7 ", then " }{TEXT 265 1 "f" }{TEXT -1 78 " must be an odd, p
eriodic function. Finally, if the third series converges to " }{TEXT 
266 1 "f" }{TEXT -1 7 ", then " }{TEXT 267 1 "f" }{TEXT -1 74 " might \+
be even, or odd, or neither. Again, in the third case the function " }
{TEXT 268 1 "f" }{TEXT -1 18 " will be periodic." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "If we know the function \+
" }{TEXT 269 1 "f" }{TEXT -1 20 " on an interval [0, " }{TEXT 270 1 "a
" }{TEXT -1 7 "] or [-" }{TEXT 271 1 "a" }{TEXT -1 2 ", " }{TEXT 272 
1 "a" }{TEXT -1 281 "], it will be important to know how to make exten
sions so that we know which type series to generate. Also, as we discu
ss the nature of convergence, we need to have reviewed these concepts:
 even or odd extensions and periodic extensions. This review is the su
bject of this section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 49 "Suppose a function is defined on an interval [0, " }
{TEXT 261 1 "L" }{TEXT -1 198 "]. We now illustrate how to extend the \+
definition of the function so that it is defined on a larger interval.
 We discuss even, odd, and periodic extensions. We examine even and od
d extensions first." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 21 "We draw the graph of " }{TEXT 299 1 "f" }{TEXT -1 1 "(
" }{TEXT 326 1 "x" }{TEXT -1 1 ")" }{TEXT 327 3 " = " }{TEXT -1 1 "4" 
}{TEXT 328 3 " x " }{TEXT -1 2 "(1" }{TEXT 343 2 "-x" }{TEXT -1 291 ")
 on the interval [0, 1]. On this graph there also will be the even and
 the odd extension. The black graph is the original function. The unio
n of the black and red graphs is the even extension of the original fu
nction. The black and green graph is the odd extension of the original
 function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "a:=1;\nf:=x->
4*x*(1-x);\nfe:=proc(x) if x<0 then f(-x)\n               else f(x) \n
            end if\n    end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 166 "fe:=proc(x) if x<0 then f(-x)\n              else f(x) \n      \+
      end if\n    end:\nfo:=proc(x) if x<0 then -f(-x)\n              \+
else f(x) \n            end if\n    end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "plot([[x,f(x),x=0..1],[x,'fe(x)',x=-1..1],[x,'fo(x)',
x=-1..1]],\n   color=[BLACK,RED,GREEN]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }{TEXT 256 10 "Definition" }{TEXT 300 4 " of " }{TEXT 
273 13 "Even Function" }{TEXT -1 2 ": " }{TEXT 276 1 "f" }{TEXT -1 27 
" is even on the interval [-" }{TEXT 280 4 "a, a" }{TEXT -1 5 "] if " 
}{TEXT 278 1 "f" }{TEXT -1 1 "(" }{TEXT 330 1 "x" }{TEXT -1 1 ")" }
{TEXT 331 4 " = f" }{TEXT -1 1 "(" }{TEXT 332 2 "-x" }{TEXT -1 28 ") f
or all x in the interval." }}{PARA 0 "" 0 "" {TEXT 257 1 " " }{TEXT 
301 10 "Definition" }{TEXT 302 4 " of " }{TEXT 274 12 "Odd Function" }
{TEXT 275 1 ":" }{TEXT -1 1 " " }{TEXT 277 1 "f" }{TEXT -1 26 " is odd
 on the interval [-" }{TEXT 281 4 "a, a" }{TEXT -1 5 "] if " }{TEXT 
279 1 "f" }{TEXT -1 1 "(" }{TEXT 333 1 "x" }{TEXT -1 1 ")" }{TEXT 334 
5 " = -f" }{TEXT -1 1 "(" }{TEXT 335 2 "-x" }{TEXT -1 28 ") for all x \+
in the interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 230 "These definitions are algebraic. There is also a geometr
ic interpretation. A function is even if its graph is symmetric about \+
the Y axis. A function is odd if its graph is symmetric about the orig
in. Look at the last graphs drawn." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 79 "There are functions that are neither even
 nor odd. You know many; here is one: " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(x^3+x^2,x=-1..1);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 197 "An important idea, howe
ver, is that every function defined on an interval symmetric about the
 origin can be written as the sum of an even and an odd function. Here
's an example for how to do that: " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "f:=x->x^3+x^2;" }}}{PARA 0 "" 0 "" {TEXT -1 47 "We ma
ke the even part of f and the odd part of " }{TEXT 303 1 "f" }{TEXT 
-1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "fe:=x->(f(x)+f(-x
))/2;\nfo:=x->(f(x)-f(-x))/2;" }}}{PARA 0 "" 0 "" {TEXT -1 53 "Adding \+
them together gets back the original function." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "fe(x)+fo(x);" }}}{PARA 0 "" 0 "" {TEXT -1 20 "Lo
ok at the plot of " }{TEXT 304 1 "f" }{TEXT -1 34 ", its even part, an
d its odd part." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plot([[x,
f(x),x=-1..1],[x,'fe(x)',x=-1..1],[x,'fo(x)',x=-1..1]],\n   color=[BLA
CK,RED,GREEN]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 282 8 "Remarks." }{TEXT -1 157 " Of course, if a funct
ion is even or if it is odd on an interval symmetric about the origin,
 this has implications to what form the Fourier Series will take." }}
{PARA 0 "" 0 "" {TEXT -1 46 "(1) Suppose that f is even on the interva
l [- " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 3 ",  " }{XPPEDIT 18 
0 "pi;" "6#%#piG" }{TEXT -1 8 "]. Then " }{XPPEDIT 18 0 "int(f(x)*sin(
n*x),x = -Pi .. Pi);" "6#-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#*&%\"n
GF+F*F+F+/F*;,$%#PiG!\"\"F4" }{TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "(2) Suppose that f is odd
 on the interval [- " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 3 ",  \+
" }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 8 "]. Then " }{XPPEDIT 18 
0 "int(f(x)*cos(n*x),x = -Pi .. Pi);" "6#-%$intG6$*&-%\"fG6#%\"xG\"\"
\"-%$cosG6#*&%\"nGF+F*F+F+/F*;,$%#PiG!\"\"F4" }{TEXT -1 5 " = 0." }}
{PARA 0 "" 0 "" {TEXT -1 56 "(3) From either of the above two remarks \+
if follows that" }}{PARA 0 "" 0 "" {TEXT -1 26 "                      \+
    " }{XPPEDIT 18 0 "int(sin(n*x)*cos(m*x),x = -Pi .. Pi);" "6#-%$int
G6$*&-%$sinG6#*&%\"nG\"\"\"%\"xGF,F,-%$cosG6#*&%\"mGF,F-F,F,/F-;,$%#Pi
G!\"\"F6" }{TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 80 "We next want to discuss periodic extensions. Re
call what a periodic function is." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 283 12 "Definition: " }{TEXT -1 16 "The function
 is " }{TEXT 284 8 "periodic" }{TEXT -1 22 " if there is a number " }
{TEXT 325 1 "P" }{TEXT -1 11 " such that " }{TEXT 340 1 "f" }{TEXT -1 
1 "(" }{TEXT 339 3 "x+P" }{TEXT -1 4 ") = " }{TEXT 337 1 "f" }{TEXT 
-1 1 "(" }{TEXT 336 1 "x" }{TEXT -1 10 ") for all " }{TEXT 338 1 "x" }
{TEXT -1 36 ". The smallest such positive number " }{TEXT 329 1 "P" }
{TEXT -1 15 " is called the " }{TEXT 285 6 "period" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 9 "Examples:
" }{TEXT -1 43 " The sine and cosine functions have period " }
{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "It is cl
ear, is it not, that if a function is specified on some interval, then
 one could ask for a periodic extension. If the function were defined \+
on an interval [0, " }{TEXT 292 1 "L" }{TEXT -1 159 "], one could ask \+
for an even periodic extension or an odd, periodic extension. In this \+
case, one would first make an even (or odd) extension, and then make a
 2" }{TEXT 344 2 " L" }{TEXT -1 133 " periodic extension. We illustrat
e with graphs. These notes begin with a particular function. You can m
odify the code to choose your " }{TEXT 289 1 "f" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 296 47 "Genera
ting even and odd extensions of functions" }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 18 "Here is an even 2 " }{TEXT 287 1 "a" }{TEXT -1 40 
" extension of a function defined on [0, " }{TEXT 288 1 "a" }{TEXT -1 
2 "]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a:=2;\n  f:=x->x;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "ff:= proc(x)\n\011\011if
 0 <= x and x < a then f(x)\n\011\011elif a <= x and x <= 2*a then f(2
*a-x)\n\011\011end if \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "plot(ff,0..2*a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f
e:= x->ff(frac(abs(x)/(2*a))*2*a);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "plot(fe,-3*a..3*a);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Here is an odd 2" }{TEXT 
290 2 " a" }{TEXT -1 40 " extension of a function defined on [0, " }
{TEXT 293 1 "a" }{TEXT -1 2 "]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "a:=1;\n  f:=x->x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
101 "ff:= proc(x)\n\011\011if 0 <= x and x < a then f(x)\n\011\011elif
 a <= x and x <= 2*a then -f(2*a-x)\n\011\011end if \nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(ff,0..2*a);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "fo:= x->sign(x)*ff(frac(abs(x)/(2*a))*2*a
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(fo,-3*a..3*a);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 295 44 "Generating periodic exte
nsions of functions." }}{PARA 0 "" 0 "" {TEXT -1 43 "Suppose that f is
 defined on the interval [" }{TEXT 294 3 "a,b" }{TEXT -1 6 "]. In " }
{TEXT 324 9 "MapleTech" }{TEXT 341 1 "," }{TEXT -1 1 " " }{TEXT 342 3 
"Vol" }{TEXT -1 112 ". 3, NO.3, Monagan and Lopez provided the followi
ng method to output a function which extends the definition of " }
{TEXT 313 1 "f" }{TEXT -1 23 " to the real line using" }}{PARA 0 "" 0 
"" {TEXT -1 7 "       " }{TEXT 314 1 "f" }{TEXT -1 1 "(" }{TEXT 315 1 
"a" }{TEXT -1 3 " + " }{TEXT 316 1 "k" }{TEXT -1 2 " (" }{TEXT 317 1 "
b" }{TEXT -1 1 "-" }{TEXT 318 1 "a" }{TEXT -1 4 ") + " }{XPPEDIT 18 0 
"delta" "6#%&deltaG" }{TEXT -1 4 ") = " }{TEXT 319 1 "f" }{TEXT -1 1 "
(" }{TEXT 320 1 "a" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "delta" "6#%&delt
aG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 12 "for integer " }
{TEXT 321 1 "k" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 174 "PeriodicExtender:=proc(f,d::range)\n  subs( \{'F' = f, 'L'=lhs(
d),\n         'D'=rhs(d)-lhs(d)\},\n  proc(x::algebraic) local y;\n   \+
   y:=floor((x-L)/D);\n      F(x-y*D);\nend)\nend:" }}}{PARA 0 "" 0 "
" {TEXT -1 79 "We try this extender for the signum function defined ov
er the interval [-1, 2]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "
sw:=PeriodicExtender(signum,-1..2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "plot('sw(x)','x'=-4..4);" }}}{PARA 0 "" 0 "" {TEXT 
-1 84 "We use the extender for the absolute value function defined on \+
the interval [-1, 1]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g:=
x->abs(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "st:=PeriodicE
xtender(g,-1..1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot('
st(x)','x'=-3..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "Finally, we consider the impli
cations for a Fourier Trigonometric series depending on characteristic
s for a function " }{TEXT 291 1 "f" }{TEXT -1 53 ". Here are four poss
ible characters for the graph of " }{TEXT 305 1 "f" }{TEXT -1 1 ":" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1a. " }
{TEXT 307 1 "f" }{TEXT -1 24 " is even and has period " }{XPPEDIT 18 
0 "2*pi;" "6#*&\"\"#\"\"\"%#piGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 4 "1b. " }{TEXT 308 1 "f" }{TEXT -1 23 " is odd and has perio
d " }{XPPEDIT 18 0 "2*pi;" "6#*&\"\"#\"\"\"%#piGF%" }{TEXT -1 2 ". " }
}{PARA 0 "" 0 "" {TEXT -1 4 "1c. " }{TEXT 309 1 "f" }{TEXT -1 12 " has
 period " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 4 "1d. " }{TEXT 310 1 "f" }{TEXT -1 12 " has period " }
{XPPEDIT 18 0 "2*pi;" "6#*&\"\"#\"\"\"%#piGF%" }{TEXT -1 54 " and alte
rnates on each half period, in the sense that" }}{PARA 0 "" 0 "" 
{TEXT -1 5 "     " }{TEXT 311 1 "f" }{TEXT -1 5 "(x + " }{XPPEDIT 18 
0 "pi;" "6#%#piG" }{TEXT -1 6 ") = - " }{TEXT 312 1 "f" }{TEXT -1 4 "(
x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Ma
tch those possibilities with the following Fourier Series for" }{TEXT 
306 3 " f:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 40 "2a. The series contains only sine terms." }}{PARA 0 "" 0 "" 
{TEXT -1 42 "2b. The series contains only cosine terms." }}{PARA 0 "" 
0 "" {TEXT -1 80 "2c. The series contains sin(nx) and cos(nx) terms, b
ut only for odd values of n." }}{PARA 0 "" 0 "" {TEXT -1 81 "2d. The s
eries contains sin(nx) and cos(nx) terms, but only for even values of \+
n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "In
 order to help see the connections in those types of series, we make e
xamples for each of the above 2 " }{TEXT 322 1 "a" }{TEXT -1 11 " thro
ugh 2 " }{TEXT 323 1 "c" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 161 "fa:=x->sum(sin(n*x)/n,n=1..4);\nfb:=x->sum(cos(n*x)/
n,n=1..4);\nfc:=x->sum(sin((2*n+1)*x)/n+cos((2*n+1)*x)/n,n=1..4);\nfd:
=x->sum(sin(2*n*x)/n+cos(2*n*x)/n,n=1..4);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "plot(fa(x),x=-Pi..3*Pi);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 24 "plot(fb(x),x=-Pi..3*Pi);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "plot(fc(x),x=-Pi..3*Pi);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "plot(fd(x),x=-Pi..3*Pi);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "EMAI
L: 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 reserv
ed" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 0" 0 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }