{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 11 1 64 0 0 0 0 0 0 0 0 0 0 1 
}{CSTYLE "Define" -1 256 "Times" 1 12 163 163 163 0 1 1 0 0 0 0 0 0 0 
1 }{CSTYLE "Emphasis" -1 257 "Times" 1 12 128 0 128 1 0 0 0 0 0 0 0 0 
0 1 }{CSTYLE "normal C" -1 258 "Times" 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 
1 }{CSTYLE "Menu" -1 259 "" 0 0 163 163 163 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 260 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
261 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 265 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
266 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 270 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
271 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 275 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
276 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 280 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
281 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 285 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
286 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 290 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
291 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 295 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
296 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 300 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
301 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 305 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
306 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 310 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
311 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 315 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
316 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 317 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 319 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 320 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
321 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 322 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 324 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 325 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
326 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 327 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 329 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 330 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
331 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 332 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 333 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 334 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 335 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
336 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 337 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 338 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 339 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 340 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
341 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 342 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 343 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 344 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 345 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
346 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 347 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 348 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 349 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 350 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
351 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 352 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 353 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 354 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 355 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
356 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 357 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 358 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 359 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 360 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
361 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 362 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 363 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 364 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 365 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
366 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 367 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 368 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 369 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 370 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
371 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 372 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 373 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 374 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 375 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
376 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 377 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 378 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 379 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 380 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
381 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 382 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 383 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 384 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 385 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
386 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 387 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "G
eneva" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Monaco" 1 9 255 0 0 1 
2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" 
-1 257 1 {CSTYLE "" -1 -1 "Geneva" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }
1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 258 1 {CSTYLE "" 
-1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 
2 2 0 1 }}
{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 348 144 "\n(c) John H. Mathews          Rus
sell W. Howell\nmathews@fullerton.edu     howell@westmont.edu\n\nCompl
imentary software to accompany the textbook:" }}{PARA 257 "" 0 "" 
{TEXT 257 82 "COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed,
 2001, ISBN: 0-7637-1425-9" }{TEXT 347 197 "\nJones and Bartlett Publi
shers, Inc.,      40  Tall  Pine  Drive,      Sudbury,  MA  01776\nTel
e.  (800) 832-0034;      FAX:  (508)  443-8000,      E-mail:  mkt@jbpu
b.com,      http://www.jbpub.com/" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 
344 1 "\n" }{TEXT 256 36 "CHAPTER 7  TAYLOR and LAURENT SERIES" }
{TEXT 338 2 "\n\n" }{TEXT 256 44 "Section 7.3a  Laurent Series Represe
ntations" }{TEXT 339 1 "\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
8 "Suppose " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 21 "  i
s not analytic in " }{XPPEDIT 18 0 "D[R](a)" "6#-&%\"DG6#%\"RG6#%\"aG
" }{TEXT -1 5 " but " }{TEXT 349 2 "is" }{TEXT -1 32 " analytic in the
 punctured disk " }{XPPEDIT 18 0 "D[R]^`*`*`(a)`" "6#*&)&%\"DG6#%\"RG%
\"*G\"\"\"%$(a)GF*" }{TEXT -1 3 " =\{" }{XPPEDIT 18 0 "`z: 0`;" "6#%%z
:~0G" }{XPPEDIT 18 0 "`` < ``;" "6#2%!GF$" }{XPPEDIT 18 0 "abs(z-alpha
) < R;" "6#2-%$absG6#,&%\"zG\"\"\"%&alphaG!\"\"%\"RG" }{TEXT -1 30 "\}
.  For example, the function " }{XPPEDIT 18 0 "f(z) = exp(z)/(z^3)" "6
#/-%\"fG6#%\"zG*&-%$expG6#F'\"\"\"*$F'\"\"$!\"\"" }{TEXT -1 23 "  is n
ot analytic when " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT -1 
22 "  but is analytic for " }{XPPEDIT 18 0 "0 < abs(z);" "6#2\"\"!-%$a
bsG6#%\"zG" }{TEXT -1 110 ".  Clearly, this function does not have a M
aclaurin series representation. If we use the Maclaurin series for " }
{XPPEDIT 18 0 "g(z) = exp(z)" "6#/-%\"gG6#%\"zG-%$expG6#F'" }{TEXT -1 
60 "  however, and formally divide each term in that series by  " }
{XPPEDIT 18 0 "z^3;" "6#*$%\"zG\"\"$" }{TEXT -1 32 ", we obtain the re
presentation  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 5 "     " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 
6 "  =   " }{XPPEDIT 18 0 "1/(z^3);" "6#*&\"\"\"F$*$%\"zG\"\"$!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(z)" "6#-%$expG6#%\"zG" }{TEXT -1 4 
"  = " }{XPPEDIT 18 0 "1/(z^3)" "6#*&\"\"\"F$*$%\"zG\"\"$!\"\"" }
{TEXT -1 3 " + " }{XPPEDIT 18 0 "1/(z^2)" "6#*&\"\"\"F$*$%\"zG\"\"#!\"
\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "1/2!/z" "6#*(\"\"\"F$-%*factoria
lG6#\"\"#!\"\"%\"zGF)" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "1/3!" "6#*&\"
\"\"F$-%*factorialG6#\"\"$!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "z/4
!" "6#*&%\"zG\"\"\"-%*factorialG6#\"\"%!\"\"" }{TEXT -1 3 " + " }
{XPPEDIT 18 0 "z^2/5!" "6#*&%\"zG\"\"#-%*factorialG6#\"\"&!\"\"" }
{TEXT -1 4 "  + " }{XPPEDIT 18 0 "z^3/6!" "6#*&%\"zG\"\"$-%*factorialG
6#\"\"'!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "`...`" "6#%$...G" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 23 "which is valid for all " }{XPPEDIT 18 0 "z" "6#%\"zG" }
{TEXT -1 11 " such that " }{XPPEDIT 18 0 "0 < abs(z)" "6#2\"\"!-%$absG
6#%\"zG" }{TEXT -1 145 ".  This example raises the question as to whet
her it might be possible to generalize the Taylor series method to fun
ctions analytic in an annulus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 6 "      " }{XPPEDIT 18 0 "A(alpha,r,R);" "6#
-%\"AG6%%&alphaG%\"rG%\"RG" }{TEXT -1 4 " = \{" }{XPPEDIT 18 0 "`z:  r
`;" "6#%&z:~~rG" }{XPPEDIT 18 0 "`` < ``;" "6#2%!GF$" }{XPPEDIT 18 0 "
abs(z-alpha) < R;" "6#2-%$absG6#,&%\"zG\"\"\"%&alphaG!\"\"%\"RG" }
{TEXT -1 9 "\}.       " }}{PARA 0 "" 0 "" {TEXT -1 90 "\011\nPerhaps w
e can represent these functions with a series that employs negative po
wers of  " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 29 "  in some way as
 we did with " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 119 "
.  As you will see shortly, this is indeed the case. We begin by defin
ing a series that allows for negative powers of  " }{XPPEDIT 18 0 "z" 
"6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 256 30 "Theorem 7.7   (Laurent series)" }
{TEXT 350 3 "   " }}{PARA 0 "" 0 "" {TEXT 385 28 "Suppose the Laurent \+
series  " }{XPPEDIT 18 0 "sum(c[n]*(z - alpha)^n,n=-infinity..infinity
)" "6#-%$sumG6$*&&%\"cG6#%\"nG\"\"\"),&%\"zGF+%&alphaG!\"\"F*F+/F*;,$%
)infinityGF0F4" }{TEXT 351 27 "  converges on an annulus  " }{XPPEDIT 
18 0 "`A(r,R,`*alpha*`)` = \{`z : r <`*abs(z  - alpha) < R\}" "6#/*(%'
A(r,R,G\"\"\"%&alphaGF&%\")GF&<#2*&%(z~:~r~<GF&-%$absG6#,&%\"zGF&F'!\"
\"F&%\"RG" }{TEXT 352 4 " .  " }}{PARA 0 "" 0 "" {TEXT 386 56 "Then th
e series converges uniformly on any subannulus   " }{XPPEDIT 18 0 "`A(
s,t,`*alpha*`)` = \{`z : s <=`*abs(z  - alpha)*` <= t`\}" "6#/*(%'A(s,
t,G\"\"\"%&alphaGF&%\")GF&<#*(%)z~:~s~<=GF&-%$absG6#,&%\"zGF&F'!\"\"F&
%&~<=~tGF&" }{TEXT 353 11 "   where   " }{XPPEDIT 18 0 "`r < s < t < R
`" "6#%.r~<~s~<~t~<~RG" }{TEXT 354 4 " .  " }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 
20 "Example on page 280." }{TEXT 340 32 "   Find the Laurent series fo
r  " }{XPPEDIT 18 0 "f(z) = exp(z)/z^3" "6#/-%\"fG6#%\"zG*&-%$expG6#F'
\"\"\"*$F'\"\"$!\"\"" }{TEXT 260 3 " .\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "f:='f': z:='z':\nf := z -> exp(z)/z^3:\n`f(z) ` = f(z
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "
" {TEXT 261 43 "The following Laurent series is valid for  " }
{XPPEDIT 18 0 "0 < abs(z)" "6#2\"\"!-%$absG6#%\"zG" }{TEXT 264 3 " .\n
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "L:='L': s:='s': S:='S': z:='z'
: Z:='Z':\ns := taylor(exp(z), z=0, 9)/z^3:\n`f(z) ` = exp(z)/z^3;\n`f
(z) ` = s;\ns := series(exp(z)/z^3, z=0, 9):\n`f(z) ` = s;\nS := serie
s(exp(Z)/Z^3, Z=0, 9):\nS := convert(S, polynom):\nLS := z -> subs(Z=z
,S):\nL[5](z) = LS(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT 262 54 "Sum up the terms to verify that
 we have things right.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "L5 := \+
z -> sum(z^(n-3)/n!, n=0..8):\nL0 := z -> sum(z^(n-3)/n!, n=0..infinit
y):\n`f(z) ` = exp(z)/z^3;\nL[5](z),` = `,Sum(z^(n-3)/n!, n=0..8) = L5
(z);\nL[infinity](z),` = `,Sum(z^(n-3)/n!, n=0..infinity) = L0(z);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 263 32 "We can plot the real functions  " }{XPPEDIT 18 0 "y = f(
x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT 265 7 "  and  " }{XPPEDIT 18 0 "y =
 L(x)" "6#/%\"yG-%\"LG6#%\"xG" }{TEXT 266 25 "  \nto get the idea that
  " }{XPPEDIT 18 0 "L[5](x)" "6#-&%\"LG6#\"\"&6#%\"xG" }{TEXT 267 26 "
  is an approximation to  " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT 268 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "plot(\{f(x),LS(
x)\}, \n  x=0.01..8, y=0..25,\n  title=`y=exp(x)/x^3 and y=L5(x)`,\n  \+
labels=[`   x`,`y `],\n  tickmarks=[8,4],\n  view=[0..8,0..25]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 32 "Theorem 7.8  (Laurent's theore
m)" }{TEXT 355 12 "   Suppose  " }{XPPEDIT 18 0 "0<=`r < R`" "6#1\"\"!
%&r~<~RG" }{TEXT 356 14 " ,  and that  " }{XPPEDIT 18 0 "f(z)" "6#-%\"
fG6#%\"zG" }{TEXT 357 30 "  is analytic in the annulus  " }{XPPEDIT 
18 0 "`A = A(r,R,`*alpha*`)` = \{`z : r <`*abs(z  - alpha) < R\}" "6#/
*(%+A~=~A(r,R,G\"\"\"%&alphaGF&%\")GF&<#2*&%(z~:~r~<GF&-%$absG6#,&%\"z
GF&F'!\"\"F&%\"RG" }{TEXT 358 4 " .  " }}{PARA 0 "" 0 "" {TEXT 387 4 "
If  " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT 359 27 "  is any number s
uch that  " }{XPPEDIT 18 0 "`r < `*rho < R" "6#2*&%%r~<~G\"\"\"%$rhoGF
&%\"RG" }{TEXT 360 17 " , then for all  " }{XPPEDIT 18 0 "z[0]*epsilon
*`A(r,R,`*alpha*`)` " "6#*,&%\"zG6#\"\"!\"\"\"%(epsilonGF(%'A(r,R,GF(%
&alphaGF(%\")GF(" }{TEXT 361 4 " ,  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6
#%\"zG" }{TEXT 362 41 "  has the Laurent series representation \n" }}
{PARA 0 "" 0 "" {TEXT -1 7 "       " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#
%\"zG" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "sum(c[n]*(z-alpha)^n,n = -i
nfinity .. infinity)" "6#-%$sumG6$*&&%\"cG6#%\"nG\"\"\"),&%\"zGF+%&alp
haG!\"\"F*F+/F*;,$%)infinityGF0F4" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 
"sum(c[-n]*(z-alpha)^(-n),n = 1 .. infinity)+sum(c[n]*(z-alpha)^n,n = \+
0 .. infinity)" "6#,&-%$sumG6$*&&%\"cG6#,$%\"nG!\"\"\"\"\"),&%\"zGF.%&
alphaGF-,$F,F-F./F,;F.%)infinityGF.-F%6$*&&F)6#F,F.),&F1F.F2F-F,F./F,;
\"\"!F6F." }{TEXT -1 6 " ,    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 363 0 "" }}{PARA 0 "" 0 "" {TEXT 368 11 "where for  " }{XPPEDIT 
18 0 "n=0,1,2,`...`" "6&/%\"nG\"\"!\"\"\"\"\"#%$...G" }{TEXT 364 20 " \+
 the coefficients  " }{XPPEDIT 18 0 "c[-n]" "6#&%\"cG6#,$%\"nG!\"\"" }
{TEXT 365 7 "  and  " }{XPPEDIT 18 0 "c[n]" "6#&%\"cG6#%\"nG" }{TEXT 
366 14 "  are given by" }}{PARA 0 "" 0 "" {TEXT -1 7 "       " }
{XPPEDIT 18 0 "c[-n]" "6#&%\"cG6#,$%\"nG!\"\"" }{TEXT -1 5 "  =  " }
{XPPEDIT 18 0 "1/(2*pi*i);" "6#*&\"\"\"F$*(\"\"#F$%#piGF$%\"iGF$!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(f(z)/((z-alpha)^(-n+1)),z = C^`+
`*` `[rho](alpha) .. `.`)" "6#-%$intG6$*&-%\"fG6#%\"zG\"\"\"),&F*F+%&a
lphaG!\"\",&%\"nGF/F+F+F//F*;*&)%\"CG%\"+GF+-&%\"~G6#%$rhoG6#F.F+%\".G
" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 5 "and  " }}{PARA 0 "" 
0 "" {TEXT -1 7 "       " }{XPPEDIT 18 0 "c[n]" "6#&%\"cG6#%\"nG" }
{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "1/(2*pi*i)" "6#*&\"\"\"F$*(\"\"#F$%
#piGF$%\"iGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(f(z)/((z-alpha
)^(n+1)),z = C^`+`*` `[rho](alpha) .. `.`)" "6#-%$intG6$*&-%\"fG6#%\"z
G\"\"\"),&F*F+%&alphaG!\"\",&%\"nGF+F+F+F//F*;*&)%\"CG%\"+GF+-&%\"~G6#
%$rhoG6#F.F+%\".G" }{TEXT -1 10 " .        " }{TEXT 367 0 "" }{TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 370 0 "" }}{PARA 0 "" 0 "" 
{TEXT 256 62 "Theorem 7.9  (Uniqueness and differentiation of Larent s
eries)" }{TEXT 369 3 "  \n" }}{PARA 0 "" 0 "" {TEXT 383 14 "Suppose th
at  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 373 30 "  is anal
ytic in the annulus  " }{XPPEDIT 18 0 "A(r,R,alpha)" "6#-%\"AG6%%\"rG%
\"RG%&alphaG" }{TEXT 374 45 " ,  and has the Laurent series representa
tion" }}{PARA 0 "" 0 "" {TEXT 384 7 "       " }{XPPEDIT 18 0 "f(z) = s
um(c[n]*(z - alpha)^n,n=-infinity..infinity) " "6#/-%\"fG6#%\"zG-%$sum
G6$*&&%\"cG6#%\"nG\"\"\"),&F'F0%&alphaG!\"\"F/F0/F/;,$%)infinityGF4F8
" }{TEXT 375 11 "  for all  " }{XPPEDIT 18 0 "`z`*epsilon*A(r,R,alpha)
" "6#*(%\"zG\"\"\"%(epsilonGF%-%\"AG6%%\"rG%\"RG%&alphaGF%" }{TEXT 
376 4 " . \n" }{TEXT 256 3 "(i)" }{TEXT 371 7 "   If  " }{XPPEDIT 18 
0 "f(z) = sum(b[n]*(z - alpha)^n,n=-infinity..infinity) " "6#/-%\"fG6#
%\"zG-%$sumG6$*&&%\"bG6#%\"nG\"\"\"),&F'F0%&alphaG!\"\"F/F0/F/;,$%)inf
inityGF4F8" }{TEXT 377 11 "  for all  " }{XPPEDIT 18 0 "`z`*epsilon*A(
r,R,alpha)" "6#*(%\"zG\"\"\"%(epsilonGF%-%\"AG6%%\"rG%\"RG%&alphaGF%" 
}{TEXT 378 10 " ,  then  " }{XPPEDIT 18 0 "b[n] = c[n]" "6#/&%\"bG6#%
\"nG&%\"cG6#F'" }{TEXT 379 65 "  for all  n. \n         (In other word
s, the Laurent series for  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }
{TEXT 380 33 "  in a given annulus is unique.)\n" }{TEXT 256 4 "(ii)" 
}{TEXT 372 11 "  For all  " }{XPPEDIT 18 0 "`z`*epsilon*A(r,R,alpha)" 
"6#*(%\"zG\"\"\"%(epsilonGF%-%\"AG6%%\"rG%\"RG%&alphaGF%" }{TEXT 381 
25 " ,  the derivatives for  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG
" }{TEXT 382 79 "  may be obtained by termwise \n         differentiat
ion of its Laurent series. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 257 "" 0 "" {TEXT 337 1 "\n" }{TEXT 256 22 "Example 7.7
, Page 286." }{TEXT 341 59 "   Find three different Laurent series rep
resentations for\n" }{XPPEDIT 18 0 "f(z) = 3/(2 + z - z^2)" "6#/-%\"fG
6#%\"zG*&\"\"$\"\"\",(\"\"#F*F'F**$F'F,!\"\"F." }{TEXT 269 23 "  invol
ving powers of  " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 294 3 " .\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f:='f': z:='z':\nf := z -> 3/(1+z)/
(2-z):\n`f(z)` = f(z);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 270 71 "Mapl
e  easily finds the Maclaurin series involving positive powers of  " }
{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 295 3 " :\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 165 "L:='L': Z:='Z':\ns0 := z -> subs(Z=z, series(f(Z), Z
=0, 6)):\n`f(z) ` = f(z);\n`f(z) ` = s0(z);\nS0 := convert(s0(Z), poly
nom):\nL0 := z -> subs(Z=z,S0):\nL[0](z) = L0(z);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 271 32 "The abo
ve series converges for  " }{XPPEDIT 18 0 "abs(z) < 1" "6#2-%$absG6#%
\"zG\"\"\"" }{TEXT 296 89 " , this requires looking at the sum of the \+
two\"geometric series\"\nthat form the parts of  " }{XPPEDIT 18 0 "L[0
](z)" "6#-&%\"LG6#\"\"!6#%\"zG" }{TEXT 272 77 " .  Maple can also find
 \"the other series\" that involves negative powers of  " }{XPPEDIT 
18 0 "z" "6#%\"zG" }{TEXT 297 57 " ,\nthis involves the mental thinkin
g that you substitute " }{XPPEDIT 18 0 "z = 1/Z" "6#/%\"zG*&\"\"\"F&%
\"ZG!\"\"" }{TEXT 298 49 " in the original series to get a new functio
n of " }{XPPEDIT 18 0 "Z" "6#%\"ZG" }{TEXT 299 25 " ,\nthen expand it \+
about  " }{XPPEDIT 18 0 "Z = 0" "6#/%\"ZG\"\"!" }{TEXT 300 23 "  and t
hen substitute  " }{XPPEDIT 18 0 "Z=1/z" "6#/%\"ZG*&\"\"\"F&%\"zG!\"\"
" }{TEXT 301 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "Z:='Z':\nS
e := series(f(1/Z), Z=0, 7):\ns1 := z -> subs(Z=1/z,Se):\n`f(z) ` = f(
z);\n`f(z) ` = s1(z);\nS1 := convert(series(f(1/Z), Z=0, 7), polynom):
\nS1 := subs(Z=1/Z,S1):\nL1 := z -> subs(Z=z,S1):\nL[1](z) = L1(z);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 273 32 "Or we can use Maple and expand  " }{XPPEDIT 18 0 "f(z)" 
"6#-%\"fG6#%\"zG" }{TEXT 302 9 "  about  " }{XPPEDIT 18 0 "z = infinit
y" "6#/%\"zG%)infinityG" }{TEXT 303 3 " .\n" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`f(z) ` = f(z);\n`f(z) ` = series(f
(z), z=infinity, 7);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT 274 32 "The above series converges for \+
 " }{XPPEDIT 18 0 "2 < abs(z)" "6#2\"\"#-%$absG6#%\"zG" }{TEXT 304 91 
" ,  this requires looking at the sum of the two \"geometric series\" \+
\nthat form the parts of " }{XPPEDIT 18 0 "S[1](z)" "6#-&%\"SG6#\"\"\"
6#%\"zG" }{TEXT 275 61 " .  This leaves an unresolved question:  \"Wha
t happens when  " }{XPPEDIT 18 0 "`1 < |z| < 2`" "6#%,1~<~|grz|gr~<~2G
" }{TEXT 305 134 " .  \nThis requires a little work.  First split up t
he functions up into their partial fraction form, and then make \nexpa
nsions about  " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 306 6 " \+
and  " }{XPPEDIT 18 0 "z = infinity" "6#/%\"zG%)infinityG" }{TEXT 307 
17 "  for each part.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "F:='F': \+
f1:='f1': f2:='f2':\nA := convert(f(z), parfrac, z): `f(z) ` = A;\nf1 \+
:= z -> 1/(1+z):  F[1](z) = f1(z);\nf2 := z -> -1/(z-2): F[2](z) = f2(
z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 
"" {TEXT 276 47 "Now form Laurent expansions for the two parts  " }
{XPPEDIT 18 0 "F[1](z)" "6#-&%\"FG6#\"\"\"6#%\"zG" }{TEXT 277 7 "  and
  " }{XPPEDIT 18 0 "F[2](z)" "6#-&%\"FG6#\"\"#6#%\"zG" }{TEXT 278 6 " \+
 of  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 308 3 " .\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 451 "S:='S': Z:='Z':\nS11 := series(f1(
z), z=0, 20):\nS11 := convert(S11, polynom):\nS[11](z) = S11;\nS12 := \+
series(f1(z), z=infinity, 18):\nS12 := series(f1(1/Z), Z=0, 18):\nS12 \+
:= convert(S12, polynom):\nS12 := subs(Z=1/z,S12):\nS[12](z) = S12;\nS
21 := series(f2(z), z=0, 12):\nS21 := convert(S21, polynom):\nS[21](z)
 = S21;\nS22 := series(f2(z), z=infinity, 12):\nS22 := series(f2(1/Z),
 Z=0, 12):\nS22 := convert(S22, polynom):\nS22 := subs(Z=1/z,S22):\nS[
22](z) = S22;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 279 20 "The Laurent series  " }{XPPEDIT 18 0 "
L[1](z) = S[11](z) + S[21](z)" "6#/-&%\"LG6#\"\"\"6#%\"zG,&-&%\"SG6#\"
#66#F*F(-&F.6#\"#@6#F*F(" }{TEXT 280 16 "  is valid for  " }{XPPEDIT 
18 0 "abs(z) < 1" "6#2-%$absG6#%\"zG\"\"\"" }{TEXT 309 3 " .\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Z:='Z':\ns1 := subs(z=Z,S11+S21):\n
L1 := z -> subs(Z=z,s1):\n`f(z) ` = f(z);\nL[1](z) = L1(z);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "plot(\{f(x),L1(x)\},x=0.0..
1,\n  title=`s = f(x), s = L1(x)`,\n  tickmarks=[5,3],\n  view=[0..1,1
.3..1.5]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
257 "" 0 "" {TEXT 281 30 "The following Laurent series  " }{XPPEDIT 
18 0 "L[2](z) = S[12](z) + S[21](z)" "6#/-&%\"LG6#\"\"#6#%\"zG,&-&%\"S
G6#\"#76#F*\"\"\"-&F.6#\"#@6#F*F2" }{TEXT 282 16 "  is valid for  " }
{XPPEDIT 18 0 "`1 < |z| < 2`" "6#%,1~<~|grz|gr~<~2G" }{TEXT 310 3 " .
\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Z:='Z':\ns2 := subs(z=Z,S12+S
21):\nL2 := z -> subs(Z=z,s2):\n`f(z) ` = f(z);\nL[2](z) = L2(z);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "plot(\{f(x),L2(x)\},x=1.01.
.1.99,\n  title=`s = f(x), s = L2(x)`,\n  tickmarks=[5,3],\n  view=[1.
.2,1.5..4]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
257 "" 0 "" {TEXT 283 4 "So  " }{XPPEDIT 18 0 "L[2](z)" "6#-&%\"LG6#\"
\"#6#%\"zG" }{TEXT 284 26 "  is an approximation to  " }{XPPEDIT 18 0 
"f(z)" "6#-%\"fG6#%\"zG" }{TEXT 311 13 "  valid for  " }{XPPEDIT 18 0 
"`1 < |z| < 2`" "6#%,1~<~|grz|gr~<~2G" }{TEXT 312 18 " .   Notice that
  " }{XPPEDIT 18 0 "L[1](z)" "6#-&%\"LG6#\"\"\"6#%\"zG" }{TEXT 285 6 "
  is \n" }{TEXT 257 3 "NOT" }{TEXT 345 27 " a good approximation for  \+
" }{XPPEDIT 18 0 "`1 < |z| < 2`" "6#%,1~<~|grz|gr~<~2G" }{TEXT 313 39 
" ,  for example it is not accurate at  " }{XPPEDIT 18 0 "z = 1.5" "6#
/%\"zG-%&FloatG6$\"#:!\"\"" }{TEXT 314 4 " . \n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 95 "`f(1.5)          L1(1.5)`;\nf(1.5) <> L1(1.5); ` `;\n
`f(1.5)  ~  L2(1.5)`;\nf(1.5), `~` , L2(1.5);" }}}{EXCHG {PARA 257 "" 
0 "" {TEXT 286 20 "The Laurent series  " }{XPPEDIT 18 0 "L[3](z) = S[1
2](z) + S[22](z)" "6#/-&%\"LG6#\"\"$6#%\"zG,&-&%\"SG6#\"#76#F*\"\"\"-&
F.6#\"#A6#F*F2" }{TEXT 287 16 "  is valid for  " }{XPPEDIT 18 0 "2 < a
bs(z)" "6#2\"\"#-%$absG6#%\"zG" }{TEXT 315 3 " .\n" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 90 "Z:='Z':\ns3 := subs(z=Z,S12+S22):\nL3 := z -> subs(
Z=z,s3):\n`f(z) ` = f(z);\nL[3](z) = L3(z);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 108 "plot(\{f(x),L3(x)\},x=2.01..4.0,\n  title=`s = f(
x),       s = L3(x)`,\n  tickmarks=[2,6],\n  view=[2..4,-5..0]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 288 4 "So  " }{XPPEDIT 18 0 "L[3](z)" "6#-&%\"LG6#\"\"$6#%\"zG" 
}{TEXT 289 26 "  is an approximation to  " }{XPPEDIT 18 0 "f(z)" "6#-%
\"fG6#%\"zG" }{TEXT 316 13 "  valid for  " }{XPPEDIT 18 0 "2< abs(z)" 
"6#2\"\"#-%$absG6#%\"zG" }{TEXT 317 17 " .  Notice that  " }{XPPEDIT 
18 0 "L[1](z)" "6#-&%\"LG6#\"\"\"6#%\"zG" }{TEXT 290 7 "  and  " }
{XPPEDIT 18 0 "L[2](z)" "6#-&%\"LG6#\"\"#6#%\"zG" }{TEXT 291 7 "  \nar
e " }{TEXT 257 3 "NOT" }{TEXT 346 26 " good approximations for  " }
{XPPEDIT 18 0 "2< abs(z)" "6#2\"\"#-%$absG6#%\"zG" }{TEXT 318 42 " ,  \+
for example they are not accurate at  " }{XPPEDIT 18 0 "z = 4.0" "6#/%
\"zG-%&FloatG6$\"#S!\"\"" }{TEXT 319 3 " .\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 146 "`f(4.0)          L1(4.0)`;\nf(4.0) <> L1(4.0);`  `;
\n`f(4.0)          L2(4.0)`;\nf(4.0) <> L2(4.0);`  `;\n`f(4.0)  ~  L3(
4.0)`;\nf(4.0), `~` , L3(4.0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 292 175 "The fourth possibility do
es not converge for any values of z and should not be considered, \nho
wever you could write down a few terms and think about why this series
 diverges.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Z:='Z':\ns4 := subs
(z=Z,S11+S22):\nL4 := z -> subs(Z=z,s4):\nL[4](z) = L4(z);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 293 
27 "In the above series, when  " }{XPPEDIT 18 0 "abs(z) < 1/2" "6#2-%$
absG6#%\"zG*&\"\"\"F)\"\"#!\"\"" }{TEXT 320 139 "  the portion of the \+
series with positive exponents will converge \nbut the portion with ne
gative exponents will diverge.  Similarly, when  " }{XPPEDIT 18 0 "abs
(z) > 1" "6#2\"\"\"-%$absG6#%\"zG" }{TEXT 321 173 "  the portion of th
e series \nwith negative exponents will converge, but the portion of t
he series with positive exponents will diverge. \nHence, the series di
verges for all  " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 322 3 " .\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`f(0.5)          L4(0.5)`;\nf(0.5)
 <> L4(0.5);`  `;\n`f(1.5)          L4(1.5)`;\nf(1.5) <> L4(1.5);`  `;
\n`f(4.0)          L4(4.0)`;\nf(4.0) <> L4(4.0);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 336 1 "\n" }
{TEXT 256 22 "Example 7.8, Page 286." }{TEXT 342 32 "   Find the Laure
nt series for  " }{XPPEDIT 18 0 "f(z) = (cos(z) - 1)/z^4" "6#/-%\"fG6#
%\"zG*&,&-%$cosG6#F'\"\"\"F-!\"\"F-*$F'\"\"%F." }{TEXT 335 3 " .\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f:='f': z:='z':\nf := z ->(cos(z) -
 1)/z^4:\n`f(z) ` = f(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT 323 43 "The following Laurent series is
 valid for  " }{XPPEDIT 18 0 "0 < abs(z)" "6#2\"\"!-%$absG6#%\"zG" }
{TEXT 334 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 254 "L:='L': s:='s': S:='S': Z:='Z':\nS := series(cos(Z
)-1, Z=0, 12)/Z^4:\ns := convert(series(cos(Z)-1, Z=0, 12), polynom)/Z
^4:\nLS := z -> subs(Z=z,expand(s)):\n`f(z) ` = f(z);\n`f(z) ` = subs(
Z=z,S);\n`f(z) ` = series((cos(z) - 1)/z^4, z=0, 12);\nL[6](z) = LS(z)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "
" {TEXT 324 54 "Sum up the terms to verify that we have things right.
\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 276 "L8 := z -> sum((-1)^n * z^(2
*n-4)/(2*n)!, n=1..6):\nL0 := z -> sum((-1)^n * z^(2*n-4)/(2*n)!, n=1.
.infinity):\n`f(z) ` = (cos(z) - 1)/z^4;\nL[8](z),` = `,\nSum((-1)^n *
 z^(2*n-4)/(2*n)!, n=1..6) = L8(z);\nL[infinity](z),` = `,\nSum((-1)^n
 * z^(2*n-4)/(2*n)!, n=1..infinity) = L0(z);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 325 57 "Compare the \+
graph of the function and its Laurent series." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 107 "plot(\{f(x),LS(x)\}, x=0.01..5.0,\n  title=`y = f(x)
,    y = L8(x)`,\n  tickmarks=[5,4],\n  view=[0..4,-20..0]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 256 22 "Example 7.9, Page 287." }{TEXT 343 32 "   Find the Laure
nt series for  " }{XPPEDIT 18 0 "f(z) = exp(-1/z^2)" "6#/-%\"fG6#%\"zG
-%$expG6#,$*&\"\"\"F-*$F'\"\"#!\"\"F0" }{TEXT 326 15 "  centered at  \+
" }{XPPEDIT 18 0 "z[0] = 0" "6#/&%\"zG6#\"\"!F'" }{TEXT 327 3 " .\n" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "f := 'f': S:='S': z:='z':\nf := z
 -> exp(-1/z^2):\nS := series(f(z), z=infinity, 12):\n`f(z) ` = f(z);
\n`f(z) ` = S;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 328 57 "We can get the Laurent expansion by se
ries substitution:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "P:='P': Z:
='Z':\nS1 := taylor(exp(Z), Z=0, 7):\nS1 := convert(S1,polynom):\n`F(Z
) ` = exp(Z);\nP[6](Z) = S1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 257 "" 0 "" {TEXT 329 23 "Make the substitution  " }
{XPPEDIT 18 0 "Z = -1/z^2" "6#/%\"ZG,$*&\"\"\"F'*$%\"zG\"\"#!\"\"F+" }
{TEXT 330 42 "  to get the Laurent series is valid for  " }{XPPEDIT 
18 0 "0 < abs(z)" "6#2\"\"!-%$absG6#%\"zG" }{TEXT 333 3 " .\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "L:='L': Z:='Z':\nL1 := z -> subs(Z=
-1/z^2,S1):\n`f(z) ` = f(z);\nL[12](z) = L1(z);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 331 72 "Or we can
 get the Laurent expansion by series expansion about infinity:\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "L2 := taylor(exp(-1/z^2), z=infinit
y, 14):\n`f(z) ` = f(z);\n`f(z) ` = L2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 107 "plot(\{f(x),L1(x)\}, x=0.5..4.0,\n  title=`y = f(x),
    y = L12(x)`,\n  tickmarks=[5,4],\n  view=[0..4,0..1.5]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 332 19 "End of Section
 7.3." }}}}{MARK "0 0 0" 37 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }