{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 0 0 0 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 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 261 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Geneva" 
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Geneva" 1 10 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 267 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 268 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 269 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Geneva" 
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Geneva" 1 10 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 275 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 276 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 277 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Geneva" 
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Geneva" 1 10 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 283 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 284 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 285 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "Geneva" 
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "Geneva" 1 10 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 291 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 292 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 293 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "Geneva" 
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 296 "Geneva" 1 10 0 0 
0 1 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 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 300 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 301 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "Geneva" 
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "Geneva" 1 10 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 306 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 307 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 308 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "Geneva" 
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 311 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "Geneva" 1 10 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 314 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 315 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
316 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 317 "Gene
va" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "Geneva" 1 10 
0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 319 "Geneva" 1 10 0 0 0 1 
0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 320 "Geneva" 1 10 0 0 0 1 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 321 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 
0 1 }{CSTYLE "" -1 322 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 323 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 324 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 325 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 326 "Geneva" 
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 327 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "Geneva" 1 10 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 330 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 331 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 332 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 333 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 334 "Geneva" 
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 335 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 336 "Geneva" 1 10 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 337 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 338 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 339 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
340 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 341 "Gene
va" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 342 "Geneva" 1 10 
0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 343 "Geneva" 1 10 0 0 0 1 
0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 344 "Geneva" 1 10 0 0 0 1 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 345 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 
0 1 }{CSTYLE "" -1 346 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 347 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "
Normal" -1 0 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 }{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 2" -1 257 1 {CSTYLE "" -1 -1 "Sy
mbol" 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" -1 258 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 }}
{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 298 144 "\n(c) John H. Mathews          Rus
sell W. Howell\nmathews@fullerton.edu     howell@westmont.edu\n\nCompl
imentary software to accompany the textbook:" }}{PARA 258 "" 0 "" 
{TEXT 257 82 "COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed,
 2001, ISBN: 0-7637-1425-9" }{TEXT 297 197 "\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 0 "" 0 "" {TEXT 
296 1 "\n" }{TEXT 256 30 "CHAPTER 6  COMPLEX INTEGRATION" }{TEXT 289 
2 "\n\n" }{TEXT 256 60 "Section 6.5  Integral Representations for Anal
ytic Functions" }{TEXT 290 193 "\n\n   We now present some major resul
ts in the theory of functions of a complex variable. The first result \+
is known as Cauchy's integral formula and shows that the value of an a
nalytic function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 307 55 " can b
e represented by a certain contour integral. The " }{XPPEDIT 18 0 "`n \+
th`;" "6#%%n~thG" }{TEXT 308 14 " derivative , " }{XPPEDIT 18 0 "f^`(n
)`;" "6#)%\"fG%$(n)G" }{XPPEDIT 18 0 "`(z)`;" "6#%$(z)G" }{TEXT 309 
313 ", will have a similar representation. In Chapter 7 we will show h
ow the Cauchy integral formulae are used to prove Taylor's theorem, an
d we will establish the power series representation for analytic funct
ions. The Cauchy integral formulae will also be a convenient tool for \+
evaluating certain contour integrals.\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 41 "Theorem 6.10  (Cauchy's \+
Integral Formula)" }{TEXT 306 4 "    " }}{PARA 0 "" 0 "" {TEXT 345 5 "
Let  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 311 46 "  be ana
lytic in the simply connected domain  " }{XPPEDIT 18 0 "D;" "6#%\"DG" 
}{TEXT 312 12 ",  and let  " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT 313 
107 "  be a simple closed positively oriented contour that lies in  D.
  \nIf   is a point that lies interior to  " }{XPPEDIT 18 0 "C" "6#%\"
CG" }{TEXT 314 11 ",  then    " }{XPPEDIT 18 0 "int(f(z)/(z - z[0]), z
=C..`   `)   =  2*pi*i*f(z[0])" "6#/-%$intG6$*&-%\"fG6#%\"zG\"\"\",&F+
F,&F+6#\"\"!!\"\"F1/F+;%\"CG%$~~~G**\"\"#F,%#piGF,%\"iGF,-F)6#&F+6#F0F
," }{TEXT 310 2 " ." }{TEXT -1 2 "  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
256 23 "Example 6.21, Page 245." }{TEXT 291 14 "  Show that   " }
{XPPEDIT 18 0 "int(exp(z)/(z - 1), z=C..`   `)   =  2*pi*i*exp(1)" "6#
/-%$intG6$*&-%$expG6#%\"zG\"\"\",&F+F,F,!\"\"F./F+;%\"CG%$~~~G**\"\"#F
,%#piGF,%\"iGF,-F)6#F,F," }{TEXT 265 11 " , \nwhere  " }{XPPEDIT 18 0 
"C" "6#%\"CG" }{TEXT 266 17 "  is the circle  " }{XPPEDIT 18 0 "C" "6#
%\"CG" }{TEXT 267 2 ": " }{XPPEDIT 18 0 "abs(z) = 2" "6#/-%$absG6#%\"z
G\"\"#" }{TEXT 268 29 "  with positive orientation.\n" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 221 "f :='f': F:='F': z:='z': Z:='Z': Zn:='Zn':\nw  \+
:= exp(z)/(z-1):\nprint(`Find `,Int(w, z=C..``));\nZn := sort([solve(d
enom(w)=0, z)]):\n`Singularity at  `, z[0] = Zn[1];\nf := Z -> subs(z=
Z, w*(z - Zn[1])):\n`Use  f(z) ` = f(z);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 17 "The integral of \+
 " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 269 14 "  taken over
  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 270 6 "  is:\n" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 115 "`f(z)` = f(z);\n`f(z0)` = f(Zn[1]);\nval := 2
*Pi*I*f(Zn[1]):\nInt(w ,z=C..``) = `2 Pi I f(z0)`;\nInt(w ,z=C..``) = \+
val;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 256 23 "Example 6.22, Page 245." }{TEXT 292 15 "   Show that  \+
 " }{XPPEDIT 18 0 "int(sin(z)/(4*z + pi), z=C..`   `) = - i*sqrt(2)*pi
/4" "6#/-%$intG6$*&-%$sinG6#%\"zG\"\"\",&*&\"\"%F,F+F,F,%#piGF,!\"\"/F
+;%\"CG%$~~~G,$**%\"iGF,-%%sqrtG6#\"\"#F,F0F,F/F1F1" }{TEXT 271 11 " ,
 \nwhere  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 272 17 "  is the circl
e  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 273 2 ": " }{XPPEDIT 18 0 "ab
s(z) = 1" "6#/-%$absG6#%\"zG\"\"\"" }{TEXT 274 29 "  with positive ori
entation.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "f :='f': F:='F': z:
='z': Z:='Z': Zn:='Zn':\nw  := sin(z)/(4*z + Pi):\nprint(`Find `,Int(w
 ,z=C..``));\nZn := sort([solve(denom(w)=0, z)]):\n`Singularity at  `,
 z[0] = Zn[1];\n`Singularity at  `, z[0] = evalf(Zn[1]);\nf := Z -> su
bs(z=Z, simplify(w*(z - Zn[1]))):\n`Use  f(z) ` = f(z);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 17 "
The integral of  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 275 
13 " taken over  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 276 6 "  is:\n
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "`f(z)` = f(z);\n`f(z0) ` = f(Z
n[1]);\n`f(z0) ` = evalc(f(Zn[1]));\nval := 2*Pi*I*f(Zn[1]):\nInt(w ,z
=C..``) = `2 Pi I f(z0)`;\nInt(w ,z=C..``) = val;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 24 "Example 6
.23,  Page 245." }{TEXT 305 15 "   Show that   " }{XPPEDIT 18 0 "int(e
xp(i*pi*z)/(2*z^2 - 5*z + 2), z=C..`   `)  = 2*pi/3" "6#/-%$intG6$*&-%
$expG6#*(%\"iG\"\"\"%#piGF-%\"zGF-F-,(*&\"\"#F-*$F/F2F-F-*&\"\"&F-F/F-
!\"\"F2F-F6/F/;%\"CG%$~~~G*(F2F-F.F-\"\"$F6" }{TEXT 299 11 " , \nwhere
  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 300 17 "  is the circle  " }
{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 301 2 ": " }{XPPEDIT 18 0 "abs(z) =
 1" "6#/-%$absG6#%\"zG\"\"\"" }{TEXT 302 58 "  with positive orientati
on.\nSolution:\nWe will find that  " }{XPPEDIT 18 0 "z[0] = 1/2" "6#/&
%\"zG6#\"\"!*&\"\"\"F)\"\"#!\"\"" }{TEXT 303 41 "  the only singularit
y that lies inside  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 304 3 " .\n
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "f :='f': F:='F': z:='z': Z:='Z
': Zn:='Zn':\nw  := exp(I*Pi*z)/(2*z^2 - 5*z + 2):\nprint(`Find `,Int(
w ,z=C..``));\nZn := sort([solve(denom(w)=0, z)]):\n`Singularities occ
ur at: ` = Zn;\n`The singularity inside C is  `,z[0] = Zn[1];\nf := Z \+
-> subs(z=Z, simplify(w*(z - Zn[1]))):\n`Use  f(z) ` = f(z);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 207 "   We now state a general resu
lt that shows how differentiation under the integral sign can be accom
plished. The proof can be found in some advanced texts. See, for insta
nce, Rolf Nevanlinna and V. Paatero, " }{TEXT 315 32 "Introduction to \+
Complex Analysis" }{TEXT -1 83 ", (Reading, Massachusetts: Addison-Wes
ley Publishing Company, 1969), Section 9.7.  " }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 256 30 "Theorem 6.11  (Leibniz's Rule)" }{TEXT 320 4 "    \+
" }}{PARA 0 "" 0 "" {TEXT 346 5 "Let  " }{XPPEDIT 18 0 "D" "6#%\"DG" }
{TEXT 323 42 "  be a simply connected domain,  and let  " }{XPPEDIT 
18 0 "`I: `;" "6#%$I:~G" }{TEXT 321 3 "   " }{TEXT -1 1 "a" }{XPPEDIT 
18 0 "`` <= ``;" "6#1%!GF$" }{TEXT -1 1 "t" }{XPPEDIT 18 0 "`` <= ``" 
"6#1%!GF$" }{TEXT -1 1 "b" }{TEXT 322 40 "  be an interval of real num
bers.  Let  " }{XPPEDIT 18 0 "f(z,t)" "6#-%\"fG6$%\"zG%\"tG" }{TEXT 
316 31 "  and its partial derivative   " }{XPPEDIT 18 0 "diff(f(z,t),z
)" "6#-%%diffG6$-%\"fG6$%\"zG%\"tGF)" }{TEXT 317 65 " ,  with respect \+
to  z,  be continuous functions for all  z  in  " }{XPPEDIT 18 0 "D" "
6#%\"DG" }{TEXT 324 29 "  and all  t  in  I.  \nThen  " }{XPPEDIT 18 
0 "F(z) = int(f(z,t),t=a..b)" "6#/-%\"FG6#%\"zG-%$intG6$-%\"fG6$F'%\"t
G/F.;%\"aG%\"bG" }{TEXT 318 37 "  is analytic for  z  in  D,  and    \+
" }{XPPEDIT 18 0 "`F '(z)` = int(diff(f(z,t),z),t=a..b)" "6#/%'F~'(z)G
-%$intG6$-%%diffG6$-%\"fG6$%\"zG%\"tGF./F/;%\"aG%\"bG" }{TEXT 319 3 " \+
. " }{TEXT -1 2 "  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "  \+
 We now show how we can generalize Theorem 6.10 to give an integral re
presentation for the " }{TEXT 325 1 " " }{XPPEDIT 18 0 "`n th`;" "6#%%
n~thG" }{TEXT 326 15 " derivative ,  " }{XPPEDIT 18 0 "f^`(n)`;" "6#)%
\"fG%$(n)G" }{XPPEDIT 18 0 "`(z)`;" "6#%$(z)G" }{TEXT -1 128 ".  Leibn
iz's rule is used in the proof, and this method of proof will be a mne
monic device for remembering the upcoming theorem." }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 256 58 "Theorem 6.12  (Cauchy's Integral Formulae for \+
Derivatives)" }{TEXT 330 5 "     " }}{PARA 0 "" 0 "" {TEXT 347 5 "Let \+
 " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 327 46 "  be analyti
c in the simply connected domain  " }{XPPEDIT 18 0 "D" "6#%\"DG" }
{TEXT 332 11 ", and let  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 333 63 
"  be a simple closed positively oriented contour that lies in  " }
{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 331 57 ". \nIf  z  is a point that \+
lies interior to  C,  then     " }{XPPEDIT 18 0 "f^`(n)`*`(z)` = (n!/(
2*pi*i))" "6#/*&)%\"fG%$(n)G\"\"\"%$(z)GF(*&-%*factorialG6#%\"nGF(*(\"
\"#F(%#piGF(%\"iGF(!\"\"" }{TEXT 328 1 " " }{XPPEDIT 18 0 "int(f(zeta)
/((z-zeta)^(n+1)),zeta = C .. ` `);" "6#-%$intG6$*&-%\"fG6#%%zetaG\"\"
\"),&%\"zGF+F*!\"\",&%\"nGF+F+F+F//F*;%\"CG%\"~G" }{TEXT 329 3 " . " }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 256 23 "Example 6.25, Page 247." }{TEXT 293 15 "   Sho
w that   " }{XPPEDIT 18 0 "int(exp(z^2)/(z - i)^4, z) = - 4*pi/(3*exp(
1))" "6#/-%$intG6$*&-%$expG6#*$%\"zG\"\"#\"\"\"*$,&F,F.%\"iG!\"\"\"\"%
F2F,,$*(F3F.%#piGF.*&\"\"$F.-F)6#F.F.F2F2" }{TEXT -1 1 " " }{TEXT 277 
10 ". \nwhere  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 278 17 "  is the \+
circle  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 279 2 ": " }{XPPEDIT 18 
0 "abs(z) = 2" "6#/-%$absG6#%\"zG\"\"#" }{TEXT 280 46 "  with positive
 orientation.\nSolution:\nSince  " }{XPPEDIT 18 0 "z = i" "6#/%\"zG%\"
iG" }{TEXT 281 29 "  is a singularity of order  " }{XPPEDIT 18 0 "n=4
" "6#/%\"nG\"\"%" }{TEXT 282 31 " ,  multiply the integrand by  " }
{XPPEDIT 18 0 "(z - i)^4" "6#*$,&%\"zG\"\"\"%\"iG!\"\"\"\"%" }{TEXT 
283 23 "  to get the function  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"z
G" }{TEXT 284 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "f :='f': \+
F:='F': z:='z': Z:='Z': Zn:='Zn':\nw  := exp(z^2)/(z - I)^4:\nprint(`F
ind `,Int(w ,z=C..``));\nZn := sort([solve(denom(w)=0, z)]):\n`Singula
rity of order  4  at  `, z[0] = Zn[1];\nf := Z -> subs(z=Z, simplify(w
*(z - Zn[1])^4)):\n`Use  f(z) ` = f(z);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 262 17 "The integral of  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"z
G" }{TEXT 285 14 "  taken over  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 
286 72 "  is obtained by applying the \nCauchy integral formula for de
rivatives.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "d3 := diff(f(z), z
$3):\nf3 := Z -> subs(z=Z, d3):\n`f '''(z) ` = f3(z);\n`f '''(z0) ` = \+
f3(Zn[1]);\nval := 2*Pi*I/3! * f3(Zn[1]):\nInt(w ,z=C..``) = `2 Pi I/3
! f '''(z0)`;\nInt(w ,z=C..``) = val;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 75 "Remark.  Sometim
es Maple will form the list of values in a different order." }}{PARA 
0 "" 0 "" {TEXT 295 79 "It is always necessary to visually inspect the
 above results before proceeding." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 263 17 "The integral of  " }{XPPEDIT 18 0 "f(z)
" "6#-%\"fG6#%\"zG" }{TEXT 287 14 "  taken over  " }{XPPEDIT 18 0 "C" 
"6#%\"CG" }{TEXT 288 55 "  is obtained by applying the Cauchy integral
 formula.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "`f(z0) ` = f(Zn[1])
;\n`f(z0) ` = evalc(f(Zn[1]));\nval := 2*Pi*I*f(Zn[1]):\nInt(w ,z=C..`
`) = `2 Pi I f(z0)`;\nInt(w ,z=C..``) = val;" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 13 "Corollary 6.2" }{TEXT 
336 7 "   If  " }{XPPEDIT 18 0 "f(z);" "6#-%\"fG6#%\"zG" }{TEXT 334 
29 "  is analytic in the domain  " }{XPPEDIT 18 0 "D;" "6#%\"DG" }
{TEXT 337 24 ", then all derivatives  " }{XPPEDIT 18 0 "`f '(z)`,`f ''
(z)`,`...`,f^`(n)`*`(z)` " "6&%'f~'(z)G%(f~''(z)G%$...G*&)%\"fG%$(n)G
\"\"\"%$(z)GF*" }{TEXT 335 30 "  exists and are analytic in  " }
{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 338 3 ".  " }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 7 "Remark." }{TEXT -1 
139 "  Corollary 6.2 is interesting, as it illustrates a big differenc
e between real and complex functions. It is possible for a real functi
on  " }{XPPEDIT 18 0 "f(z);" "6#-%\"fG6#%\"zG" }{TEXT -1 29 "  to have
 the property that  " }{XPPEDIT 18 0 "`f '(z)`;" "6#%'f~'(z)G" }{TEXT 
-1 32 "  exists everywhere in a domain " }{XPPEDIT 18 0 "D;" "6#%\"DG
" }{TEXT -1 7 ", but  " }{XPPEDIT 18 0 "`f ''(z)`;" "6#%(f~''(z)G" }
{TEXT -1 67 " exists nowhere. This corollary states that if a complex \+
function  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 24 " has
 the property that  " }{XPPEDIT 18 0 "`f '(z)`;" "6#%'f~'(z)G" }{TEXT 
-1 31 " exists everywhere in a domain " }{XPPEDIT 18 0 "D;" "6#%\"DG" 
}{TEXT -1 20 ", then, remarkably, " }{TEXT 339 3 "all" }{TEXT -1 17 " \+
derivatives of  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 
10 " exist in " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 3 ".  " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 13 "C
orollary 6.3" }{TEXT 342 7 "   If  " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG
6$%\"xG%\"yG" }{TEXT 340 40 "  is a harmonic function at each point  \+
" }{XPPEDIT 18 0 "`(x,y)`;" "6#%&(x,y)G" }{TEXT 344 17 "  in the domai
n  " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 343 33 ",  then all partial d
erivatives  " }{XPPEDIT 18 0 "diff(u(x,y),x),diff(u(x,y),y),diff(u(x,y
),x,x),diff(u(x,y),x,y),diff(u(x,y),y,y)" "6'-%%diffG6$-%\"uG6$%\"xG%
\"yGF)-F$6$-F'6$F)F*F*-F$6%-F'6$F)F*F)F)-F$6%-F'6$F)F*F)F*-F$6%-F'6$F)
F*F*F*" }{TEXT 341 37 "  exist and are harmonic functions.  " }{TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 264 19 "End of Section 6.5." }}}}{MARK "0 0 0" 20 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }