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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 274 144 "\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 273 197 "\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 
272 1 "\n" }{TEXT 256 36 "CHAPTER 7  TAYLOR and LAURENT SERIES" }
{TEXT 270 2 "\n\n" }{TEXT 256 32 "Section 7.1  Uniform Convergence" }
{TEXT 269 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 707 "   Throughout this te
xt we have compared and contrasted properties of complex functions wit
h functions whose domain and range lie entirely within the reals.  The
re are many similarities, such as the standard differentiation formula
s.  However, there are also some surprises, and in this chapter we wil
l encounter one of the hallmarks distinguishing complex functions from
 their real counterparts.\n\011\n   It is possible for a function defi
ned on the real numbers to be differentiable everywhere and yet not be
 expressible as a power series (see Exercise 7.2.20 at the end of Sect
ion 7.2).  In the complex case, however, things are much simpler!  It \+
turns out that if a complex function is analytic in the disk " }
{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 26 ", its Taylor series about \+
" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 139 " will converge t
o the function at every point in this disk. Thus, analytic functions a
re locally nothing more than glorified polynomials.  " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 256 36 "Definition 7.1:  Uniform convergence" 
}{TEXT -1 4 "    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "The sequence  " }{XPPEDIT 18 0 "\{S[n](z)\}" "6#<#-&%\"SG
6#%\"nG6#%\"zG" }{TEXT -1 2 "  " }{TEXT 257 19 "converges uniformly" }
{TEXT -1 5 " to  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 
31 "  on the set  T  if for every  " }{XPPEDIT 18 0 "epsilon > 0" "6#2
\"\"!%(epsilonG" }{TEXT -1 37 " ,  there exists a positive integer  " 
}{XPPEDIT 18 0 "N[epsilon]" "6#&%\"NG6#%(epsilonG" }{TEXT -1 26 "  (wh
ich depends only on  " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT 
-1 17 ")  such that if  " }{XPPEDIT 18 0 "N <= n" "6#1%\"NG%\"nG" }
{TEXT -1 11 " ,  then   " }{XPPEDIT 18 0 "abs(S[n](z) - f(z)) < epsilo
n" "6#2-%$absG6#,&-&%\"SG6#%\"nG6#%\"zG\"\"\"-%\"fG6#F.!\"\"%(epsilonG
" }{TEXT -1 13 "   for all   " }{XPPEDIT 18 0 "`z`*epsilon*`T`" "6#*(%
\"zG\"\"\"%(epsilonGF%%\"TGF%" }{TEXT -1 4 " .  " }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 256 23 "Example 7.1,  Page 262." }{TEXT 275 2 "  " }
{TEXT -1 14 "The sequence  " }{XPPEDIT 18 0 "\{S[n](z)\};" "6#<#-&%\"S
G6#%\"nG6#%\"zG" }{TEXT -1 4 " =  " }{XPPEDIT 18 0 "\{exp(z)+1/n\};" "
6#<#,&-%$expG6#%\"zG\"\"\"*&F)F)%\"nG!\"\"F)" }{TEXT -1 38 " converges
 uniformly to the function  " }{XPPEDIT 18 0 "f(z) = exp(z);" "6#/-%\"
fG6#%\"zG-%$expG6#F'" }{TEXT -1 46 "  on the entire complex plane beca
use for any " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 5 ">0
 , " }{XPPEDIT 18 0 "abs(S[n](z) - f(z)) < epsilon" "6#2-%$absG6#,&-&%
\"SG6#%\"nG6#%\"zG\"\"\"-%\"fG6#F.!\"\"%(epsilonG" }{TEXT -1 22 " is s
atisfied for all " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 5 " for " }
{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 1 ">" }{XPPEDIT 18 0 "N[epsilon
];" "6#&%\"NG6#%(epsilonG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "N[ep
silon]" "6#&%\"NG6#%(epsilonG" }{TEXT -1 29 " is any integer greater t
han " }{XPPEDIT 18 0 "1/epsilon;" "6#*&\"\"\"F$%(epsilonG!\"\"" }
{TEXT -1 56 ". We leave the details for showing this as an exercise. \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "There is a useful proced
ure known as the Weierstrass M-test that can help determine whether an
 infinite series is uniformly convergent." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 33 "Theorem 7.1  (Weierstras
s M-test)" }{TEXT -1 6 "      " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 29 "Suppose the infinite series  " }{XPPEDIT 
18 0 "sum(u[k](z),k=0..infinity)" "6#-%$sumG6$-&%\"uG6#%\"kG6#%\"zG/F*
;\"\"!%)infinityG" }{TEXT -1 39 "  has the property that for each  k, \+
  " }{XPPEDIT 18 0 "abs(u[k](z)) < M[k]" "6#2-%$absG6#-&%\"uG6#%\"kG6#
%\"zG&%\"MG6#F+" }{TEXT -1 13 "   for all   " }{XPPEDIT 18 0 "`z`*epsi
lon*`T`" "6#*(%\"zG\"\"\"%(epsilonGF%%\"TGF%" }{TEXT -1 11 " .  \nIf  \+
  " }{XPPEDIT 18 0 "sum(M[k],k=0..infinity)" "6#-%$sumG6$&%\"MG6#%\"kG
/F);\"\"!%)infinityG" }{TEXT -1 22 "   converges,  then   " }{XPPEDIT 
18 0 "sum(u[k](z),k=0..infinity)" "6#-%$sumG6$-&%\"uG6#%\"kG6#%\"zG/F*
;\"\"!%)infinityG" }{TEXT -1 31 "   converges uniformly on  T.  " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Th
eorem 7.2 gives an interesting application of the Weierstrass M-test.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 
11 "Theorem 7.2" }{TEXT -1 30 "   Suppose the power series   " }
{XPPEDIT 18 0 "sum(c[k]*(z - alpha)^k,k=0..infinity)" "6#-%$sumG6$*&&%
\"cG6#%\"kG\"\"\"),&%\"zGF+%&alphaG!\"\"F*F+/F*;\"\"!%)infinityG" }
{TEXT -1 30 "   has radius of convergence  " }{XPPEDIT 18 0 "rho > 0" 
"6#2\"\"!%$rhoG" }{TEXT -1 4 " .  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 15 "Then for each  " }{XPPEDIT 18 0 "r" "6#%
\"rG" }{TEXT -1 4 " ,  " }{XPPEDIT 18 0 "`0 < r`<rho" "6#2%&0~<~rG%$rh
oG" }{TEXT -1 57 "   ,  the series converges uniformly on the closed d
isk  " }{XPPEDIT 18 0 "D[r](alpha) = \{`z :  `*abs(z  - alpha) <= r\}
" "6#/-&%\"DG6#%\"rG6#%&alphaG<#1*&%&z~:~~G\"\"\"-%$absG6#,&%\"zGF/F*!
\"\"F/F(" }{TEXT -1 4 " .  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 256 13 "Corollary 7.1" }{TEXT -1 13 "   For eac
h  " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "`0
 < r`<1" "6#2%&0~<~rG\"\"\"" }{TEXT -1 65 " ,  the geometric series co
nverges uniformly on the closed disk  " }{XPPEDIT 18 0 "D[r](0) = \{`z
 :  `*abs(z) <= r\}" "6#/-&%\"DG6#%\"rG6#\"\"!<#1*&%&z~:~~G\"\"\"-%$ab
sG6#%\"zGF/F(" }{TEXT -1 3 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 256 11 "Theorem 7.3" }{TEXT -1 12 "   Suppo
se  " }{XPPEDIT 18 0 "\{S[n](z)\}" "6#<#-&%\"SG6#%\"nG6#%\"zG" }{TEXT 
-1 91 "  is a sequence of continuous functions defined on a set T cont
aining the contour C.   If  " }{XPPEDIT 18 0 "\{S[n](z)\}" "6#<#-&%\"S
G6#%\"nG6#%\"zG" }{TEXT -1 26 "  converges uniformly to  " }{XPPEDIT 
18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 23 "  on the set  T,  then\n" 
}{TEXT 256 3 "(i)" }{TEXT -1 4 "\011   " }{XPPEDIT 18 0 "f(z)" "6#-%\"
fG6#%\"zG" }{TEXT -1 28 "  is continuous on  T,  and\n" }{TEXT 256 4 "
(ii)" }{TEXT -1 3 "\011  " }{XPPEDIT 18 0 "limit(int(S[k](z),z=C..`.`)
 ,k=infinity) = int(limit(S[k](z) ,k=infinity) ,z=C..`.`) " "6#/-%&lim
itG6$-%$intG6$-&%\"SG6#%\"kG6#%\"zG/F0;%\"CG%\".G/F.%)infinityG-F(6$-F
%6$-&F,6#F.6#F0/F.F6/F0;F3F4" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(f(
z),z=C..`.`) " "6#-%$intG6$-%\"fG6#%\"zG/F);%\"CG%\".G" }{TEXT -1 4 " \+
.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 13 "Corollary 7.2" }{TEXT -1 18 "   If the series  " }{XPPEDIT 18 
0 "sum(c[k]*(z - alpha)^k,k=0..infinity)" "6#-%$sumG6$*&&%\"cG6#%\"kG
\"\"\"),&%\"zGF+%&alphaG!\"\"F*F+/F*;\"\"!%)infinityG" }{TEXT -1 26 " \+
 converges uniformly to  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }
{TEXT -1 62 "  on the set  T,  and  C  is a contour contained in  T,  \+
then\n" }{XPPEDIT 18 0 "sum(int(c[k]*(z - alpha)^k,z=C..`.`) ,k=0..inf
inity)= int(sum(c[k]*(z - alpha)^k,k=0..infinity),z=C..`.`)" "6#/-%$su
mG6$-%$intG6$*&&%\"cG6#%\"kG\"\"\"),&%\"zGF/%&alphaG!\"\"F.F//F2;%\"CG
%\".G/F.;\"\"!%)infinityG-F(6$-F%6$*&&F,6#F.F/),&F2F/F3F4F.F//F.;F;F</
F2;F7F8" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "int(f(z),z=C..`.`)" "6#-%
$intG6$-%\"fG6#%\"zG/F);%\"CG%\".G" }{TEXT -1 2 " ." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 22 "Exam
ple 7.2, Page 266." }{TEXT 271 14 "  Show that   " }{XPPEDIT 18 0 "Log
(1- z) = - sum(z^n/n, n=1..infinity)" "6#/-%$LogG6#,&\"\"\"F(%\"zG!\"
\",$-%$sumG6$*&)F)%\"nGF(F1F*/F1;F(%)infinityGF*" }{TEXT 263 13 " ,  f
or all  " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 264 6 "  in  " }
{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 265 2 ": " }{XPPEDIT 18 0 "abs(z) <
 1" "6#2-%$absG6#%\"zG\"\"\"" }{TEXT 266 4 " . \n" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 210 "f:='f': p:='p': P:='P': s:='s': t:='t': z:='z': Z:
='Z':\nf := z -> log(1-z):\nt := taylor(f(Z), Z=0, 11):\ns := subs(Z=z
,t):\np := z -> subs(Z=z,convert(t, polynom)):\n`f(z)  ` = f(z);\n`f(z
)  ` = s;\nP[10](z) = p(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Or
 we could use Maple's \"unapply\" procedure." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 203 "f:='f': p:='p': P:='P': s:='s': t:='t': z:='z': Z:='
Z':\nf := z -> log(1-z):\ns := taylor(f(z), z=0, 11):\np:=unapply(conv
ert(taylor(f(z),z=0,11),polynom),z):\n`f(z)  ` = f(z);\n`f(z)  ` = s;
\nP[10](z) = p(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 260 54 "Sum up the terms to verify that we hav
e things right.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "f:='f': n:='n
': s:='s': S:='S': z:='z':\nf := z -> log(1-z):\nS10 := z -> sum(-1/n*
z^n, n=1..10):\nS := z -> sum(-1/n*z^n, n=1..infinity):\n`f(z) ` = f(z
);\ns[10](z),` = `,Sum(-1/n*z^n, n=1..10) = S10(z);\ns[infinity](z),` \+
= `,Sum(-1/n*z^n, n=1..infinity)  = S(z);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 261 27 "The real varia
ble plot of  " }{XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-%\"fG6#%\"xG" }
{TEXT 267 7 "  and  " }{XPPEDIT 18 0 "y = s[10](x)" "6#/%\"yG-&%\"sG6#
\"#56#%\"xG" }{TEXT 268 6 "  is:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
159 "plot(\{f(x),S10(x)\}, \n  x=-0.999..0.999, y=-3..0.7,\n  title=`y
=ln(1-z)  and   y=s10(x)`,\n  labels=[`      x`,`y  `],\n  tickmarks=[
5,7],\n  view=[-1..1,-3..0.7]);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 
262 19 "End of Section 7.1." }}}}{MARK "0 0 0" 26 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }