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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 275 144 "\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 274 197 "\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 269 2 "\n\n" }{TEXT 256 55 "Section 7.5  Applications of Taylor \+
and Laurent Series\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 250 " \+
  In this section we show how Taylor and Laurent series can be used to
 derive important properties of analytic functions. We begin by showin
g that the zeros of an analytic function must be \"isolated\" unless t
he function is identically zero. A point " }{XPPEDIT 18 0 "alpha" "6#%
&alphaG" }{TEXT -1 11 " of a set  " }{XPPEDIT 18 0 "T;" "6#%\"TG" }
{TEXT -1 12 "  is called " }{TEXT 276 8 "isolated" }{TEXT -1 24 " if t
here exists a disk " }{XPPEDIT 18 0 "D[r](alpha)" "6#-&%\"DG6#%\"rG6#%
&alphaG" }{TEXT -1 7 " about " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }
{TEXT -1 44 " that does not contain any other points of  " }{XPPEDIT 
18 0 "T" "6#%\"TG" }{TEXT -1 3 ".  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 256 12 "Theorem 7.13" }{TEXT -1 12 "   Suppose  " }{XPPEDIT 18 
0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 18 "  is analytic at  " }
{XPPEDIT 18 0 "z=alpha" "6#/%\"zG%&alphaG" }{TEXT -1 12 "  and that  \+
" }{XPPEDIT 18 0 "f(alpha)=0" "6#/-%\"fG6#%&alphaG\"\"!" }{TEXT -1 8 "
 .  If  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 64 "  is \+
\nnot identically zero, then there exists a punctured disk  " }
{XPPEDIT 18 0 "D[r](alpha)^`*`" "6#)-&%\"DG6#%\"rG6#%&alphaG%\"*G" }
{TEXT -1 12 "  in which  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }
{TEXT -1 17 "  has no zeros.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 
34 "Corollary 7.11  (L'Hopital's rule)" }{TEXT -1 12 "   Suppose  " }
{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "g(z)" "6#-%\"gG6#%\"zG" }{TEXT -1 19 "  are analytic at
  " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" }{TEXT -1 4 " .  " }
}{PARA 0 "" 0 "" {TEXT -1 5 "\nIf  " }{XPPEDIT 18 0 "f(alpha) = 0" "6#
/-%\"fG6#%&alphaG\"\"!" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "g(alpha)
 = 0" "6#/-%\"gG6#%&alphaG\"\"!" }{TEXT -1 7 "  but  " }{XPPEDIT 18 0 
"`g '`(alpha) <> 0" "6#0-%$g~'G6#%&alphaG\"\"!" }{TEXT -1 11 " ,  then
   " }{XPPEDIT 18 0 "limit(f(z)/g(z),z=alpha)" "6#-%&limitG6$*&-%\"fG6
#%\"zG\"\"\"-%\"gG6#F*!\"\"/F*%&alphaG" }{TEXT -1 3 " = " }{XPPEDIT 
18 0 "limit(`f '`(z)/`g '`(z),z=alpha) = `f '`(alpha)/`g '`(alpha)" "6
#/-%&limitG6$*&-%$f~'G6#%\"zG\"\"\"-%$g~'G6#F+!\"\"/F+%&alphaG*&-F)6#F
2F,-F.6#F2F0" }{TEXT -1 3 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 
40 "Theorem 7.14  (Division of power series)" }{TEXT -1 11 " \nSuppose
  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "g(z)" "6#-%\"gG6#%\"zG" }{TEXT -1 19 "  are analytic at
  " }{XPPEDIT 18 0 "z=alpha" "6#/%\"zG%&alphaG" }{TEXT -1 36 "  with p
ower series representations\n" }{XPPEDIT 18 0 "f(z) = sum(a[n]*(z-alph
a)^n,n=0..infinity)" "6#/-%\"fG6#%\"zG-%$sumG6$*&&%\"aG6#%\"nG\"\"\"),
&F'F0%&alphaG!\"\"F/F0/F/;\"\"!%)infinityG" }{TEXT 277 9 "   for   " }
{XPPEDIT 18 0 "`z`*epsilon*D[R](alpha)" "6#*(%\"zG\"\"\"%(epsilonGF%-&
%\"DG6#%\"RG6#%&alphaGF%" }{TEXT 279 9 "   and   " }{XPPEDIT 18 0 "g(z
) = sum(b[n]*(z-alpha)^n,n=0..infinity)" "6#/-%\"gG6#%\"zG-%$sumG6$*&&
%\"bG6#%\"nG\"\"\"),&F'F0%&alphaG!\"\"F/F0/F/;\"\"!%)infinityG" }
{TEXT 278 9 "   for   " }{XPPEDIT 18 0 "`z`*epsilon*D[R](alpha)" "6#*(
%\"zG\"\"\"%(epsilonGF%-&%\"DG6#%\"RG6#%&alphaGF%" }{TEXT 280 2 " ." }
{TEXT -1 5 "\nIf  " }{XPPEDIT 18 0 "g(alpha) <> 0" "6#0-%\"gG6#%&alpha
G\"\"!" }{TEXT -1 23 " ,  then the quotient  " }{XPPEDIT 18 0 "f(z)/g(
z)" "6#*&-%\"fG6#%\"zG\"\"\"-%\"gG6#F'!\"\"" }{TEXT -1 39 "   has the \+
power series representation\n" }{XPPEDIT 18 0 "f(z)/g(z) = sum(c[n]*(z
-alpha)^n,n=0..infinity)" "6#/*&-%\"fG6#%\"zG\"\"\"-%\"gG6#F(!\"\"-%$s
umG6$*&&%\"cG6#%\"nGF)),&F(F)%&alphaGF-F5F)/F5;\"\"!%)infinityG" }
{TEXT -1 51 " ,  where the coefficients satisfy the equations  \n" }
{XPPEDIT 18 0 "a[n] = b[0]*c[n] + b[1]*c[n-1] + `...`+b[n-1]*c[1] + b[
n]*c[0]" "6#/&%\"aG6#%\"nG,,*&&%\"bG6#\"\"!\"\"\"&%\"cG6#F'F.F.*&&F+6#
F.F.&F06#,&F'F.F.!\"\"F.F.%$...GF.*&&F+6#,&F'F.F.F8F.&F06#F.F.F.*&&F+6
#F'F.&F06#F-F.F." }{TEXT -1 49 " . \nIn other words, the series for th
e quotient  " }{XPPEDIT 18 0 "f(z)/g(z)" "6#*&-%\"fG6#%\"zG\"\"\"-%\"g
G6#F'!\"\"" }{TEXT -1 71 "  can be obtained by the familiar process of
 \ndividing the series for  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" 
}{TEXT -1 21 "  by the series for  " }{XPPEDIT 18 0 "g(z)" "6#-%\"gG6#
%\"zG" }{TEXT -1 46 "  using the standard long division algorithm. " }
}}{EXCHG {PARA 257 "" 0 "" {TEXT 273 1 "\n" }{TEXT 256 23 "Example 7.1
5, Page 300." }{TEXT 270 56 "  Find the firt few terms of the \nMaclau
rin series for  " }{XPPEDIT 18 0 "f(z) = sec(z)" "6#/-%\"fG6#%\"zG-%$s
ecG6#F'" }{TEXT 266 27 "  and use it to compute    " }{XPPEDIT 18 0 "f
^` (4)`*`(0)`" "6#*&)%\"fG%%~(4)G\"\"\"%$(0)GF'" }{TEXT 260 3 " .\n" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "f:='f': p:='p': P:='P': S:='S': z
:='z': Z:='Z':\nf := z -> sec(z):\nS := series(f(Z), Z=0, 14):\np := z
 -> subs(Z=z,convert(S, polynom)):\n`f(z) ` = f(z);\n`f(z) ` = subs(Z=
z,S);\nP[12](z) = p(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Or we \+
could use Maple's \"unapply\" procedure." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 201 "f:='f': p:='p': P:='P': s:='s': t:='t': z:='z': Z:='
Z':\nf := z -> sec(z):\nS := taylor(f(z), z=0, 14):\np:=unapply(conver
t(taylor(f(z),z=0,14),polynom),z):\n`f(z)  ` = f(z);\n`f(z)  ` = S;\nP
[12](z) = p(z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 457 "D4 := s
implify(diff(f(Z), Z$4)):\nf4 := z -> subs(Z=z, D4):\nS4 := simplify(d
iff(p(Z), Z$4)):\np4 := z -> subs(Z=z, S4):\n`f'(z) ` = simplify(diff(
f(z), z));\n`p'(z) ` = simplify(diff(p(z), z)); ` `;\n`f''(z) ` = simp
lify(diff(f(z), z$2));\n`p''(z) ` = simplify(diff(p(z), z$2)); ` `;\n`
f'''(z) ` = simplify(diff(f(z), z$3));\n`p'''(z) ` = simplify(diff(p(z
), z$3)); ` `;\n`f''''(z) ` = f4(z);\n`p''''(z) ` = p4(z); ` `;\n`f'''
'(0) ` = eval(f4(0));\n`p''''(0) ` = p4(0);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 256 23 "Theorem 7.15  (Riemann)" }{TEXT -1 17 "   Suppose t
hat  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 18 "  is anal
ytic in  " }{XPPEDIT 18 0 "D[r](alpha)^`*`" "6#)-&%\"DG6#%\"rG6#%&alph
aG%\"*G" }{TEXT -1 8 " . \nIf  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"z
G" }{TEXT -1 17 "  is bounded in  " }{XPPEDIT 18 0 "D[r](alpha)^`*`" "
6#)-&%\"DG6#%\"rG6#%&alphaG%\"*G" }{TEXT -1 17 " ,  then either  " }
{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 18 "  is analytic at \+
 " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" }{TEXT -1 6 "  or  " 
}{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 35 "  has a removabl
e \nsingularity at  " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" }
{TEXT -1 3 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 256 12 "Theorem 7.16" }{TEXT -1 17 "   The function  " }
{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 30 "  has a pole of o
rder  k  at  " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" }{TEXT 
-1 20 "  \nif and only if   " }{XPPEDIT 18 0 "limit(abs(f(z)), z=alpha
) = infinity" "6#/-%&limitG6$-%$absG6#-%\"fG6#%\"zG/F-%&alphaG%)infini
tyG" }{TEXT -1 3 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 256 12 "Theorem 7.18" }{TEXT -1 17 "   The funct
ion  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 34 "  has an \+
essential singularity at " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alpha
G" }{TEXT -1 19 "  \nif and only if  " }{XPPEDIT 18 0 "limit(abs(f(z))
, z=alpha) " "6#-%&limitG6$-%$absG6#-%\"fG6#%\"zG/F,%&alphaG" }{TEXT 
-1 18 "  does not exist. " }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 23 "E
xample 7.16, Page 302." }{TEXT 271 26 "  Show that the function  " }
{XPPEDIT 18 0 "g(z) = exp(-1/z^2)" "6#/-%\"gG6#%\"zG-%$expG6#,$*&\"\"
\"F-*$F'\"\"#!\"\"F0" }{TEXT 261 25 "  \nis NOT continuous at  " }
{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 267 3 " .\n" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 347 "g:='g': G:='G': x:='x': y:='y': z:='z':\nG :
= z -> exp(-1/z^2):\n`g(z) ` = G(z); ` `;\n`g(x + I*0) ` = evalc(G(x +
 I*0)),`  and  `,\n`g(0 + I*y) ` = evalc(G(0 + I*y));\nLimit(g(x),   x
=0, right),` = `,\nLimit(G(x),   x=0, right) = limit(G(x),   x=0, righ
t);\nLimit(g(0 + I*y),   y=0, right),` = `,\nLimit(G(I*y),   y=0, righ
t) = limit(G(I*y), y=0, right);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 262 36 "Since the limits are NOT \+
the same,  " }{XPPEDIT 18 0 "g(z) = exp(-1/z^2)" "6#/-%\"gG6#%\"zG-%$e
xpG6#,$*&\"\"\"F-*$F'\"\"#!\"\"F0" }{TEXT 263 24 "  is NOT continuous \+
at  " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 268 22 " .\nREMARK
.  Although  " }{XPPEDIT 18 0 "g(z)= exp(-1/z^2)" "6#/-%\"gG6#%\"zG-%$
expG6#,$*&\"\"\"F-*$F'\"\"#!\"\"F0" }{TEXT 264 132 "  is known to be i
nfinitely differentiable,\nit does not have a Maclaurin series expansi
on.  However, it does have a Laurent series.\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 126 "f:='f': p:='p': S:='S': z:='z':\nf := z -> exp(-1/z^
2):\ns := z -> series(f(z), z=infinity, 14):\n`f(z) ` = f(z);\n`f(z) `
 = s(z);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 265 19 "End of Section 7.5
." }}}}{MARK "0 0 0" 27 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }