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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 284 144 "\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 283 197 "\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 
278 1 "\n" }{TEXT 256 65 "CHAPTER 4  SEQUENCES, JULIA and MANDELBROT S
ETS, and Power Series" }{TEXT 285 2 "\n\n" }{TEXT 256 35 "Section 4.4 \+
 Power Series Functions" }{TEXT 274 16 "\n\nThe function  " }{XPPEDIT 
18 0 "f(z) = sum(a[n]*z^n, n=0..infinity)" "6#/-%\"fG6#%\"zG-%$sumG6$*
&&%\"aG6#%\"nG\"\"\")F'F/F0/F/;\"\"!%)infinityG" }{TEXT 281 14 "  is c
alled a " }{TEXT 257 12 "power series" }{TEXT 282 2 ". " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 256 12 "Theorem 4.14" }{TEXT -1 17 "   Suppose
 that  " }{XPPEDIT 18 0 "f(z) = Sum(c[n]*(z-alpha)^n,n = 0 .. infinity
);" "6#/-%\"fG6#%\"zG-%$SumG6$*&&%\"cG6#%\"nG\"\"\"),&F'F0%&alphaG!\"
\"F/F0/F/;\"\"!%)infinityG" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Then the set of points " }
{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 58 " for which the series conve
rges is one of the following:  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 287 2 "i." }{TEXT -1 20 "  The single point  " }
{XPPEDIT 18 0 "z = alpha;" "6#/%\"zG%&alphaG" }{TEXT -1 4 ".\n\011\n" 
}{TEXT 288 3 "ii." }{TEXT -1 12 "  The disk  " }{XPPEDIT 18 0 "D[rho](
alpha)" "6#-&%\"DG6#%$rhoG6#%&alphaG" }{TEXT -1 5 "  =  " }{XPPEDIT 
18 0 "\{`z: `*abs(z-alpha) < rho\};" "6#<#2*&%$z:~G\"\"\"-%$absG6#,&%
\"zGF'%&alphaG!\"\"F'%$rhoG" }{TEXT -1 4 " ,  " }}{PARA 0 "" 0 "" 
{TEXT -1 65 "      along with part (either none, some, or all) of the \+
circle  " }{XPPEDIT 18 0 "C[rho](alpha);" "6#-&%\"CG6#%$rhoG6#%&alphaG
" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "\{`z: `*abs(z-alpha) <= rho\};" 
"6#<#1*&%$z:~G\"\"\"-%$absG6#,&%\"zGF'%&alphaG!\"\"F'%$rhoG" }{TEXT 
-1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 2 "\011\n" }{TEXT 289 4 "iii." }
{TEXT -1 29 "  The entire complex plane.  " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 65 "Another way to phrase case (ii) is to say that the power \+
series  " }{XPPEDIT 18 0 "f(z) = Sum(c[n]*(z-alpha)^n,n = 0 .. infinit
y);" "6#/-%\"fG6#%\"zG-%$SumG6$*&&%\"cG6#%\"nG\"\"\"),&F'F0%&alphaG!\"
\"F/F0/F/;\"\"!%)infinityG" }{TEXT -1 17 "   converges if  " }
{XPPEDIT 18 0 "abs(z-alpha) < rho;" "6#2-%$absG6#,&%\"zG\"\"\"%&alphaG
!\"\"%$rhoG" }{TEXT -1 20 " , and diverges if  " }{XPPEDIT 18 0 "rho <
 abs(z-alpha);" "6#2%$rhoG-%$absG6#,&%\"zG\"\"\"%&alphaG!\"\"" }{TEXT 
-1 4 ".   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 19 "We call the number " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT 
-1 6 " the r" }{TEXT 286 20 "adius of convergence" }{TEXT -1 165 " of \+
the power series. If we are in case (i), we say that the radius of con
vergence is zero, and that the radius of convergence is infinity if we
 are in case (iii).  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 36 "T
heorem 4.14 (Radius of convergence)" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Given the power series \+
function  " }{XPPEDIT 18 0 "Sum(c[n]*(z-alpha)^n,n = 0 .. infinity);" 
"6#-%$SumG6$*&&%\"cG6#%\"nG\"\"\"),&%\"zGF+%&alphaG!\"\"F*F+/F*;\"\"!%
)infinityG" }{TEXT -1 17 " ,  we can find  " }{XPPEDIT 18 0 "rho;" "6#
%$rhoG" }{TEXT -1 62 ",  its radius of convergence, by any of the foll
owing methods:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 290 2 "i." }{TEXT -1 24 "  Cauchy's root test:   " }{XPPEDIT 18 
0 "rho = 1/Limit(abs(c[n])^`1/n`,n = infinity);" "6#/%$rhoG*&\"\"\"F&-
%&LimitG6$)-%$absG6#&%\"cG6#%\"nG%$1/nG/F1%)infinityG!\"\"" }{TEXT -1 
37 "    .  (Provided the limit exists.)  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 291 3 "ii." }{TEXT -1 29 "  d'Alembert's \+
ratio test:   " }{XPPEDIT 18 0 "rho = 1/Limit(abs(c[n+1])/abs(c[n]),n \+
= infinity)" "6#/%$rhoG*&\"\"\"F&-%&LimitG6$*&-%$absG6#&%\"cG6#,&%\"nG
F&F&F&F&-F,6#&F/6#F2!\"\"/F2%)infinityGF7" }{TEXT -1 34 " .  (Provided
 the limit exists.)  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT 279 1 "\n" }{TEXT 256 23 "Example 4.21,
 Page 153." }{TEXT 275 47 " Find the radius of convergence of the seri
es  " }{XPPEDIT 18 0 "f(z) = sum(((n+2)/(3n+1) )^n*(z - 4 )^n, n=0..in
finity)" "6#/-%\"fG6#%\"zG-%$sumG6$*&)*&,&%\"nG\"\"\"\"\"#F0F0,&*&\"\"
$F0F/F0F0F0F0!\"\"F/F0),&F'F0\"\"%F5F/F0/F/;\"\"!%)infinityG" }{TEXT 
266 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 381 "c:='c': C:='C': n:='n
': P:='P':\nC := n -> ((n+2)/(3*n+1))^n:\n### WARNING: calls to `C` fo
r generating C code should be replaced by codegen[C]\n`The general ter
m is  `, c[n]= C(n); ` `;\n`The  n-th  root is:`;\n### WARNING: calls \+
to `C` for generating C code should be replaced by codegen[C]\nP := C(
n)^(1/n):\nabs(c[n])^(1/n) = P;\nP := simplify(P, assume=positive):\na
bs(c[n])^(1/n) = P;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 260 73 "Use the root test and find the li
mit and then the radius of convergence  " }{XPPEDIT 18 0 "R" "6#%\"RG
" }{TEXT 273 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "L:='L': R:
='R':\nL0 := limit(P, n=infinity):\nR := 1/L0:\n### WARNING: calls to \+
`C` for generating C code should be replaced by codegen[C]\nc[n] = C(n
);\nabs(c[n])^(1/n) = P;\n`L = `, limit(abs(c[n])^(1/n), n=infinity) =
 L0;\n`The radius of convergence is  R = `, 1/L = R;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 23 "Exam
ple 4.23, Page 153." }{TEXT 276 48 "  Find the radius of convergence o
f the series  " }{XPPEDIT 18 0 "f(z)  =  sum(z^n/n!, n=0..infinity)" "
6#/-%\"fG6#%\"zG-%$sumG6$*&)F'%\"nG\"\"\"-%*factorialG6#F-!\"\"/F-;\"
\"!%)infinityG" }{TEXT 267 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
440 "c:='c': C:='C': n:='n': Q:='Q':\nC := n ->1/n!:\n### WARNING: cal
ls to `C` for generating C code should be replaced by codegen[C]\n`The
 general term is  `, c[n] = C(n); ` `;\n`The ratio of consecutive term
s is:`;\n### WARNING: calls to `C` for generating C code should be rep
laced by codegen[C]\n### WARNING: calls to `C` for generating C code s
hould be replaced by codegen[C]\nQ := C(n+1)/C(n):\nc[n+1]/c[n] = Q;\n
Q := simplify(Q):\nc[n+1]/c[n] = Q;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 261 74 "Use the ratio test a
nd find the limit and then the radius of convergence  " }{XPPEDIT 18 
0 "R" "6#%\"RG" }{TEXT 268 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
181 "L:='L': R:='R':\nL0 := limit(Q, n=infinity):\nR := limit(1/Q, n=i
nfinity):\nc[n+1]/c[n] = Q;\n`L = `, limit(c[n+1]/c[n], n=infinity) = \+
L0;\n`The radius of convergence is  R = `, 1/L = R;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 262 48 "The \+
sum of the infinite series can be computed.\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 218 "S:='S':\n`The general term is  `, c[n] = C(n)*z^n;\n
`A partial sum is:`;\nS[5](z) = sum(C(n)*z^n, n=0..5); ` `;\n`The sum \+
of the infinite series is:`;\n`f(z) = `, Sum(C(n)*z^n, n=0..infinity) \+
= sum(C(n)*z^n, n=0..infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 23 "Example 4.24, Page 156." 
}{TEXT 277 16 "   Show that    " }{XPPEDIT 18 0 "1/(1-z)^2 = sum((n+1)
*z^n,n=0..infinity)" "6#/*&\"\"\"F%*$,&F%F%%\"zG!\"\"\"\"#F)-%$sumG6$*
&,&%\"nGF%F%F%F%)F(F0F%/F0;\"\"!%)infinityG" }{TEXT 269 24 "  .\nUse t
he fact that   " }{XPPEDIT 18 0 "1/(1-z) = sum(z^n,n=0..infinity)" "6#
/*&\"\"\"F%,&F%F%%\"zG!\"\"F(-%$sumG6$)F'%\"nG/F-;\"\"!%)infinityG" }
{TEXT 270 39 "  and perform termwise differentiation." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "s:='s': S:='S':
 S1:='S1': z:='z': Z:='Z':\ns  := sum(Z^n, n=0..infinity):\nS  := z ->
 subs(Z=z, s):\nS1 := z -> subs(Z=z, diff(S(Z),Z)):\n`S(z) ` = S(z);\n
`S'(z) = f(z) ` = S1(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT 263 28 "Or sum the series directly.\n" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 590 "c:='c': C:='C': n:='n': S:='S': \+
z:='z':\nC := n ->(n+1):\n### WARNING: calls to `C` for generating C c
ode should be replaced by codegen[C]\n`The general term is  `, c[n] = \+
C(n)*z^n;\n`A partial sum is:`;\n### WARNING: calls to `C` for generat
ing C code should be replaced by codegen[C]\nS[5](z) = sum(C(n)*z^n, n
=0..5); ` `;\n`The sum of the infinite series is:`;\n### WARNING: call
s to `C` for generating C code should be replaced by codegen[C]\n### W
ARNING: calls to `C` for generating C code should be replaced by codeg
en[C]\n`f(z) = `, Sum(C(n)*z^n, n=0..infinity) = sum(C(n)*z^n, n=0..in
finity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 
"" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 23 "Example 4.25, \+
Page 156." }{TEXT 280 46 "  Termwise differentiate the Bessel function
  " }{XPPEDIT 18 0 "J[0](z)" "6#-&%\"JG6#\"\"!6#%\"zG" }{TEXT 271 12 "
  and  get  " }{XPPEDIT 18 0 "- J[1](z)" "6#,$-&%\"JG6#\"\"\"6#%\"zG!
\"\"" }{TEXT 272 3 " .\n" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 298 "J:='J': s:='s': S:='S': t:='t': T:='T': z:='z':\nS :
= series(BesselJ(0,z), z=0, 14):\nT := series(BesselJ(1,z), z=0, 13):
\ns := convert(S, polynom):\nt := convert(T, polynom):\ns1 := diff(s, \+
z):\n`S(z) ` = s, `...`;\n`T(z) ` = t, `...`;;\n`S '(z) ` = s1, `...`;
;\n`Does  T(z) = - S '(z) ?`;\nevalb(t = - s1);" }}}{EXCHG {PARA 257 "
" 0 "" {TEXT 264 66 "Or use Maple's built in Bessel functions and use \+
differentiation.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "J[0](z) = Bes
selJ(0,z);\ndiff(J[0](z),z) = diff(BesselJ(0,z) , z);" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 265 19 "End of Section 4.4." }}}}{MARK "0 0 0
" 19 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }