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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 282 144 "\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 281 197 "\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 
277 1 "\n" }{TEXT 256 29 "CHAPTER 2   COMPLEX FUNCTIONS" }{TEXT 274 2 
"\n\n" }{TEXT 256 34 "Section 2.5  Branches of Functions" }{TEXT 275 
288 "\n\n   In Section 2.3 we defined the principal square root functi
on and investigated some of its properties. We left some unanswered qu
estions concerning the choices of square roots.  We now look into this
 problem because it is similar to situations involving other elementar
y functions.   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 224 "   In our definition of a function in Section 2.1 we spe
cified that each value of the independent variable in the domain is ma
pped onto one and only one value of the dependent variable.  As a resu
lt, one often talks about a " }{TEXT 284 22 "single-valued function" }
{TEXT -1 23 ", which emphasizes the " }{TEXT 285 8 "only one" }{TEXT 
-1 125 " part of the definition and allows us to distinguish such func
tions from multiple-valued functions, which we now introduce.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 280 8 "\n   Let " }{XPPEDIT 18 0 
"w = f(z);" "6#/%\"wG-%\"fG6#%\"zG" }{TEXT 289 43 " denote a function \+
whose domain is the set " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT 290 28 
" and whose range is the set " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT 
291 3 ".  " }{TEXT -1 0 "" }{TEXT 286 3 "If " }{XPPEDIT 18 0 "w;" "6#%
\"wG" }{TEXT 292 70 "is a value in the range, then there is an associa
ted inverse relation " }{XPPEDIT 18 0 "z = g(w);" "6#/%\"zG-%\"gG6#%\"
wG" }{TEXT 293 28 " that assigns to each value " }{XPPEDIT 18 0 "w;" "
6#%\"wG" }{TEXT 294 26 " the value (or values) of " }{XPPEDIT 18 0 "z;
" "6#%\"zG" }{TEXT 295 4 " in " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 
296 24 " for which the equation " }{XPPEDIT 18 0 "f(z) = w;" "6#/-%\"f
G6#%\"zG%\"wG" }{TEXT 297 14 " holds true.  " }{TEXT -1 0 "" }{TEXT 
287 12 "But unless  " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 298 21 "  t
akes on the value " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT 299 17 " at mo
st once in " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 300 28 ", then the in
verse relation " }{XPPEDIT 18 0 "g;" "6#%\"gG" }{TEXT 301 45 " is nece
ssarily many valued, and we say that " }{XPPEDIT 18 0 "g;" "6#%\"gG" }
{TEXT 302 6 " is a " }{TEXT 303 20 "multivalued function" }{TEXT 304 
3 ".  " }{TEXT -1 0 "" }{TEXT 288 42 "For example, the inverse of the \+
function  " }{XPPEDIT 18 0 "`w = f(z) ` = z^2;" "6#/%*w~=~f(z)~G*$%\"z
G\"\"#" }{TEXT 305 31 "  is the square root function  " }{XPPEDIT 18 
0 "`z = g(w)` = w^`1/2`;" "6#/%)z~=~g(w)G)%\"wG%$1/2G" }{TEXT 306 30 "
.  We see that for each value " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 
307 12 " other than " }{XPPEDIT 18 0 "z = 0;" "6#/%\"zG\"\"!" }{TEXT 
308 17 ", the two points " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT 309 5 
" and " }{XPPEDIT 18 0 "-z;" "6#,$%\"zG!\"\"" }{TEXT 310 32 " are mapp
ed onto the same point " }{XPPEDIT 18 0 "w = f(z)" "6#/%\"wG-%\"fG6#%
\"zG" }{TEXT 311 9 "; hence  " }{XPPEDIT 18 0 "g;" "6#%\"gG" }{TEXT 
312 39 " is in general a two-valued function.  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 493 "   The study of limits, \+
continuity, and derivatives loses all meaning if an arbitrary or ambig
uous assignment of function values is made.  For this reason we did no
t allow multivalued functions to be considered when we defined these c
oncepts.  When working with inverse functions, it is necessary to care
fully specify one of the many possible inverse values when constructin
g an inverse function.  The idea is the same as determining implicit f
unctions in calculus. If the values of a function" }{TEXT 313 2 "  " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 314 2 "  " }{TEXT -1 143 "are dete
rmined by an equation that they satisfy rather than by an explicit for
mula, then we say that the function is defined implicitly or that" }
{TEXT 315 2 "  " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 316 2 "  " }
{TEXT -1 6 "is an " }{TEXT 317 17 "implicit function" }{TEXT -1 67 ". \+
 In the theory of complex variables we study a similar concept.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "   Let " }
{XPPEDIT 18 0 "w = f(z)" "6#/%\"wG-%\"fG6#%\"zG" }{TEXT -1 35 " be a m
ultiple-valued function.  A " }{TEXT 318 6 "branch" }{TEXT -1 3 " of" 
}{TEXT 319 2 "  " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 320 2 "  " }
{TEXT -1 31 "is any single-valued function  " }{XPPEDIT 18 0 "f[0];" "
6#&%\"fG6#\"\"!" }{TEXT -1 23 " that is continuous in " }{TEXT 321 4 "
some" }{TEXT -1 62 " domain (except, perhaps, on the boundary), and at
 each point " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 46 " in the doma
in, assigns one of the values of  " }{XPPEDIT 18 0 "f(z);" "6#-%\"fG6#
%\"zG" }{TEXT -1 3 ".  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 283 102 "Load Maple's  \"conformal mapping\" pro
cedure.\nMake sure this is done only ONCE during a Maple  session." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 22 "E
xample 2.20, Page 79." }{TEXT 276 58 "   Consider the two branches of \+
the square root function:\n" }{XPPEDIT 18 0 "w = f[1](z)" "6#/%\"wG-&%
\"fG6#\"\"\"6#%\"zG" }{TEXT 265 5 "  =  " }{XPPEDIT 18 0 "sqrt(z)  =  \+
sqrt(r)*(cos(theta/2) + i*sin(theta/2))" "6#/-%%sqrtG6#%\"zG*&-F%6#%\"
rG\"\"\",&-%$cosG6#*&%&thetaGF,\"\"#!\"\"F,*&%\"iGF,-%$sinG6#*&F2F,F3F
4F,F,F," }{TEXT 260 10 " ,    and\n" }{XPPEDIT 18 0 "w = f[2](z)" "6#/
%\"wG-&%\"fG6#\"\"#6#%\"zG" }{TEXT 266 5 "  =  " }{XPPEDIT 18 0 "- sqr
t(z)  =  sqrt(r)*(cos((2*pi + theta)/2) + i*sin((2*pi + theta)/2)" "6#
/,$-%%sqrtG6#%\"zG!\"\"*&-F&6#%\"rG\"\"\",&-%$cosG6#*&,&*&\"\"#F.%#piG
F.F.%&thetaGF.F.F6F)F.*&%\"iGF.-%$sinG6#*&,&*&F6F.F7F.F.F8F.F.F6F)F.F.
F." }{TEXT 261 4 " .\n\n" }{TEXT 256 3 "(a)" }{TEXT 278 29 " Find the \+
image of the disk  " }{XPPEDIT 18 0 "abs(z) <= 4" "6#1-%$absG6#%\"zG\"
\"%" }{TEXT 267 10 "  in the  " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 
273 29 "-plane  slit along \nthe ray  " }{XPPEDIT 18 0 "y=0" "6#/%\"yG
\"\"!" }{TEXT 268 3 " , " }{XPPEDIT 18 0 "x<=0" "6#1%\"xG\"\"!" }
{TEXT 269 21 "  under the mapping  " }{XPPEDIT 18 0 "w = sqrt(z)" "6#/
%\"wG-%%sqrtG6#%\"zG" }{TEXT 264 3 " .\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 222 "f1:='f1': z:='z':\nf1 := z -> z^(1/2):\n`f1(z) ` = f
1(z);\nconformal(f1(Re(z)*exp(I*Im(z))), z=0.01-I*3.14..4+I*3.14,\n  t
itle=`w = f1(z) = z^(1/2)`,\n  grid=[13,13], numxy=[50,50],\n  scaling
=constrained,\n  view=[-2..2,-2..2]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 256 3 "(b)" }{TEXT 279 29 " Find the image of the disk  " }
{XPPEDIT 18 0 "abs(z) <= 4" "6#1-%$absG6#%\"zG\"\"%" }{TEXT 270 38 "  \+
in the z-plane slit along \nthe ray  " }{XPPEDIT 18 0 "y=0" "6#/%\"yG
\"\"!" }{TEXT 271 3 " , " }{XPPEDIT 18 0 "x<=0" "6#1%\"xG\"\"!" }
{TEXT 272 21 "  under the mapping  " }{XPPEDIT 18 0 "w =  - sqrt(z)" "
6#/%\"wG,$-%%sqrtG6#%\"zG!\"\"" }{TEXT 262 3 " .\n" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 226 "f2:='f2': z:='z':\nf2 := z -> - z^(1/2):\n`f2(z) `
 = f2(z);\nconformal(f2(Re(z)*exp(I*Im(z))), z=0.01-I*3.14..4+I*3.14,
\n  title=`w = f2(z) = - z^(1/2)`,\n  grid=[13,13], numxy=[50,50],\n  \+
scaling=constrained,\n  view=[-2..2,-2..2]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 263 19 "End of Section 2.5." }}}}{MARK "0
 0 0" 21 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 
}