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{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 
293 1 "\n" }{TEXT 256 31 "CHAPTER 5  ELEMENTARY FUNCTIONS" }{TEXT 288 
2 "\n\n" }{TEXT 256 55 "Section 5.2  Branches of the Complex Logarithm
 Function" }{TEXT 289 5 "\n\n   " }{TEXT -1 33 "In Section 5.1 we show
ed that if " }{XPPEDIT 18 0 "w;" "6#%\"wG" }{TEXT -1 42 " is a nonzero
 complex number the equation " }{XPPEDIT 18 0 "w = exp(z);" "6#/%\"wG-
%$expG6#%\"zG" }{TEXT -1 53 " has infinitely many solutions. Because t
he function " }{XPPEDIT 18 0 "`exp(z)`;" "6#%'exp(z)G" }{TEXT -1 85 " \+
is a many-to-one function, its inverse (the logarithm) is necessarily \+
multivalued.  " }{TEXT 299 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 300 124 "Load Maple's  \"cylinderplot\"  and
 \"conformal mapping\" procedures.\nMake sure this is done only ONCE d
uring a Maple  session.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(p
lots):" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 256 38 "Definition 5.2:  Multivalued loga
rithm" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 5 "For  " }{XPPEDIT 18 0 "z <> 0;" "6#0%\"zG\"\"!" }
{TEXT -1 39 ",  we define the multivalued function  " }{XPPEDIT 18 0 "
log;" "6#%$logG" }{TEXT -1 55 "  as the inverse of the exponential fun
ction; that is, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 5 "     " }{XPPEDIT 18 0 "`log(z) = w`;" "6#%+log(z)~=~wG" }
{TEXT -1 22 "    if and only if    " }{XPPEDIT 18 0 "`z = exp(w)`;" "6
#%+z~=~exp(w)G" }{TEXT -1 3 ".  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 256 49 "Definition 5.3:  Principal value \+
of the logarithm" }{TEXT -1 4 "    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 5 "For  " }{XPPEDIT 18 0 "z <> 0;" "6#0%\"zG
\"\"!" }{TEXT -1 41 ",  we define the single-valued function  " }
{XPPEDIT 18 0 "Log;" "6#%$LogG" }{TEXT -1 45 ",  the principal value o
f the logarithm, by  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 5 "     " }{XPPEDIT 18 0 "`Log(z) = ln|z| + i Arg(z)`;" "6
#%:Log(z)~=~ln|grz|gr~+~i~Arg(z)G" }{TEXT -1 3 ".  " }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 32 "   The domain for the function  " }
{XPPEDIT 18 0 "Log;" "6#%$LogG" }{TEXT -1 51 "  is the set of all nonz
ero complex numbers in the " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 
47 "-plane, and its range is the horizontal strip  " }{XPPEDIT 18 0 "`
\{w: `;" "6#%%|frw:~G" }{XPPEDIT 18 0 "-pi;" "6#,$%#piG!\"\"" }{TEXT 
-1 1 "<" }{XPPEDIT 18 0 "`Im(w)` <= pi;" "6#1%&Im(w)G%#piG" }{XPPEDIT 
18 0 "`\}`;" "6#%\"|hrG" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 
7 "in the " }{XPPEDIT 18 0 "w;" "6#%\"wG" }{TEXT -1 31 "-plane.  We st
ress again that  " }{XPPEDIT 18 0 "Log;" "6#%$LogG" }{TEXT -1 58 "  is
 a single-valued function and corresponds to setting  " }{XPPEDIT 18 
0 "n = 0;" "6#/%\"nG\"\"!" }{TEXT -1 66 "  in the above definition.  A
s we saw in Chapter 2, the function  " }{XPPEDIT 18 0 "Arg;" "6#%$ArgG
" }{TEXT -1 52 "  is discontinuous at each point along the negative " 
}{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 33 "-axis, hence so is the fun
ction  " }{XPPEDIT 18 0 "Log" "6#%$LogG" }{TEXT -1 61 ".  In fact, bec
ause any branch of the multi-valued function  " }{XPPEDIT 18 0 "arg;" 
"6#%$argG" }{TEXT -1 121 "  is discontinuous along some ray, a corresp
onding branch of the logarithm will have a discontinuity along that sa
me ray." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 21 "Exam
ple for Page 170." }{TEXT 290 2 "  " }}{PARA 0 "" 0 "" {TEXT 301 4 "  \+
  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "r:='r': t:='t': u:='u': U:='
U': \nv:='v': V:='V': w:='w': x:='x': y:='y':\nw := log(x + I*y):\n`Lo
g(z) ` = w;\nw := evalc(log(x + I*y)):\n`Log(z) ` = w;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "u := proc(x,y)  ln(x^2+y^2)/2  end
:\nv := proc(x,y)  arctan(y,x)  end:\n`Log(z) ` = u(x,y) + I*v(x,y);\n
U := proc(r,t)  ln(r)  end:\nV := proc(r,t)  t  end:\n`z ` = r*exp(I*t
);\n`Log(z) ` = U(r,t) + I*V(r,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 22 "To plot the function  \+
" }{XPPEDIT 18 0 "U = log(abs(z))" "6#/%\"UG-%$logG6#-%$absG6#%\"zG" }
{TEXT 269 25 "  we need to solve \nfor  " }{XPPEDIT 18 0 "r" "6#%\"rG
" }{TEXT 270 20 "  as a function of  " }{XPPEDIT 18 0 "u" "6#%\"uG" }
{TEXT 271 14 " ,  that is:  " }{XPPEDIT 18 0 "r = exp(u)" "6#/%\"rG-%$
expG6#%\"uG" }{TEXT 272 28 "  and use a \"cylinderplot.\"\n" }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "cylinderplot(e
xp(u) ,t=0..2*Pi,u=-2..1,\n  title=`u(x,y) = ln|z| = ln(r)`,\n  style=
patchnogrid,\n  grid=[50,50],\n  lightmodel=light1,\n  axes=normal);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 22 "Example 5.3, Page 170." 
}{TEXT 291 9 "   Find  " }{XPPEDIT 18 0 "Log(1 + i)" "6#-%$LogG6#,&\"
\"\"F'%\"iGF'" }{TEXT 274 7 "  and  " }{XPPEDIT 18 0 "log(1 + i)" "6#-
%$logG6#,&\"\"\"F'%\"iGF'" }{TEXT 273 3 " .\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 98 "w:='w': z:='z':\nz := 1+I:\n`z ` = z;\nw := log(z):\n
`Log(z) ` = w;\nw := evalc(log(z)):\n`Log(z) ` = w;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 40 "The be
st we can do for the multivalued  " }{XPPEDIT 18 0 "log(1 + i)" "6#-%$
logG6#,&\"\"\"F'%\"iGF'" }{TEXT 275 13 "  is to add  " }{XPPEDIT 18 0 
"i*2*pi*n" "6#**%\"iG\"\"\"\"\"#F%%#piGF%%\"nGF%" }{TEXT 262 3 " .\n" 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "w:='w
': z:='z':\nz := 1+I:\n`z ` = z;\nw := log(z) + I*2*Pi*n:\n`log(z) ` =
 w;\nw := evalc(log(z)) + I*2*Pi*n:\n`log(z) ` = w;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 20 "The tr
ansformation  " }{XPPEDIT 18 0 "w = Log(z)" "6#/%\"wG-%$LogG6#%\"zG" }
{TEXT 277 22 "  maps the punctured  " }{XPPEDIT 18 0 "z" "6#%\"zG" }
{TEXT 276 27 "-plane  slit along the ray\n" }{XPPEDIT 18 0 "y=0" "6#/%
\"yG\"\"!" }{TEXT 278 3 " , " }{XPPEDIT 18 0 "x<=0" "6#1%\"xG\"\"!" }
{TEXT 279 46 " ,  one-to-one and onto the horizontal strip  " }
{XPPEDIT 18 0 "-pi" "6#,$%#piG!\"\"" }{TEXT 264 3 " < " }{XPPEDIT 18 
0 "Im(w)" "6#-%#ImG6#%\"wG" }{TEXT 280 2 "< " }{XPPEDIT 18 0 "pi" "6#%
#piG" }{TEXT 265 10 "  in the  " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT 
281 55 "-plane.\n\nThe following graph is the image of the disk  " }
{XPPEDIT 18 0 "abs(z)<=100" "6#1-%$absG6#%\"zG\"$+\"" }{TEXT 284 22 " \+
 slit along the ray  " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT 
283 3 " , " }{XPPEDIT 18 0 "x<=0" "6#1%\"xG\"\"!" }{TEXT 282 3 " .\n" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "f:='f': z:='z':\nf := z -> log(z
):\n`f(z) ` = f(z);\nconformal(f(Re(z)*exp(I*Im(z))), z=0.01-I*3.14..1
00+I*3.14,\n  title=`w = log(z)`,\n  grid=[13,13],numxy=[13,13],\n  sc
aling=constrained,\n  labels=[`  u`,`v `],\n  tickmarks=[5,7],\n  view
=[-4.7..4.7,-3.2..3.2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 22 "E
xample 5.4, Page 170." }{TEXT 292 3 "   " }{TEXT 256 3 "(a)" }{TEXT 
294 8 "  Find  " }{XPPEDIT 18 0 "Log(-e)" "6#-%$LogG6#,$%\"eG!\"\"" }
{TEXT 285 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "w:='w': z:='z
':\nz := -exp(1):\n`z ` = z;\nw := log(z):\n`Log(z) ` = w;\nw := evalc
(log(z)):\n`Log(z) ` = w;\nw := ln(abs(z)) + I*argument(z):\n`ln|z| + \+
I Arg(z) ` = w;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 256 3 "(b)" }{TEXT 295 8 "  Find  " }{XPPEDIT 
18 0 "Log(-1)" "6#-%$LogG6#,$\"\"\"!\"\"" }{TEXT 286 3 " .\n" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 120 "w:='w': z:='z':\nz := -1:\n`z ` = z;\nw \+
:= log(z):\n`Log(z) ` = w;\nw := ln(abs(z)) + I*argument(z):\n`ln|z| +
 I Arg(z) ` = w;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 320 1 "\n" }{TEXT 256 22 "Example 5.5, Page 171.
" }{TEXT 321 17 "    Verify that  " }{XPPEDIT 18 0 "Log(z[1]*z[2]) <> \+
Log(z[1]) + Log(z[2])" "6#0-%$LogG6#*&&%\"zG6#\"\"\"F+&F)6#\"\"#F+,&-F
%6#&F)6#F+F+-F%6#&F)6#F.F+" }{TEXT 319 3 " .\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 441 "z:='z':\nz1 := -sqrt(3) + I:  z[1] = z1;\nz2 := - 1 \+
+ I*sqrt(3):  z[2] = z2;\nw1 := log(z1) + log(z2):\nlog(z1) = evalc(lo
g(z1));\nlog(z2) = evalc(log(z2));\nLog(z[1]) + Log(z[2]) = w1;\nw1 :=
evalc(w1):\nLog(z[1]) + Log(z[2]) = w1; ` `;\nz3 := z1*z2:\nw3 := log(
z3):\nLog(z[1]*z[2]) = w3;\nz4 := evalc(z3):\nw4 := log(z4):\nLog(z[1]
*z[2]) = w4;\nw4 := evalc(w4):\nLog(z[1]*z[2]) = w4; ` `;\n`Does  Log(
z1) + Log(z2) = Log(z1 z2)  ?`;\nw1 = w4;\nevalb(w1 = w4);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 256 53 "Theorem 5.2 (A fact concerning princip
al logarithm)  " }{TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 14 "The identity  " }{TEXT 302 4 "Log(" }
{XPPEDIT 18 0 "z[1]*z[2];" "6#*&&%\"zG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT 
303 8 ") = Log(" }{XPPEDIT 18 0 "z[1];" "6#&%\"zG6#\"\"\"" }{TEXT 304 
8 ") + Log(" }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\"#" }{TEXT 305 1 ")
" }{TEXT -1 32 "   holds true if and only if    " }{XPPEDIT 18 0 "-pi;
" "6#,$%#piG!\"\"" }{TEXT -1 3 " < " }{TEXT 306 4 "Arg(" }{XPPEDIT 18 
0 "z[1];" "6#&%\"zG6#\"\"\"" }{TEXT 307 8 ") + Arg(" }{XPPEDIT 18 0 "z
[2];" "6#&%\"zG6#\"\"#" }{XPPEDIT 18 0 "`)` <= pi;" "6#1%\")G%#piG" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 51 "Theorem \+
5.3 (Facts about the multivalued logarithm)" }{TEXT -1 3 "   " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }
{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "z[2]" "6#&%\"zG6#\"\"#" }{TEXT -1 54 " be nonzero complex number
s. The multivalued function " }{XPPEDIT 18 0 "log" "6#%$logG" }{TEXT 
-1 46 " obeys the familiar properties of logarithms: " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 322 2 "i." }{TEXT -1 7 "     \+
  " }{TEXT 308 4 "log(" }{XPPEDIT 18 0 "z[1]*z[2];" "6#*&&%\"zG6#\"\"
\"F'&F%6#\"\"#F'" }{TEXT 309 8 ") = log(" }{XPPEDIT 18 0 "z[1];" "6#&%
\"zG6#\"\"\"" }{TEXT 310 8 ") + log(" }{XPPEDIT 18 0 "z[2];" "6#&%\"zG
6#\"\"#" }{TEXT 311 5 ") ,  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 323 3 "ii." }{TEXT -1 6 "      " }{TEXT 312 4 "Log(" }
{XPPEDIT 18 0 "z[1]/z[2];" "6#*&&%\"zG6#\"\"\"F'&F%6#\"\"#!\"\"" }
{TEXT 313 8 ") = Log(" }{XPPEDIT 18 0 "z[1];" "6#&%\"zG6#\"\"\"" }
{TEXT 314 8 ") - Log(" }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\"#" }
{TEXT 315 11 ") ,  and   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 324 5 "iii. " }{TEXT -1 4 "    " }{TEXT 316 4 "Log(" }
{XPPEDIT 18 0 "1/z;" "6#*&\"\"\"F$%\"zG!\"\"" }{TEXT 317 11 ") =  - Lo
g(" }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT 318 5 ").   " }{TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 256 26 "Page 172,  Properties of  " }{XPPEDIT 18 0 "Log(z)" "6#-
%$LogG6#%\"zG" }{TEXT 256 2 " ." }{TEXT 296 1 "\n" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 159 "z1 := x1 + I*y1:\nz2 := x2 + I*y2:\nw1 := log(z1):
\nw2 := log(z2):\nv3 := v1 + v2:\n`Log(z1) + Log(z2) ` = w3;\nw3 := ev
alc(w1 + w2):\n`Log(z1) + Log(z2) ` = w3; ` `;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 235 "z3 := z1*z2:\nz4 := evalc(z1*z2):\nw4 := log(
z3):\n`Log(z1 z2) ` = w4;\nw4 := log(z4):\n`Log(z1 z2) ` = w4;\nw4 := \+
evalc(w4):\n`Log(z1 z2) ` = w4;\nw4 := simplify(w4):\n`Log(z1 z2) ` = \+
w4; ` `;\n`Does  Log(z1 z2) = Log(z1) + Log(z1) ?`;\nw4 = w3;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
266 25 "Therefore  we will have  " }{XPPEDIT 18 0 "Log(z[1]*z[2]) = Lo
g(z[1]) + Log(z[2])" "6#/-%$LogG6#*&&%\"zG6#\"\"\"F+&F)6#\"\"#F+,&-F%6
#&F)6#F+F+-F%6#&F)6#F.F+" }{TEXT 267 16 "  provided that " }{XPPEDIT 
18 0 "arctan(x[1]*y[2] + y[1]*x[2] , x[1]*x[2] - y[1]*y[2]) =  arctan(
y[1], x[1]) +  arctan(y[2] , x[2])" "6#/-%'arctanG6$,&*&&%\"xG6#\"\"\"
F,&%\"yG6#\"\"#F,F,*&&F.6#F,F,&F*6#F0F,F,,&*&&F*6#F,F,&F*6#F0F,F,*&&F.
6#F,F,&F.6#F0F,!\"\",&-F%6$&F.6#F,&F*6#F,F,-F%6$&F.6#F0&F*6#F0F," }
{TEXT 287 53 " .\nThose pesky  arctan's  must be taken seriously !  " 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 268 19 "End of Section 5.2." }}}}{MARK "0 0 0" 31 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }