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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 280 144 "\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 279 197 "\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 
278 1 "\n" }{TEXT 256 31 "CHAPTER 5  ELEMENTARY FUNCTIONS" }{TEXT 273 
2 "\n\n" }{TEXT 256 45 "Section 5.1  The Complex Exponential Function
" }{TEXT 274 3 "\n\n " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " \+
  How should complex-valued functions such as " }{XPPEDIT 18 0 "exp(z)
;" "6#-%$expG6#%\"zG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "log(z);" "6#-%$
logG6#%\"zG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(z);" "6#-%$sinG6#%\"
zG" }{TEXT -1 94 ", etc., be defined? Clearly, any responsible definit
ion should satisfy the following criteria:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 2 "i." }{TEXT -1 119 "  The functi
ons so defined must give the same values as the corresponding function
s for real variables when the number " }{XPPEDIT 18 0 "z;" "6#%\"zG" }
{TEXT -1 20 " is a real number.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 291 3 "ii." }{TEXT -1 131 " As far as possible, \+
the properties of these new functions must correspond with their real \+
counterparts. For example, we would want" }}{PARA 0 "" 0 "" {TEXT -1 
7 "       " }{XPPEDIT 18 0 "exp(z[1]+z[2]) = exp(z[1])*exp(z[2]);" "6#
/-%$expG6#,&&%\"zG6#\"\"\"F+&F)6#\"\"#F+*&-F%6#&F)6#F+F+-F%6#&F)6#F.F+
" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 35 "to be valid regardless of whether  " }{XPPEDIT 18 0 "z[1]
;" "6#&%\"zG6#\"\"\"" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "z[2];" "6#
&%\"zG6#\"\"#" }{TEXT -1 23 "  were real or complex." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "   These requirements \+
may seem like a tall order to fill. There is a procedure, however, tha
t offers promising results. It is to put the expansion of the real fun
ctions " }{XPPEDIT 18 0 "exp(z);" "6#-%$expG6#%\"zG" }{TEXT -1 2 ", " 
}{XPPEDIT 18 0 "log(z);" "6#-%$logG6#%\"zG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "sin(z);" "6#-%$sinG6#%\"zG" }{TEXT -1 94 ", etc., as po
wer series into complex form. This will be our stategy for the next fe
w sections." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "   Recall that \+
the real exponential function can be represented by the power series \+
" }{XPPEDIT 18 0 "exp(x) = Sum(x^n/n!,n = 0 .. infinity);" "6#/-%$expG
6#%\"xG-%$SumG6$*&)F'%\"nG\"\"\"-%*factorialG6#F-!\"\"/F-;\"\"!%)infin
ityG" }{TEXT -1 62 " .  Thus it is only natural to define the complex \+
exponential " }{XPPEDIT 18 0 "exp(z);" "6#-%$expG6#%\"zG" }{TEXT -1 
18 ", also written as " }{XPPEDIT 18 0 "`exp(z)`" "6#%'exp(z)G" }
{TEXT -1 23 ", in the following way:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 49 "Definition 5.1:  The complex e
xponential function" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 292 6 "      " }{XPPEDIT 18 0 "exp(z)" "6#-%
$expG6#%\"zG" }{TEXT 283 5 "  =  " }{XPPEDIT 18 0 "`exp(z)`;" "6#%'exp
(z)G" }{TEXT 284 6 "  =   " }{XPPEDIT 18 0 "sum(z^n/n!,n = 0 .. infini
ty);" "6#-%$sumG6$*&)%\"zG%\"nG\"\"\"-%*factorialG6#F)!\"\"/F);\"\"!%)
infinityG" }{TEXT 282 3 "  \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 79 "Clearly this definition agrees with that of the real exponentia
l function when " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 20 " is a re
al number.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 103 "Load Maple's \+
 \"conformal mapping\" procedure.\nMake sure this is done only ONCE du
ring a Maple  session.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl
ots):" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 256 8 "Theorem." }{TEXT 275 72 "   The exponential fun
ction is a solution to the differential equation  " }{XPPEDIT 18 0 "`f
 '(z)` = f(z)" "6#/%'f~'(z)G-%\"fG6#%\"zG" }{TEXT 269 30 "  with the i
nitial condition  " }{XPPEDIT 18 0 "f(0) = 1" "6#/-%\"fG6#\"\"!\"\"\"
" }{TEXT 270 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "f:='f': z:
='z': Z:='Z':\nf := z -> exp(z):\nf1 := z -> subs(Z=z, diff(f(Z),Z)):
\n`f(z) ` = f(z);\n`f '(z) ` = f1(z);\n`f(0) ` = f(0);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 57 "
To see its real and imaginary parts, use complex expand.\n" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 84 "x:='x': y:='y': Z:='Z':\nZ := x + I*y:\n`ex
p(z) ` = exp(Z);\n`exp(z) ` = evalc(exp(Z));" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 46 "Now verify tha
t the \"rules of exponents\" hold." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 483 "x:='x': y:='y': z:='z': \nw1 := ev
alc(exp(x[1]+I*y[1])):\nw2 := evalc(exp(x[2]+I*y[2])):\nw3 := w1*w2: e
xp(z[1])*exp(z[2]) = w3; ` `;\nw3 := evalc(w1*w2): exp(z[1])*exp(z[2])
 = w3; ` `;\nw3 := expand(w3): exp(z[1])*exp(z[2]) = w3; ` `;\nz4 := (
x[1]+I*y[1])+(x[2]+I*y[2]):\nw4 := exp(z4): exp(z[1]*z[2]) = w4; ` `;
\nw4 := evalc(exp(z4)): exp(z[1]*z[2]) = w4; ` `;\nw4 := expand(w4, tr
ig): exp(z[1]*z[2]) = w4; ` `;\n`Does  exp(z1 z2) = exp(z1) exp(z2)  ?
`;\nw3 = w4; ` `;\nevalb(expand(w3 = w4));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 256 11 "Theorem 5.1" }{TEXT 285 3 "   " }{TEXT -1 14 "The func
tion  " }{XPPEDIT 18 0 "`exp(z)` = exp(z);" "6#/%'exp(z)G-%$expG6#%\"z
G" }{TEXT -1 36 "  is an entire function satisfying:\n" }}{PARA 0 "" 
0 "" {TEXT 293 2 "i." }{TEXT -1 4 "    " }{XPPEDIT 18 0 "`exp'(z) = ex
p(z)` = exp(z)" "6#/%1exp'(z)~=~exp(z)G-%$expG6#%\"zG" }{TEXT -1 32 " \+
      (Using Leibniz notation, " }{XPPEDIT 18 0 "Diff(exp(z),z) = exp(
z);" "6#/-%%DiffG6$-%$expG6#%\"zGF*-F(6#F*" }{TEXT -1 5 ".)  \n" }}
{PARA 0 "" 0 "" {TEXT 294 3 "ii." }{TEXT -1 3 "   " }{TEXT 286 4 "exp(
" }{XPPEDIT 18 0 "z[1]+z[2];" "6#,&&%\"zG6#\"\"\"F'&F%6#\"\"#F'" }
{TEXT 287 8 ") = exp(" }{XPPEDIT 18 0 "z[1];" "6#&%\"zG6#\"\"\"" }
{TEXT 288 5 ")exp(" }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\"#" }{TEXT 
289 1 ")" }{TEXT -1 17 "       (That is, " }{XPPEDIT 18 0 "exp(z[1]+z[
2]) = exp(z[1])*exp(z[2])" "6#/-%$expG6#,&&%\"zG6#\"\"\"F+&F)6#\"\"#F+
*&-F%6#&F)6#F+F+-F%6#&F)6#F.F+" }{TEXT -1 3 ".)\n" }}{PARA 0 "" 0 "" 
{TEXT 295 5 "iii. " }{TEXT -1 4 " If " }{XPPEDIT 18 0 "theta;" "6#%&th
etaG" }{TEXT -1 26 " is a real number, then   " }{XPPEDIT 18 0 "exp(i*
theta) = cos(theta)+i*sin(theta);" "6#/-%$expG6#*&%\"iG\"\"\"%&thetaGF
),&-%$cosG6#F*F)*&F(F)-%$sinG6#F*F)F)" }{TEXT -1 4 ".   " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 256 22 "Example 5.1, Page 164." }{TEXT 276 15 
"   The points  " }{XPPEDIT 18 0 "z[n] = 5/4 + i*(11*pi/6 + 2*n*pi)" "
6#/&%\"zG6#%\"nG,&*&\"\"&\"\"\"\"\"%!\"\"F+*&%\"iGF+,&*(\"#6F+%#piGF+
\"\"'F-F+*(\"\"#F+F'F+F3F+F+F+F+" }{TEXT 267 69 "  for \nn =..., -2, -
1, 0, 1, 2, ...  are mapped onto a single point  " }{XPPEDIT 18 0 "w[0
] = exp(z[n])" "6#/&%\"wG6#\"\"!-%$expG6#&%\"zG6#%\"nG" }{TEXT 268 3 "
 .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Z0 := 5/4 + I*11*Pi/6:\nfor
 k from 0 to 5 do\n  exp(Z0+I*2*Pi*k)=evalc(exp(Z0+I*2*Pi*k)); \nod;" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 22 "Example 5.2, Page 166." 
}{TEXT 277 33 "   Show that the transformation  " }{XPPEDIT 18 0 "w = \+
exp(z)" "6#/%\"wG-%$expG6#%\"zG" }{TEXT 271 23 "  maps the rectangle  \+
\n" }{XPPEDIT 18 0 "`R = \{(x,y):  a <= x <= b and c <= y <= d\}`" "6#
%JR~=~|fr(x,y):~~a~<=~x~<=~b~and~c~<=~y~<=~d|hrG" }{TEXT 272 119 "  on
to a portion of an annular region bounded by two rays.\nFor illustrati
on we use   R = \{(x,y):  -1 <= x <= 1  and  - " }{TEXT 262 1 "p" }
{TEXT 263 12 "/3 <= y <=  " }{TEXT 264 1 "p" }{TEXT 265 6 "/4\} .\n" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "f:='f': z:='z':\nf := z -> exp(z)
:\n`f(z) ` = f(z);\nconformal(f(z), z=-1-I*Pi/3..1+I*Pi/4,\n  title=`w
 = exp(z)`,\n  grid=[8,8],numxy=[50,50],\n  scaling=constrained,\n  la
bels=[`u`,`v `],\n  tickmarks=[3,5],\n  view=[0..2.8,-2.4..2.0]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 266 19 "End of Section 5.1." }}}}{MARK "0
 0 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 
}