{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "Define" -1 256 "Times" 1 12 163 163 163 0 1 1 0 0 0 0 0 0 0 
1 }{CSTYLE "Emphasis" -1 257 "Times" 1 12 128 0 128 1 0 0 0 0 0 0 0 0 
0 1 }{CSTYLE "normal C" -1 258 "Times" 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 
1 }{CSTYLE "Menu" -1 259 "" 0 0 163 163 163 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 260 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
261 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 265 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
266 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 270 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
271 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 275 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
276 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 280 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
281 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 10 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 1 10 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 
{CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Mona
co" 1 9 255 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "R3 Font 0" -1 257 1 {CSTYLE "" -1 -1 "Geneva" 1 12 0 0 0 1 2 
2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" 
-1 258 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }
1 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 257 "" 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 257 "" 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 257 "" 0 "" {TEXT 
275 1 "\n" }{TEXT 256 31 "CHAPTER 5  ELEMENTARY FUNCTIONS" }{TEXT 270 
2 "\n\n" }{TEXT 256 30 "Section 5.3  Complex Exponents" }{TEXT 271 1 "
\n" }}{PARA 0 "" 0 "" {TEXT -1 144 "    In Section 1.5 we indicated th
at the complex numbers are complete in the sense that it is possible t
o make sense out of expressions such as " }{XPPEDIT 18 0 "sqrt(1+i);" 
"6#-%%sqrtG6#,&\"\"\"F'%\"iGF'" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "(-1
)^i;" "6#),$\"\"\"!\"\"%\"iG" }{TEXT -1 219 " left without appealing t
o a number system beyond the framework of complex numbers. We will do \+
this by taking note of some rudimentary properties of the complex expo
nential and logarithm, and then using our imagination." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 33 "Definiti
on 5.4:  Complex exponent" }{TEXT 284 3 "   " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "c;" "6#%\"
cG" }{TEXT -1 32 " be a complex number. We define " }{XPPEDIT 18 0 "z^
c;" "6#)%\"zG%\"cG" }{TEXT -1 5 " as  " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 6 "      " }{XPPEDIT 18 0 "z^c;" "6#)%\"z
G%\"cG" }{TEXT -1 5 "  =  " }{TEXT 285 4 "exp(" }{XPPEDIT 18 0 "c*log(
z);" "6#*&%\"cG\"\"\"-%$logG6#%\"zGF%" }{TEXT 286 1 ")" }{TEXT -1 4 ".
   " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 
"" {TEXT 276 1 "\n" }{TEXT 256 22 "Example 5.7, Page 178." }{TEXT 272 
2 "  " }{TEXT 256 3 "(a)" }{TEXT 277 31 "  Find the principal value of
  " }{XPPEDIT 18 0 "sqrt(1 + i)" "6#-%%sqrtG6#,&\"\"\"F'%\"iGF'" }
{TEXT 263 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "c:='c': C:='C
': w:='w': z:='z': Z:='Z':\nZ := 1 + I:\nC := 1/2:\nz = Z;\nc = C;\nw \+
:= exp(C*log(Z)):\nz^c = w;\nw := evalc(w):\nz^c = w;\nw := exp(C*eval
c(log(Z))):\nz^c = w;\nw := evalf(w):\nz^c = w;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 3 "(b)" }
{TEXT 278 31 "  Find the principal value of  " }{XPPEDIT 18 0 "(-1)^i
" "6#),$\"\"\"!\"\"%\"iG" }{TEXT 264 3 " .\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 128 "c:='c': C:='C': w:='w': z:='z': Z:='Z':\nZ := -1:\nC
 := I:\nw := Z^C:\nz^c = w;\nw := exp(C*log(Z)):\nz^c = w;\nw := evalf
(w):\nz^c = w;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 256 22 "Example 5.8, Page 179." }{TEXT 273 22 
"  Find the values of  " }{XPPEDIT 18 0 "2^(1/9 + i/50)" "6#)\"\"#,&*&
\"\"\"F'\"\"*!\"\"F'*&%\"iGF'\"#]F)F'" }{TEXT 265 27 " .\nThe principa
l value is:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "c:='c': C:='C': w
:='w': z:='z': Z:='Z':\nZ := 2:\nC := 1/9 + I/50:\nw := Z^C:\nz^c = w;
\nw := exp(C*log(Z)):\nz^c = w;\nw := evalc(w):\nz^c = w;\nw := evalf(
w):\nz^c = w;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 260 29 "Some of the other values of  " }
{XPPEDIT 18 0 "2^(1/9 + i/50)" "6#)\"\"#,&*&\"\"\"F'\"\"*!\"\"F'*&%\"i
GF'\"#]F)F'" }{TEXT 266 7 "  are:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
162 "n:='n': z:='z': Z:='Z':\nZ := n -> evalf(exp((1/9+I/50)*(log(2)+2
*Pi*I*n))):\nz[n] = exp((1/9+I/50)*(log(2)+2*Pi*I*n)); ` `;\nfor n fro
m -5 to 5 do  print(Z(n))  od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 222 "pts := [[ Re(Z(j)), Im(Z(j))] $j=-9..9]:\nplot(pts, x=-2..3, \+
\n  title=`Values of  2^(1/9 + I/50).`,\n  scaling=constrained,\n  lab
els=[` x`,`y `],\n  style=point,symbol=circle,\n  tickmarks=[5,4],\n  \+
view=[-2.0..2.6,-1.5..2.6]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 22 "Example 5.9, Page 180." }
{TEXT 274 2 "  " }{TEXT 256 3 "(a)" }{TEXT 279 22 "  Find the values o
f  " }{XPPEDIT 18 0 "(i^2)^i" "6#)*$%\"iG\"\"#F%" }{TEXT 267 29 " .  T
he principal values is:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "c:='c
': C:='C': w:='w': z:='z': Z:='Z':\nZ := I^2:\nC := I:\nw := Z^C:\nz^c
 = w;\nw := exp(C*log(Z)):\nz^c = w;\nw := evalf(w):\nz^c = w;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 256 3 "(b)" }{TEXT 280 22 "  Find the values of  " }{XPPEDIT 18 
0 "(i)^(2i)" "6#)%\"iG*&\"\"#\"\"\"F$F'" }{TEXT 268 29 " . \nThe princ
ipal values is:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "c:='c': C:='C
': w:='w': z:='z': Z:='Z':\nZ := I:\nC := 2*I:\nw := Z^C:\nz^c = w;\nw
 := exp(C*log(Z)):\nz^c = w;\nw := evalf(w):\nz^c = w;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 261 
127 "Remark. The principal value is the same, but the general values a
re different.\nSome of the other values are given in the sets:\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "Z1 := n -> evalf(exp(I*(log(I^2)+2
*Pi*I*n))):\nZ2 := n -> evalf(exp(2*I*(log(I)+2*Pi*I*n))):\n`Portions \+
of the solution sets are:`;\nZ1s := \{[ Re(Z1(j)), Im(Z1(j))] $j=-2..2
\}:\n`(i^2)^i ` = Z1s;\nZ2s := \{[ Re(Z2(j)), Im(Z2(j))] $j=-2..2\}:\n
`i^(2i) ` = Z2s;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT 262 62 "Remark.  The two sets are diffe
rent.  Therefore, in general,  " }{XPPEDIT 18 0 "(z^a)^b <> z^(ab)" "6
#0))%\"zG%\"aG%\"bG)F&%#abG" }{TEXT 269 2 " ." }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 19 "End of Sectio
n 5.3." }{TEXT -1 0 "" }}}}{MARK "0 0 0" 26 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }