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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 283 144 "\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 282 197 "\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 
279 1 "\n" }{TEXT 256 29 "CHAPTER 2   COMPLEX FUNCTIONS" }{TEXT 276 1 
"\n" }{TEXT 256 27 "Section 2.3  The Mappings  " }{XPPEDIT 18 0 "w = z
^n" "6#/%\"wG)%\"zG%\"nG" }{TEXT 256 7 "  and  " }{XPPEDIT 18 0 "w = z
^`1/n`;" "6#/%\"wG)%\"zG%$1/nG" }{TEXT 260 20 "\n\n     The mapping  \+
" }{XPPEDIT 18 0 "w  =   z^2" "6#/%\"wG*$%\"zG\"\"#" }{TEXT 267 6 "  o
r  " }{XPPEDIT 18 0 "w  =   x^2 - y^2  + i *2*x*y" "6#/%\"wG,(*$%\"xG
\"\"#\"\"\"*$%\"yGF(!\"\"**%\"iGF)F(F)F'F)F+F)F)" }}{PARA 0 "" 0 "" 
{TEXT 266 55 "can be expressed in polar coordinates by the function  \+
" }{XPPEDIT 18 0 "f(z)  =  r^2 * exp(i *2 *theta)" "6#/-%\"fG6#%\"zG*&
%\"rG\"\"#-%$expG6#*(%\"iG\"\"\"F*F0%&thetaGF0F0" }{TEXT 261 23 " . \n
\n     The mapping  " }{XPPEDIT 18 0 "w = sqrt(z)" "6#/%\"wG-%%sqrtG6#
%\"zG" }{TEXT 268 58 "  can be expressed in polar coordinates \nby the
 function  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 275 3 " = \+
" }{XPPEDIT 18 0 "f(r* exp(i*theta)) =  sqrt(r) * exp(i*theta/2)" "6#/
-%\"fG6#*&%\"rG\"\"\"-%$expG6#*&%\"iGF)%&thetaGF)F)*&-%%sqrtG6#F(F)-F+
6#*(F.F)F/F)\"\"#!\"\"F)" }{TEXT 262 4 " .  " }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 120 "Load Maple's  \"elimi
nate\" and \"conformal mapping\" procedures.\nMake sure this is done o
nly ONCE during a Maple  session.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "readlib(eliminate):\nwith(plots):" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 256 38 "Definition 2.1:  Principal Square Root" 
}{TEXT 285 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 286 3 "The" }{TEXT -1 9 " function" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "g(w);" "6#-%
\"gG6#%\"wG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "w^`1/2`;" "6#)%\"wG%$1/
2G" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "abs(w)*exp(i*`Arg(z)/2`);" "6#*&
-%$absG6#%\"wG\"\"\"-%$expG6#*&%\"iGF(%)Arg(z)/2GF(F(" }{TEXT -1 8 ", \+
 for  " }{XPPEDIT 18 0 "w <> 0;" "6#0%\"wG\"\"!" }{TEXT -1 8 ",       \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "is ca
lled the " }{TEXT 287 30 "principal square root function" }{TEXT -1 1 
"." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 22 "Example 2.12, Page 63." 
}{TEXT 277 22 "  The transformation  " }{XPPEDIT 18 0 "w = z^2" "6#/%
\"wG*$%\"zG\"\"#" }{TEXT 274 38 "  maps lines onto lines or parabolas.
\n" }{TEXT 256 3 "(a)" }{TEXT 280 39 "  Find the image of the vertical
 line  " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT 273 3 " .\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "x:='x':y:='y':u:='u':v:='v':U:='U'
:V:='V':\neqns1 := \{u = x^2 - y^2, v = 2*x*y\}: eqns1;\n`Substitute  \+
x=a  in the previous equations.`;\neqns2 := subs(x=a, eqns1): eqns2;\n
`Eliminate  y  in the previous equations.`;\neqns3 := eliminate(eqns2,
 y): eqns3;\n`Solve for  u  in the previous equations.`;\nsolset := [s
olve(eqns3[2][1], u)]:\n`u ` = solset[1];\nu1 := v -> expand(solset[1]
):\n`u ` = u1(v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 263 39 "Hence, the image of the vertical line  \+
" }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT 272 16 "  is a parabol
a." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 3 "(b)" }{TEXT 281 39 "  Find \+
the image of the vertical line  " }{XPPEDIT 18 0 "y = b" "6#/%\"yG%\"b
G" }{TEXT 271 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 389 "x:='x':y:='y':u:='u':v:='v':U:='U':V:='V':\neqns1
 := \{u = x^2 - y^2, v = 2*x*y\}: eqns1;\n`Substitute  y=b  in the pre
vious equations.`;\neqns2 := subs(y=b, eqns1): eqns2;\n`Eliminate  x  \+
in the previous equations.`;\neqns3 := eliminate(eqns2, x): eqns3;\n`S
olve for  u  in the previous equations.`;\nsolset := [solve(eqns3[2][1
], u)]:\n`u ` = solset[1];\nu2 := v -> expand(solset[1]):\n`u ` = u2(v
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 264 39 "Hence, the image of the vertical line  " }{XPPEDIT 18 0 
"y = b" "6#/%\"yG%\"bG" }{TEXT 270 17 "  is a parabola. " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "f:='f': z:='z':\nf := z -> z^2:\n`
f(z) ` = f(z);\nconformal(f(z), z=0..0.5+2*I,\n  title=`w = z^2`,\n  g
rid=[11,11],numxy=[50,50],\n  scaling=constrained,\n  labels=[`u     `
,`  v`],\n  view=[-4.1..0.3,-0.1..2.1]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 256 22 "Example 2.13, Page 65." }{TEXT 278 24 "   The transfor
mation   " }{XPPEDIT 18 0 "w = sqrt(z)" "6#/%\"wG-%%sqrtG6#%\"zG" }
{TEXT 269 43 "  maps  lines  onto  lines  or hyperbolas.\n" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 208 "f:='f': z:='z':\nf := z -> z^(1/2):\n`f(z)
 ` = f(z);\nconformal(f(z), z=-4..4+4*I,\n  title=`w = z^(1/2)`,\n  gr
id=[9,9],numxy=[50,50],\n  scaling=constrained,\n  labels=[`u `,`v   `
],\n  view=[-0.1..2.5,-0.1..2.5]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 36 "Definiti
on 2.2:  Principal n-th root" }{TEXT 288 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 3 "The" }{TEXT -1 9 " functi
on" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "    \+
 " }{XPPEDIT 18 0 "g(w);" "6#-%\"gG6#%\"wG" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "w^`1/n`;" "6#)%\"wG%$1/nG" }{TEXT -1 3 " = " }{XPPEDIT 
18 0 "abs(w)*exp(i*`Arg(z)/n`);" "6#*&-%$absG6#%\"wG\"\"\"-%$expG6#*&%
\"iGF(%)Arg(z)/nGF(F(" }{TEXT -1 8 ",  for  " }{XPPEDIT 18 0 "w <> 0;
" "6#0%\"wG\"\"!" }{TEXT -1 8 ",       " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 14 "is called the " }{TEXT 290 28 "princ
ipal n-th root function" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 19 "End of Section 2
.3." }}}}{MARK "0 0 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }