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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 276 144 "\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 275 197 "\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 
273 1 "\n" }{TEXT 256 26 "CHAPTER 1  COMPLEX NUMBERS" }{TEXT 269 2 "\n
\n" }{TEXT 256 45 "Section 1.6  The Topology of Complex Numbers\n" }
{TEXT 270 126 "\nIn this section we investigate some basic ideas conce
rning sets of points in the plane.\nThe first concept is that of a cur
ve." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT 256 18 "Definition:  Curve" }}{PARA 257 "" 0 "" {TEXT 277 2 "  \+
" }}{PARA 257 "" 0 "" {TEXT 287 2 "A " }{TEXT 257 5 "curve" }{TEXT 
278 27 " in the complex plane is:  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "C" "6#%\"CG" }
{TEXT 280 3 ":  " }{XPPEDIT 18 0 "z(t)  =  x(t)  + i y(t)" "6#/-%\"zG6
#%\"tG,&-%\"xG6#F'\"\"\"*&%\"iGF,-%\"yG6#F'F,F," }{TEXT 279 10 "    fo
r   " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{XPPEDIT 18 0 "`` <= ``;" "6#1%!G
F$" }{TEXT 281 1 " " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT 283 1 " " }
{XPPEDIT 18 0 "`` <= ``" "6#1%!GF$" }{TEXT 282 1 " " }{XPPEDIT 18 0 "b
;" "6#%\"bG" }{TEXT 284 1 "." }{TEXT -1 4 "    " }}}{EXCHG {PARA 257 "
" 0 "" {TEXT 274 1 "\n" }{TEXT 256 22 "Example 1.22, Page 40." }{TEXT 
271 8 "   If   " }{XPPEDIT 18 0 "z[0] = x[0] + i y[0]" "6#/&%\"zG6#\"
\"!,&&%\"xG6#F'\"\"\"*&%\"iGF,&%\"yG6#F'F,F," }{TEXT 263 9 "   and   \+
" }{XPPEDIT 18 0 "z[1] = x[1] + i y[1]" "6#/&%\"zG6#\"\"\",&&%\"xG6#F'
F'*&%\"iGF'&%\"yG6#F'F'F'" }{TEXT 264 63 " are two given points, then \+
the straight line segment joining  " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#
\"\"!" }{TEXT 265 6 "  to  " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" 
}{TEXT 266 10 "  is  C:  " }{XPPEDIT 18 0 "z(t)  =  (x[0] + (x[1] - x[
0])*t) + i (y[0] + (y[1] - y[0])*t)" "6#/-%\"zG6#%\"tG,(&%\"xG6#\"\"!
\"\"\"*&,&&F*6#F-F-&F*6#F,!\"\"F-F'F-F--%\"iG6#,&&%\"yG6#F,F-*&,&&F:6#
F-F-&F:6#F,F4F-F'F-F-F-" }{TEXT 285 10 "    for   " }{XPPEDIT 18 0 "0;
" "6#\"\"!" }{XPPEDIT 18 0 "`` <= ``;" "6#1%!GF$" }{XPPEDIT 18 0 "t;" 
"6#%\"tG" }{XPPEDIT 18 0 "`` <= ``" "6#1%!GF$" }{XPPEDIT 18 0 "1;" "6#
\"\"\"" }{TEXT 286 1 "." }{TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "t:='t':x0:='x0':x1:='x1'
:y0:='y0':y1:='y1':z:='z':\nz := t -> x0 + (x1-x0)*t + I*(y0 + (y1-y0)
*t):\n`Equation of a line segment:`;\n`z(t) ` = z(t), `   for  0 <= t \+
<= 1`; ` `;\n`Initial  point    z(0) ` = z(0);\n`Terminal point    z(1
) ` = z(1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
257 "" 0 "" {TEXT 260 1 "\n" }{TEXT 256 22 "Extra Eample, Page 40." }
{TEXT 272 64 "  Find the equation of the line segment with the initial
 point  " }{XPPEDIT 18 0 "z[0] = -3 + 2*i" "6#/&%\"zG6#\"\"!,&\"\"$!\"
\"*&\"\"#\"\"\"%\"iGF-F-" }{TEXT 267 26 "  and the terminal point  " }
{XPPEDIT 18 0 "z[1] = 1 + i" "6#/&%\"zG6#\"\"\",&F'F'%\"iGF'" }{TEXT 
268 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 338 "t:='t':x0:='x0':x1:
='x1':y0:='y0':y1:='y1':z:='z':\nz0 := - 3 + 2*I:\nz1 := 1 + I:\nx0 :=
 Re(z0): y0 := Im(z0): x1 := Re(z1): y1 := Im(z1):\nz := t -> x0 + (x1
-x0)*t + I*(y0 + (y1-y0)*t):\n`Equation of a line segment:`;\n`z(t) ` \+
= z0 + (z1 - z0)*t, `   for  0 <= t <= 1`; ` `;\n`Initial  point    z(
0) ` = z(0);\n`Terminal point    z(1) ` = z(1);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 261 70 "The graph
 for this line segment can is drawn with the plot subroutine." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "pl
ot([evalf(Re(z(t))),evalf(Im(z(t))), t=0..1],\ntitle=`Line segment bet
ween z0 and z1.`,\nscaling=constrained, color=red,\nlabels=[`  x`,`  y
`],\nview=[-3.5..1.5,-1.0..3.50]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 262 19 "End of Section 1.6." }}}}{MARK "0 0 0
" 25 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }