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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 297 144 "\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 296 197 "\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 
294 1 "\n" }{TEXT 256 29 "CHAPTER 2   COMPLEX FUNCTIONS" }{TEXT 288 2 
"\n\n" }{TEXT 256 43 "Section 2.6 The Reciprocal Transformation  " }
{XPPEDIT 18 0 "w = 1/z" "6#/%\"wG*&\"\"\"F&%\"zG!\"\"" }{TEXT 267 17 "
\n\n   The mapping " }{XPPEDIT 18 0 "w = 1/z" "6#/%\"wG*&\"\"\"F&%\"zG
!\"\"" }{TEXT 299 15 " is called the " }{TEXT 300 25 "reciprocal trans
formation" }{TEXT 301 14 " and maps the " }{XPPEDIT 18 0 "z;" "6#%\"zG
" }{TEXT 302 31 " plane one-to-one and onto the " }{XPPEDIT 18 0 "w;" 
"6#%\"wG" }{TEXT 304 28 " plane except for the point " }{XPPEDIT 18 0 
"z = 0;" "6#/%\"zG\"\"!" }{TEXT 305 36 ", which has no image, and the \+
point " }{XPPEDIT 18 0 "w = 0;" "6#/%\"wG\"\"!" }{TEXT 306 42 ", which
 has no preimage or inverse image. " }{TEXT -1 0 "" }{TEXT 303 28 "Usi
ng exponential notation  " }{XPPEDIT 18 0 "w = rho*exp(i*phi);" "6#/%
\"wG*&%$rhoG\"\"\"-%$expG6#*&%\"iGF'%$phiGF'F'" }{TEXT 308 19 ",  we s
ee that if  " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT 309 3 " = " }
{XPPEDIT 18 0 "r*exp(i*theta) <> 0;" "6#0*&%\"rG\"\"\"-%$expG6#*&%\"iG
F&%&thetaGF&F&\"\"!" }{TEXT 310 17 ",  then we have  " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "      " }{XPPEDIT 18 0 
"w;" "6#%\"wG" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "rho*exp(i*phi)" "6#
*&%$rhoG\"\"\"-%$expG6#*&%\"iGF%%$phiGF%F%" }{TEXT -1 5 "  =  " }
{XPPEDIT 18 0 "1/z" "6#*&\"\"\"F$%\"zG!\"\"" }{TEXT -1 5 "  =  " }
{XPPEDIT 18 0 "1/r;" "6#*&\"\"\"F$%\"rG!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "exp(i*theta);" "6#-%$expG6#*&%\"iG\"\"\"%&thetaGF(" }
{TEXT -1 9 ".        " }}{PARA 0 "" 0 "" {TEXT 307 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 105 "   It is convenient to extend the system of complex
 numbers by joining to it an \"ideal\" point denoted by " }{XPPEDIT 
18 0 "infinity;" "6#%)infinityG" }{TEXT -1 16 " and called the " }
{TEXT 311 17 "point at infinity" }{TEXT -1 29 ". This new set is calle
d the " }{TEXT 312 22 "extended complex plane" }{TEXT -1 2 ". " }
{TEXT 313 57 "The reciprocal transformation maps the \"extended comple
x " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 286 51 "-plane\" one-to-one an
d onto the \"extended complex  " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT 
287 11 "-plane\".   " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
257 "" 0 "" {TEXT 298 102 "Load Maple's  \"conformal mapping\" procedu
re.\nMake sure this is done only ONCE during a Maple  session." }
{MPLTEXT 1 0 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):
" }{TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 260 1 "\n" }{TEXT 
256 22 "Example 2.22, Page 86." }{TEXT 289 47 "   Show that the image \+
of the right half plane " }{XPPEDIT 18 0 "Re(z) > 1/2" "6#2*&\"\"\"F%
\"\"#!\"\"-%#ReG6#%\"zG" }{TEXT 314 28 "  under the transformation  " 
}{XPPEDIT 18 0 "w = 1/z" "6#/%\"wG*&\"\"\"F&%\"zG!\"\"" }{TEXT 268 15 
"  is the disk  " }{XPPEDIT 18 0 "abs(w - 1) < 1" "6#2-%$absG6#,&%\"wG
\"\"\"F)!\"\"F)" }{TEXT 269 3 " .\n" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 321 "u:='u': v:='v': x:='x': X:='X': Y:='Y': Z:='Z
':\nassume(X, real); assume(Y, real);\nZ := X + I*Y:\nineq0 := Re(Z) >
 1/2:\nineq1 := subs(X='x',ineq0):\nineq2 := subs(\{x=u/(u^2 + v^2),y=
-v/(u^2 + v^2)\}, ineq1):\nineq3 := 2*(u^2 + v^2)*1/2 < 2*(u^2 + v^2)*
u/(u^2+v^2):\nineq4 := (1-2*u < 1-2*u) + ineq3:\nineq1; ineq2; ineq3; \+
ineq4;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "
" 0 "" {TEXT 261 29 "Thus, the image is the disk  " }{XPPEDIT 18 0 "ab
s(w - 1) < 1" "6#2-%$absG6#,&%\"wG\"\"\"F)!\"\"F)" }{TEXT 285 2 " ." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "f:='f': z:='z':\nf := z -
> 1/z:\n`f(z) ` = f(z);\nconformal(f(z), z=0.5-5*I..10+5*I,\n  title=`
Image of Re z > 1/2 under  w = 1/z`,\n  grid=[30,31], numxy=[200,200],
\n  scaling=constrained,\n  labels=[`  u`,`v   `],\n  tickmarks=[5,5],
\n  view=[-0.1..2.1,-1.1..1.1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 262 1 "\n" }{TEXT 256 22 "Examp
le 2.23, Page 87." }{TEXT 290 57 "  Find the image of the portion of t
he right half plane  " }{XPPEDIT 18 0 "Re(z) > 1/2" "6#2*&\"\"\"F%\"\"
#!\"\"-%#ReG6#%\"zG" }{TEXT 315 3 "   " }{TEXT 270 29 "that lies insid
e the circle  " }{XPPEDIT 18 0 "abs(z-1/2) < 1" "6#2-%$absG6#,&%\"zG\"
\"\"*&F)F)\"\"#!\"\"F,F)" }{TEXT 271 21 "  under the mapping  " }
{XPPEDIT 18 0 "w = 1/z" "6#/%\"wG*&\"\"\"F&%\"zG!\"\"" }{TEXT 272 3 " \+
. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 315 "f:='f': F:='F': z:='z':\nf := z -> 1/z:\n`f(z) ` =
 f(z);\nF := z -> subs(Z=z+1/2,f(Z)):\n`F(z) ` = F(z);\nconformal(F(Re
(z)*exp(I*Im(z))), z=0.01..1+I*2*Pi,\n  title=`Image of |z-1/2|<1 unde
r  w = 1/z`,\n  grid=[25,25],numxy=[100,100],\n  scaling=constrained,
\n  labels=[` u`,` v`],\n  tickmarks=[5,5],\n  view=[-3..1,-2..2]);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 263 171 "Using the result from Example 2.16 and the above inform
ation.  We conclude that the image is the crescent shaped region in th
e  w-plane  which is the portion of the disk  " }{XPPEDIT 18 0 "abs(w \+
- 1) < 1" "6#2-%$absG6#,&%\"wG\"\"\"F)!\"\"F)" }{TEXT 273 32 "  that l
ies outside the circle  " }{XPPEDIT 18 0 "abs(w + 2/3) = 4/3" "6#/-%$a
bsG6#,&%\"wG\"\"\"*&\"\"#F)\"\"$!\"\"F)*&\"\"%F)F,F-" }{TEXT 274 2 " .
" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 295 1 "\n" }{TEXT 256 22 "Example \+
2.24, Page 88." }{TEXT 291 1 "\n" }{TEXT 256 3 "(a)" }{TEXT 292 41 "  \+
Find the images of the vertical lines  " }{XPPEDIT 18 0 "x = a" "6#/%
\"xG%\"aG" }{TEXT 275 21 "  under the mapping  " }{XPPEDIT 18 0 "w=1/z
" "6#/%\"wG*&\"\"\"F&%\"zG!\"\"" }{TEXT 276 3 " .\n" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 226 "a:='a': u:='u': v:='v': x:='x': y:='y':\neq1 := x
 = a:\neq2 := subs(\{x=u/(u^2 + v^2),y=-v/(u^2 + v^2)\}, eq1):\neq3 :=
 (u^2 + v^2)/a*eq2:\neq4 := (- u/a = - u/a) + eq3:\neq5 := (1/(2*a)^2 \+
= 1/(2*a)^2) + eq4:\neq1; eq2; eq3; eq4; eq5;" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 264 31 "Thus, the i
mage is the circle  " }{XPPEDIT 18 0 "abs(w - 1/(2a)) = 1/abs(2a)" "6#
/-%$absG6#,&%\"wG\"\"\"*&F)F)*&\"\"#F)%\"aGF)!\"\"F.*&F)F)-F%6#*&F,F)F
-F)F." }{TEXT 277 17 " ,  with center  " }{XPPEDIT 18 0 "w[0] = 1/(2a)
" "6#/&%\"wG6#\"\"!*&\"\"\"F)*&\"\"#F)%\"aGF)!\"\"" }{TEXT 278 14 "  a
nd radius  " }{XPPEDIT 18 0 "1/abs(2a)" "6#*&\"\"\"F$-%$absG6#*&\"\"#F
$%\"aGF$!\"\"" }{TEXT 279 2 " ." }}}{EXCHG {PARA 257 "" 0 "" {TEXT 
256 3 "(b)" }{TEXT 293 43 "  Find the images of the horizontal lines  \+
" }{XPPEDIT 18 0 "y=b" "6#/%\"yG%\"bG" }{TEXT 280 21 "  under the mapp
ing  " }{XPPEDIT 18 0 "w =1/z" "6#/%\"wG*&\"\"\"F&%\"zG!\"\"" }{TEXT 
281 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "b:='b': u:='u': v:=
'v': x:='x': y:='y':\neq1 := y = b:\neq2 := subs(\{x=u/(u^2 + v^2),y=-
v/(u^2 + v^2)\}, eq1):\neq3 := (u^2 + v^2)/b*eq2:\neq4 := (v/b = v/b) \+
+ eq3:\neq5 := (1/(2*b)^2 = 1/(2*b)^2) + eq4:\neq1; eq2; eq3; eq4; eq5
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 265 31 "Thus, the image is the circle  " }{XPPEDIT 18 0 "abs(w +
 i/(2b))  =  1/abs(2b)" "6#/-%$absG6#,&%\"wG\"\"\"*&%\"iGF)*&\"\"#F)%
\"bGF)!\"\"F)*&F)F)-F%6#*&F-F)F.F)F/" }{TEXT 282 15 " ,with center  " 
}{XPPEDIT 18 0 "w[0] = - i/(2b)" "6#/&%\"wG6#\"\"!,$*&%\"iG\"\"\"*&\"
\"#F+%\"bGF+!\"\"F/" }{TEXT 283 14 "  and radius  " }{XPPEDIT 18 0 "1/
abs(2b)" "6#*&\"\"\"F$-%$absG6#*&\"\"#F$%\"bGF$!\"\"" }{TEXT 284 2 " .
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "f:='f': z:='z':\nf := \+
z -> 1/z:\n`f(z) ` = f(z);\ne := 0.000001:\nconformal(f(z), z=-2-2*I..
e+2+e+2*I,\n  title=`Image of lines under  w = 1/z`,\n  grid=[10,10], \+
numxy=[50,50],\n  scaling=constrained,\n  labels=[`    u`,`v   `],\n  \+
tickmarks=[5,5],\n  view=[-1.1..1.1,-1.1..1.1]);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT 266 19 "End of Section 2.6." }}}}{MARK 
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