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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 281 1 "\n" }{TEXT 256 26 "CHAPTER 1 COMPLEX NUMBERS" }{TEXT 275 2 "\n \n" }{TEXT 256 55 "Section 1.4 The Geometry of Complex Numbers, Conti nued" }{TEXT 276 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 50 "\n In Secti on 1.3 we saw that a complex number " }{XPPEDIT 18 0 "z = x+i*y;" "6#/ %\"zG,&%\"xG\"\"\"*&%\"iGF'%\"yGF'F'" }{TEXT -1 344 " could be viewed \+ as a vector in the xy-plane whose tail is at the origin and whose head is at the point (x,y). A vector can be uniquely specified by giving i ts magnitude (i.e., its length) and direction (i.e., the angle it make s with the positive x-axis). In this section, we focus on these two ge ometric aspects of complex numbers.\n\011\n Let " }{XPPEDIT 18 0 " r;" "6#%\"rG" }{TEXT -1 19 " be the modulus of " }{XPPEDIT 18 0 "z;" " 6#%\"zG" }{TEXT -1 8 " (i.e., " }{XPPEDIT 18 0 "r = abs(z);" "6#/%\"rG -%$absG6#%\"zG" }{TEXT -1 11 "), and let " }{XPPEDIT 18 0 "theta;" "6# %&thetaG" }{TEXT -1 66 " be the angle that the line from the origin to the complex number " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 52 " mak es with the positive x -axis. (Note: The number " }{XPPEDIT 18 0 "thet a;" "6#%&thetaG" }{TEXT -1 17 " is undefined if " }{XPPEDIT 18 0 "z = \+ 0;" "6#/%\"zG\"\"!" }{TEXT -1 7 ". Then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(1-25) " }{XPPEDIT 18 0 "z = r (cos(theta)+i*sin(theta));" "6#/%\"zG-%\"rG6#,&-%$cosG6#%&thetaG\"\"\" *&%\"iGF--%$sinG6#F,F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 256 37 "Definition 1.9: Polar Represen tation" }{TEXT 284 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " The identity " }{XPPEDIT 18 0 "z;" "6#%\"zG" } {TEXT -1 6 " = (" }{XPPEDIT 18 0 "r*cos(theta),r*sin(theta);" "6$*&% \"rG\"\"\"-%$cosG6#%&thetaGF%*&F$F%-%$sinG6#F)F%" }{TEXT -1 6 ") = \+ " }{XPPEDIT 18 0 "`r `(cos(theta)+i*sin(theta));" "6#-%#r~G6#,&-%$cosG 6#%&thetaG\"\"\"*&%\"iGF+-%$sinG6#F*F+F+" }{TEXT -1 16 " is known as \+ a " }{TEXT 285 20 "polar representation" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 17 ", and the values " }{XPPEDIT 18 0 "r ;" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 12 " are called " }{TEXT 286 17 "polar coordinates" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 21 "Example \+ 1.7, Page 23." }{TEXT 306 32 " Find several polar forms of " } {XPPEDIT 18 0 "z = 1+i;" "6#/%\"zG,&\"\"\"F&%\"iGF&" }{TEXT 305 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 351 "z1 := 1 + I:\nz2 := sqrt(2)*cos(Pi/4) + I*sqrt(2)*sin(Pi/4):\nz3 \+ := sqrt(2)*cos(9*Pi/4) + I*sqrt(2)*sin(9*Pi/4):\nz4 := sqrt(2)*cos(-7* Pi/4) + I*sqrt(2)*sin(-7*Pi/4):\n`z ` = z1; ` `;\n`A few polar forms f or z.`;\n`sqrt(2)cos(Pi/4) + i sqrt(2)sin(Pi/4)` = z2;\n`sqrt(2)cos(9P i/4) + i sqrt(2)sin(9Pi/4)` = z3;\n`sqrt(2)cos(-7Pi/4) + i sqrt(2)sin( -7Pi/4)` = z4;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 258 "" 0 "" {TEXT 256 18 "Definition 1.10: " }{XPPEDIT 308 0 "arg(z);" "6#-%$argG6#%\"zG" }{TEXT 303 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "z <> 0" "6#0%\"zG\"\"!" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "arg(z) = \{t heta, where, z = r(cos(theta)+i*sin(theta))\}" "6#/-%$argG6#%\"zG<%%&t hetaG%&whereG/F'-%\"rG6#,&-%$cosG6#F)\"\"\"*&%\"iGF3-%$sinG6#F)F3F3" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arg(z);" "6#-%$argG6#%\"zG" }{TEXT -1 14 ", we say that " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 " is " }{TEXT 304 11 " an argument" }{TEXT -1 5 " of " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 3 "An " }{TEXT 257 8 "argument" }{TEXT 299 5 " of " } {XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 300 6 " is " }{XPPEDIT 18 0 "thet a = arg(z)" "6#/%&thetaG-%$argG6#%\"zG" }{TEXT 292 6 " or " } {XPPEDIT 18 0 "theta = arctan(y/x)" "6#/%&thetaG-%'arctanG6#*&%\"yG\" \"\"%\"xG!\"\"" }{TEXT 293 17 " provided that " }{XPPEDIT 18 0 "x <> 0" "6#0%\"xG\"\"!" }{TEXT 294 6 ".\nThe " }{TEXT 257 16 "exponential \+ form" }{TEXT 301 5 " of " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 295 6 " is " }{XPPEDIT 18 0 "z = r * e^(i*theta)" "6#/%\"zG*&%\"rG\"\"\")% \"eG*&%\"iGF'%&thetaGF'F'" }{TEXT 296 11 " , where " }{XPPEDIT 18 0 "r = abs(z)" "6#/%\"rG-%$absG6#%\"zG" }{TEXT 297 7 " and " } {XPPEDIT 18 0 "theta = arg(z)" "6#/%&thetaG-%$argG6#%\"zG" }{TEXT 298 3 ". " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 256 21 "Example 1.8, Page 24." }{TEXT 287 11 " Because " }{XPPEDIT 18 0 "1+i = sqrt(2)(cos(pi/4)+i*sin(pi/ 4));" "6#/,&\"\"\"F%%\"iGF%--%%sqrtG6#\"\"#6#,&-%$cosG6#*&%#piGF%\"\"% !\"\"F%*&F&F%-%$sinG6#*&F2F%F3F4F%F%" }{TEXT 288 9 ", we have" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {XPPEDIT 18 0 "arg(1+i) = \{pi/4+2*n*pi, where*n*is*an*integer\};" "6# /-%$argG6#,&\"\"\"F(%\"iGF(<$,&*&%#piGF(\"\"%!\"\"F(*(\"\"#F(%\"nGF(F- F(F(*,%&whereGF(F2F(%#isGF(%#anGF(%(integerGF(" }{TEXT -1 4 ". " } {TEXT 289 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 256 18 "Definition 1.11: " }{XPPEDIT 307 0 "A rg(z);" "6#-%$ArgG6#%\"zG" }{TEXT 291 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "z <> 0;" " 6#0%\"zG\"\"!" }{TEXT -1 26 " be a complex number. Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Arg(z) = theta;" " 6#/-%$ArgG6#%\"zG%&thetaG" }{TEXT -1 11 ", provided " }{XPPEDIT 18 0 " z = r(cos(theta)+i*sin(theta));" "6#/%\"zG-%\"rG6#,&-%$cosG6#%&thetaG \"\"\"*&%\"iGF--%$sinG6#F,F-F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-p i;" "6#,$%#piG!\"\"" }{TEXT -1 1 "<" }{XPPEDIT 18 0 "theta <= pi;" "6# 1%&thetaG%#piG" }{TEXT -1 10 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "theta" "6#%&t hetaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Arg(z);" "6#-%$ArgG6#%\"zG" } {TEXT -1 14 ", we say that " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" } {TEXT -1 4 " is " }{TEXT 309 12 "the argument" }{TEXT -1 5 " of " } {XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 21 "Example 1.9, P age 24." }{TEXT -1 2 " " }{XPPEDIT 18 0 "Arg(1+i) = pi/4;" "6#/-%$Arg G6#,&\"\"\"F(%\"iGF(*&%#piGF(\"\"%!\"\"" }{TEXT -1 4 " . " }{TEXT 290 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 1 "\n" }{TEXT 256 22 "Example 1.10, Page 24." }{TEXT 277 26 " Find the polar form of " }{XPPEDIT 18 0 "z" "6#%\"zG" } {TEXT 264 19 " , by computing " }{XPPEDIT 18 0 "abs(z)" "6#-%$absG6 #%\"zG" }{TEXT 265 8 " and " }{XPPEDIT 18 0 "arg(z)" "6#-%$argG6#% \"zG" }{TEXT 266 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "z1 := \+ - sqrt(3) - I:\n`z1 ` = z1; ` `;\nr := abs(z1):\nt := argument(z1):\n ` r ` = r, theta = t;\nz2 := r*(cos(t) + I*sin(t)):\n`z2 = 2cos(-5Pi/6 ) + i 2sin(-5Pi/6)` = z2; ` `;\n`Does z1 = z2 ?`;\nz1 = z2;\nevalb(z1 = z2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 1 "\n" }{TEXT 256 22 "Example 1.11, Page 25." }{TEXT 278 10 " Write " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 267 10 " in t he " }{XPPEDIT 18 0 "z = r * e^(i*theta)" "6#/%\"zG*&%\"rG\"\"\")%\"e G*&%\"iGF'%&thetaGF'F'" }{TEXT 268 8 " form.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "z1 := 4*I:\n`z1 ` = z1; ` `;\nr := abs(z1):\nt := a rgument(z1):\n` r ` = r, theta = t;\nz2 := r*exp(I*t):\n`z2 = 4 exp(iP i/2)` = z2; ` `;\n`Does z1 = z2 ?`;\nz1 = z2;\nevalb(z1 = z2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 1 "\n" }{TEXT 256 22 "Example 1.12, Page 26." }{TEXT 279 10 " Gi ven " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 269 10 " , find " } {XPPEDIT 18 0 "abs(z)" "6#-%$absG6#%\"zG" }{TEXT 270 7 " and " } {XPPEDIT 18 0 "1/z" "6#*&\"\"\"F$%\"zG!\"\"" }{TEXT 271 3 " .\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "Z := 1 + I: `z ` = Z; z:='z':\nR \+ := abs(Z): `r ` = R; r:='r': ` `;\ncz := conjugate(Z):\nw1 := 1/R^2*c onjugate(Z):\nw2 := 1/Z:\nconjugate(z) = cz; ` `;\nconjugate(z)/r^2 = w2;\n`1/z ` = w2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Theorem \+ 1.3" }{TEXT 310 8 " If " }{XPPEDIT 18 0 "z[1];" "6#&%\"zG6#\"\"\" " }{TEXT 312 3 " = " }{XPPEDIT 18 0 "r[1]*exp(i*theta[1]);" "6#*&&%\"r G6#\"\"\"F'-%$expG6#*&%\"iGF'&%&thetaG6#F'F'F'" }{XPPEDIT 18 0 "`` <> \+ 0;" "6#0%!G\"\"!" }{TEXT 311 7 " and " }{XPPEDIT 18 0 "z[2];" "6#&% \"zG6#\"\"#" }{TEXT 313 3 " = " }{XPPEDIT 18 0 "r[2]*exp(i*theta[2]); " "6#*&&%\"rG6#\"\"#\"\"\"-%$expG6#*&%\"iGF(&%&thetaG6#F'F(F(" } {XPPEDIT 18 0 "`` <> 0;" "6#0%!G\"\"!" }{TEXT 314 16 ", then as sets \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " \+ " }{XPPEDIT 18 0 "arg(z[1]*z[2]) = arg(z[1])+arg(z[2])" "6#/-%$argG 6#*&&%\"zG6#\"\"\"F+&F)6#\"\"#F+,&-F%6#&F)6#F+F+-F%6#&F)6#F.F+" } {TEXT -1 6 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 22 "Example 1.13, Page 28." }{TEXT 280 9 " \+ Given " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }{TEXT 272 7 " and \+ " }{XPPEDIT 18 0 "z[2]" "6#&%\"zG6#\"\"#" }{TEXT 273 13 " , compute \+ " }{XPPEDIT 18 0 "z[1]/z[2];" "6#*&&%\"zG6#\"\"\"F'&F%6#\"\"#!\"\"" } {TEXT 274 29 " using polar computations. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 420 "z1 := 8*I: `z1 ` = z1;\nr1 := abs(z1):\nt1 := argum ent(z1):\nz1 = r1*exp(I*t1):\n`z1 = 8 exp(iPi/2)` = z1; ` `;\nz2 := 1 \+ + I*sqrt(3): `z2 ` = z2;\nr2 := abs(z2):\nt2 := argument(z2):\nz2 = r 2*exp(I*t2):\n`z2 = 2 exp(iPi/3)` = z2; ` `;\nw1 := z1/z2:\n`w1 = z1/ z2 ` = w1;\nw1 := evalc(w1):\n`w1 = z1/z2 ` = w1; ` `;\nw2 := r1/r2*e xp(I*(t1 - t2)):\n`w2 = r1/r2 exp(iPi/2-iPi/3) ` = w2; ` `;\n`Does w1 = w2 ?`;\nw1 = w2;\nevalb(w1 = w2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 20 "End of Section 1.4 ." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }