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0 1 }{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 "Monaco" 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 "R 3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Monaco" 1 9 0 0 255 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 3" -1 258 1 {CSTYLE "" -1 -1 "Symbol" 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" -1 259 1 {CSTYLE "" -1 -1 "Gene va" 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 }} {SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple Worksheets, 2001" }{TEXT 316 144 "\n(c) John H. Mathews Rus sell W. Howell\nmathews@fullerton.edu howell@westmont.edu\n\nCompl imentary software to accompany the textbook:" }}{PARA 259 "" 0 "" {TEXT 257 82 "COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9" }{TEXT 315 197 "\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 301 1 "\n" }{TEXT 256 29 "CHAPTER 2 COMPLEX FUNCTIONS" }{TEXT 291 2 "\n\n" }{TEXT 256 44 "Section 2.1 Functions of a Complex Variable" } {TEXT 292 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " A " }{TEXT 317 23 "complex valued function" }{TEXT 318 1 " " }{TEXT 319 1 " " }{XPPEDIT 320 0 "f;" "6#%\"fG" }{TEXT -1 26 " of the complex variable " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 47 " is a rule that assigns to each complex number " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 10 " in a set " }{XPPEDIT 18 0 "D;" "6#%\"D G" }{TEXT -1 33 " one and only one complex number " }{XPPEDIT 18 0 "w; " "6#%\"wG" }{TEXT -1 11 ". We write " }{XPPEDIT 18 0 "w = f(z);" "6#/ %\"wG-%\"fG6#%\"zG" }{TEXT -1 10 " and call " }{XPPEDIT 18 0 "w;" "6#% \"wG" }{TEXT -1 5 " the " }{TEXT 321 9 "image of " }{XPPEDIT 322 0 "z; " "6#%\"zG" }{TEXT 323 7 " under " }{XPPEDIT 324 0 "f;" "6#%\"fG" } {TEXT -1 10 ". The set " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 15 " \+ is called the " }{TEXT 325 10 "domain of " }{XPPEDIT 326 0 "f;" "6#%\" fG" }{TEXT -1 28 ", and the set of all images " }{XPPEDIT 18 0 "\{w = \+ f(z), z*epsilon*D\};" "6#<$/%\"wG-%\"fG6#%\"zG*(F)\"\"\"%(epsilonGF+% \"DGF+" }{TEXT -1 15 " is called the " }{TEXT 327 9 "range of " } {XPPEDIT 328 0 "f;" "6#%\"fG" }{TEXT -1 37 ". As we saw in section 1.6 , the term " }{TEXT 329 6 "domain" }{TEXT -1 84 " is also used to indi cate a connected open set. When speaking about the domain of a " } {TEXT 330 8 "function" }{TEXT -1 139 ", however, mathematicians mean o nly the set of points on which the function is defined. This is a dist inction worth noting.\n\011\n Just as " }{XPPEDIT 18 0 "z;" "6#%\" zG" }{TEXT -1 51 " can be expressed by its real and imaginary parts, \+ " }{XPPEDIT 18 0 "z = x+i*y;" "6#/%\"zG,&%\"xG\"\"\"*&%\"iGF'%\"yGF'F' " }{TEXT -1 11 ", we write " }{XPPEDIT 18 0 "f(z) = u+i*v;" "6#/-%\"fG 6#%\"zG,&%\"uG\"\"\"*&%\"iGF*%\"vGF*F*" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v;" " 6#%\"vG" }{TEXT -1 37 " are the real and imaginary parts of " } {XPPEDIT 18 0 "w;" "6#%\"wG" }{TEXT -1 49 ", respectively. This gives \+ us the representation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "w;" "6#%\"wG" }{TEXT -1 4 " \+ = " }{XPPEDIT 18 0 "f(z);" "6#-%\"fG6#%\"zG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "f(x,y);" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "f(x,y);" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "u+i*v;" "6#,&%\"uG\"\"\"*&%\"iGF%%\"vGF%F%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " Since " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 11 " depend on " }{XPPEDIT 18 0 "x;" "6# %\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 75 ", they can be considered to be real valued functions of the real vari ables " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 10 "; that is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "u = u(x, y);" "6#/%\"uG-F$6$%\"xG%\"yG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "v = v(x,y);" "6#/%\"vG-F$6$%\"xG%\"yG" }{TEXT -1 6 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Combining these id eas it is customary to write a complex function f in the form " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " } {XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "u( x,y)+i*v(x,y);" "6#,&-%\"uG6$%\"xG%\"yG\"\"\"*&%\"iGF)-%\"vG6$F'F(F)F) " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Definition." }{TEXT 293 4 " A " }{TEXT 257 8 "function " }{TEXT 304 2 " " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 305 2 " " }{TEXT 257 23 "of the complex variable" }{TEXT 303 2 " " } {XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 263 18 " can be written:\n" }} {PARA 0 "" 0 "" {TEXT 332 5 " " }{XPPEDIT 18 0 "f(x + i y) = u(x ,y) + i *v(x,y)" "6#/-%\"fG6#,&%\"xG\"\"\"*&%\"iGF)%\"yGF)F),&-%\"uG6 $F(F,F)*&F+F)-%\"vG6$F(F,F)F)" }{TEXT 262 4 " .\n\n" }{TEXT 256 11 "De finition." }{TEXT 294 6 " The " }{TEXT 257 21 "polar coordinate form " }{TEXT 306 27 " of a complex function is:\n" }}{PARA 0 "" 0 "" {TEXT 333 6 " " }{XPPEDIT 18 0 "f(r*exp(i*theta));" "6#-%\"fG6#*& %\"rG\"\"\"-%$expG6#*&%\"iGF(%&thetaGF(F(" }{TEXT 334 5 " = " } {XPPEDIT 18 0 "u(r,theta)+i*v(r,theta);" "6#,&-%\"uG6$%\"rG%&thetaG\" \"\"*&%\"iGF)-%\"vG6$F'F(F)F)" }{TEXT 260 71 " .\n\nThere are two appr oaches to defining a complex function in Maple.\n\n" }{TEXT 256 9 "Met hod 1." }{TEXT 295 7 " Make " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"x G%\"yG" }{TEXT 264 36 " a function of two real variables " } {XPPEDIT 18 0 "(x,y)" "6$%\"xG%\"yG" }{TEXT 265 4 " .\n\n" }{TEXT 256 9 "Method 2." }{TEXT 296 7 " Make " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6# %\"zG" }{TEXT 273 38 " a function of the complex variable " } {XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 272 2 " ." }}}{EXCHG {PARA 0 "" 0 " " {TEXT 302 1 "\n" }{TEXT 256 21 "Example 2.1, Page 49." }{TEXT 297 9 " Write " }{XPPEDIT 18 0 "f(z) = z^4" "6#/-%\"fG6#%\"zG*$F'\"\"%" } {TEXT 290 10 " in the " }{XPPEDIT 18 0 "f = u+ i v" "6#/%\"fG,&%\"uG \"\"\"*&%\"iGF'%\"vGF'F'" }{TEXT 289 9 " form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 9 "Method 1." }{TEXT 307 7 " Make " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT 266 36 " a function of two real variables " }{XPPEDIT 18 0 "(x,y)" "6$%\"xG %\"yG" }{TEXT 267 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " > " 0 "" {MPLTEXT 1 0 180 "f:='f': x:='x': y:='y': z:='z':\nf := proc( x,y)\n local z,w;\n z := x + I*y;\n w := expand(z^4);\nend:\n`f(z) \+ ` = z^4;\n`f(x,y) ` = f(x,y); ` `;\n`At z = 1 + 2i: `;\n`f(1,2) ` = f (1,2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 9 "Method 2." }{TEXT 308 7 " Make " }{XPPEDIT 18 0 "f( z)" "6#-%\"fG6#%\"zG" }{TEXT 271 17 " a function of " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 270 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "F:='F': x:='x': y:='y': z:='z':\nF := proc(z)\n local w;\n w := expand(z^4);\nend:\n`F(z) ` = F(z);\n` F(x + I y) ` = F(x + I*y); ` `;\n`At z = 1 + 2i: `;\n`F(1 + I 2) ` = \+ F(1 + I*2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 21 "Example 2.2, P age 50." }{TEXT 298 9 " Write " }{XPPEDIT 18 0 "f(z) = conjugate(z) \+ *Re(z) + z^2 + Im(z)" "6#/-%\"fG6#%\"zG,(*&-%*conjugateG6#F'\"\"\"- %#ReG6#F'F-F-*$F'\"\"#F--%#ImG6#F'F-" }{TEXT 287 10 " in the " } {XPPEDIT 18 0 "f = u + i v" "6#/%\"fG,&%\"uG\"\"\"*&%\"iGF'%\"vGF'F '" }{TEXT 288 8 " form.\n" }}{PARA 0 "" 0 "" {TEXT 257 9 "Method 1." }{TEXT 309 7 " Make " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT 268 36 " a function of two real variables " }{XPPEDIT 18 0 "` (x,y)`" "6#%&(x,y)G" }{TEXT 269 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "f:='f': x:='x': y:='y':\nf := \+ proc(x,y)\n local w;\n w := (x - I*y)*x + (x + I*y)^2 + y;\nend:\n`f (x,y) ` = f(x,y);\n`f(x,y) ` = evalc(f(x,y)); ` `;\n`At z = 1 + 2i: ` ;\n`f(1,2) ` = f(1,2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 257 9 "Method 2." }{TEXT 310 7 " Make " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 275 17 " a function of " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 274 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "F:='F': x:='x': y: ='y': z:='z':\nF := proc(z)\n local w;\n w := conjugate(z)*Re(z) + z ^2 + Im(z);\nend:\n`F(z) ` = F(z);\n`F(x + I y) ` = (x-I*y)*x + (x+I*Y )^2 + y; ` `;\n`At z = 1 + 2i: `;\n`F(1 + I 2) ` = F(1 + I*2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 21 "Example 2.3, Page 50." }{TEXT 299 12 " Express " }{XPPEDIT 18 0 "f(z) = 4*x^2 + i *4 *y^2" "6 #/-%\"fG6#%\"zG,&*&\"\"%\"\"\"*$%\"xG\"\"#F+F+*(%\"iGF+F*F+%\"yGF.F+" }{TEXT 285 26 " by a formula involving " }{XPPEDIT 18 0 "z" "6#%\"zG " }{TEXT 286 7 " and " }{XPPEDIT 18 0 "conjugate(z);" "6#-%*conjugat eG6#%\"zG" }{TEXT 331 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 9 "Method 1." }{TEXT 311 7 " Make " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT 278 36 " a function of two r eal variables " }{XPPEDIT 18 0 "`(x,y)`" "6#%&(x,y)G" }{TEXT 279 2 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "f:='f': x:='x': y:='y':\nf := proc(x,y)\n local w;\nw := 4*x^2 + I*4*y^2;\nend:\n`f(x,y) ` = f(x,y); ` `;\n`At z = 1 + 2i: `;\n`f(1,2 ) ` = f(1,2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 9 "Method 2." }{TEXT 312 7 " Make " } {XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 277 17 " a function of \+ " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 276 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "F:='F': w:='w': z: ='z': Z:='Z':\nw := subs(\{x=(Z+conjugate(Z))/2, y=(Z-conjugate(Z))/(2 *I)\},f(x,y)):\nF := z -> subs(Z=z, expand(w)):\n`f(x,y) ` = f(x,y);\n `F(z) ` = F(z); ` `;\n`At z = 1 + 2i: `;\n`F(1 + I 2) ` = F(1+I*2); ` `;\n`F(1 + I 2) ` = evalc(F(1+I*2));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 256 21 "Example 2.5, Page 51." }{TEXT 300 12 " Express " } {XPPEDIT 18 0 "f(z) = z^5 + 4 z^2 - 6" "6#/-%\"fG6#%\"zG,(*$F'\" \"&\"\"\"*&\"\"%F+*$F'\"\"#F+F+\"\"'!\"\"" }{TEXT 284 33 " in the pol ar coordinate form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 9 "Method 1." }{TEXT 313 7 " Make " }{XPPEDIT 18 0 "f(x, y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT 282 36 " a function of two real var iables " }{XPPEDIT 18 0 "(x,y)" "6$%\"xG%\"yG" }{TEXT 283 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "F: ='F': x:='x': y:='y': z:='z':\nF := proc(z)\n local w;\n w := z^5 + \+ 4*z^2 - 6;\nend:\n`F(z) ` = z^5 + 4*z^2 - 6;\n`F(x + I y) ` = F(x + I* y);` `;\n`At z = 1 + i: `;\n`F(1 + I) ` = F(1 + I);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 9 "Method \+ 2." }{TEXT 314 7 " Make " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" } {TEXT 281 17 " a function of " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 280 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 268 "f:='f': r:='r': t:='t': z:='z':\nf := proc(r,t)\n l ocal w;\n w := subs(\{z^2=r^2*cos(2*t) + I*r^2*sin(2*t), \n z^5=r^ 5*cos(5*t) + I*r^5*sin(5*t)\}, F(z));\nend:\n`F(z) ` = z^5 + 4*z^2 - 6 ;\n`f(r,t) ` = f(r,t); ` `;\n`At z = 1 + i: `;\n`f(sqrt(2),Pi/4) ` = f(sqrt(2),Pi/4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 19 "End of Sect ion 2.1." }}}}{MARK "0 0 0" 22 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }