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va" 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 }} {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 288 1 "\n" }{TEXT 256 26 "CHAPTER 1 COMPLEX NUMBERS" }{TEXT 280 2 "\n \n" }{TEXT 256 43 "Section 1.2 The Algebra of Complex Numbers" } {TEXT 281 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 117 " We have seen tha t complex numbers came to be viewed as ordered pairs of real numbers. \+ That is, a complex number " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 4 " is " }{TEXT 299 14 "defined to be " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 18 0 "z = (x, y);" " 6#/%\"zG6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The reason we say " }{TEXT 300 7 "order ed" }{TEXT -1 120 " pair is because we are thinking of a point in the \+ plane. The point (2, 3), for example, is not the same as (3, 2). The \+ " }{TEXT 301 5 "order" }{TEXT -1 19 " in which we write " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 103 " in the equation makes a difference. Clearly, then, two complex numbers are equal if and only if their " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 23 " coordinates are equal " }{TEXT 302 3 "and" } {TEXT -1 7 " their " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 40 " coor dinates are equal. In other words,\n" }}{PARA 0 "" 0 "" {TEXT -1 8 " \+ " }{XPPEDIT 18 0 "(x, y) = (u, v);" "6#/6$%\"xG%\"yG6$%\"uG%\"vG " }{TEXT -1 9 " iff " }{XPPEDIT 18 0 "x = u;" "6#/%\"xG%\"uG" } {TEXT -1 3 " " }{TEXT 303 3 "and" }{TEXT -1 3 " " }{XPPEDIT 18 0 " y = v;" "6#/%\"yG%\"vG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 370 "If we are to have a meaningful numbe r system, there needs to be a method for combining these ordered pairs . We need to define algebraic operations in a consistent way so that t he sum, difference, product, and quotient of any two ordered pairs wil l again be an ordered pair. The key to defining how these numbers shou ld be manipulated is to follow Gauss' lead and equate " }{XPPEDIT 18 0 "\"(x, y)\";" "6#Q'(x,~y)6\"" }{TEXT -1 6 " with " }{XPPEDIT 18 0 " \"x+iy\";" "6#Q%x+iy6\"" }{TEXT -1 19 ". Then, by letting " }{XPPEDIT 18 0 "z[1] = (x[1], y[1]);" "6#/&%\"zG6#\"\"\"6$&%\"xG6#F'&%\"yG6#F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z[2] = (x[2], y[2]);" "6#/&%\"zG6# \"\"#6$&%\"xG6#F'&%\"yG6#F'" }{TEXT -1 38 " be arbitrary complex numbe rs, we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "\011 " }{XPPEDIT 18 0 "z[1]+z[2] = (x[1], y[1])+(x[2], y[2]) " "6#/,&&%\"zG6#\"\"\"F(&F&6#\"\"#F(,&6$&%\"xG6#F(&%\"yG6#F(F(6$&F/6#F +&F26#F+F(" }{TEXT -1 4 " \n" }}{PARA 0 "" 0 "" {TEXT -1 5 "\011 \+ " }{XPPEDIT 18 0 "z[1]+z[2] = x[1]+i*y[1]+x[2]+i*y[2];" "6#/,&&%\"zG6# \"\"\"F(&F&6#\"\"#F(,*&%\"xG6#F(F(*&%\"iGF(&%\"yG6#F(F(F(&F.6#F+F(*&F1 F(&F36#F+F(F(" }{TEXT -1 4 " \n" }}{PARA 0 "" 0 "" {TEXT -1 5 "\011 \+ " }{XPPEDIT 18 0 "z[1]+z[2] = x[1]+x[2]+i*(y[1]+y[2]);" "6#/,&&%\"z G6#\"\"\"F(&F&6#\"\"#F(,(&%\"xG6#F(F(&F.6#F+F(*&%\"iGF(,&&%\"yG6#F(F(& F66#F+F(F(F(" }{TEXT -1 4 " \n" }}{PARA 0 "" 0 "" {TEXT -1 5 "\011 \+ " }{XPPEDIT 18 0 "z[1]+z[2] = (x[1]+x[2], y[1]+y[2]);" "6#/,&&%\"zG6 #\"\"\"F(&F&6#\"\"#F(6$,&&%\"xG6#F(F(&F/6#F+F(,&&%\"yG6#F(F(&F56#F+F( " }{TEXT -1 4 " \n" }}{PARA 0 "" 0 "" {TEXT -1 10 "Thus, if " } {XPPEDIT 18 0 "z[1] = (x[1], y[1]);" "6#/&%\"zG6#\"\"\"6$&%\"xG6#F'&% \"yG6#F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z[2] = (x[2], y[2]);" "6 #/&%\"zG6#\"\"#6$&%\"xG6#F'&%\"yG6#F'" }{TEXT -1 76 " are arbitrary co mplex numbers, the following definitions should make sense." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 25 "Definition 1.1 : Addition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 327 32 "Formula (1-6), Page 7. " }{XPPEDIT 18 0 "z[1]+z[2] = \+ (x[1]+x[2], y[1]+y[2]);" "6#/,&&%\"zG6#\"\"\"F(&F&6#\"\"#F(6$,&&%\"xG6 #F(F(&F/6#F+F(,&&%\"yG6#F(F(&F56#F+F(" }{TEXT -1 4 " \n" }}{PARA 0 " " 0 "" {TEXT 256 28 "Definition 1.2: Subtraction" }{TEXT 304 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 328 24 "Formula \+ (1-7), Page 7. " }{TEXT -1 6 "\011 " }{XPPEDIT 18 0 "z[1]-z[2] = \+ (x[1]-x[2], y[1]-y[2]);" "6#/,&&%\"zG6#\"\"\"F(&F&6#\"\"#!\"\"6$,&&%\" xG6#F(F(&F06#F+F,,&&%\"yG6#F(F(&F66#F+F," }{TEXT -1 4 " \n" }}{PARA 0 "" 0 "" {TEXT 298 171 "The rules for addition, subtraction, multipli cation and division of complex numbers\nare extensions of the rules fo r real numbers. They obey familiar algebraic properties." }}}{EXCHG {PARA 257 "" 0 "" {TEXT 289 1 "\n" }{TEXT 256 21 "Example 1.1, Page 7 ." }{TEXT 282 10 " Find " }{XPPEDIT 18 0 "z[1] + z[2]" "6#,&&%\"zG 6#\"\"\"F'&F%6#\"\"#F'" }{TEXT 270 9 " and " }{XPPEDIT 18 0 "z[1] \+ - z[2]" "6#,&&%\"zG6#\"\"\"F'&F%6#\"\"#!\"\"" }{TEXT 279 3 " .\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "z:='z':\nZ1 := 3 + 7*I:\nZ2 := 5 - 6*I:\nz[1] = Z1;\nz[2] = Z2; ` `;\nz[1] + z[2] = Z1 + Z2;\nz[1] - z[ 2] = Z1 - Z2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 31 "Definition 1.3: Multiplication" }{TEXT 324 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 329 32 "Formula (1-8), Page 8. " }{TEXT -1 0 "" }{XPPEDIT 18 0 "z [1]*z[2] = (x[1]*x[2]-y[1]*y[2], x[1]*y[2]+x[2]*y[1]);" "6#/*&&%\"zG6# \"\"\"F(&F&6#\"\"#F(6$,&*&&%\"xG6#F(F(&F06#F+F(F(*&&%\"yG6#F(F(&F66#F+ F(!\"\",&*&&F06#F(F(&F66#F+F(F(*&&F06#F+F(&F66#F(F(F(" }{TEXT -1 3 " \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 " " {TEXT 260 1 "\n" }{TEXT 256 21 "Example 1.2, Page 8." }{TEXT 283 10 " Find " }{XPPEDIT 18 0 "z[1]*z[2]" "6#*&&%\"zG6#\"\"\"F'&F%6# \"\"#F'" }{TEXT 278 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 84 "z:='z':\nZ1 := 3 + 7*I:\nZ2 := 5 - 6*I:\nz[1 ] = Z1;\nz[2] = Z2; ` `;\nz[1]*z[2] = Z1*Z2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 25 "Definition 1. 4: Division" }{TEXT 325 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 330 31 "Formula (1-9), Page 9. " }{TEXT -1 1 "\011" }{XPPEDIT 18 0 "z[1]/z[2] = ((x[1]*x[2]+y[1]*y[2])/(x[2]^2 +y[2]^2), (-x[1]*y[2]+x[2]*y[1])/(x[2]^2+y[2]^2));" "6#/*&&%\"zG6#\"\" \"F(&F&6#\"\"#!\"\"6$*&,&*&&%\"xG6#F(F(&F26#F+F(F(*&&%\"yG6#F(F(&F86#F +F(F(F(,&*$&F26#F+F+F(*$&F86#F+F+F(F,*&,&*&&F26#F(F(&F86#F+F(F,*&&F26# F+F(&F86#F(F(F(F(,&*$&F26#F+F+F(*$&F86#F+F+F(F," }{TEXT -1 3 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 261 1 "\n" }{TEXT 256 21 "Example 1.3, Page 9." }{TEXT 284 10 " Find " }{XPPEDIT 18 0 "z[1]/z[2]" "6#*&&%\"zG6#\"\"\"F'&F%6#\"\"# !\"\"" }{TEXT 277 4 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "z:='z ':\nZ1 := 3 + 7*I:\nZ2 := 5 - 6*I:\nz[1] = Z1;\nz[2] = Z2; ` `;\nz[1] /z[2] = Z1/Z2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 262 1 "\n" }{TEXT 256 30 "Derivation for Multi plication," }{TEXT 290 51 " Formula (1-8), Page 8. In general we can \+ derive:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "x:='x': y:='y': z:='z ':\nZ1:='Z1': Z1 := x[1] + I*y[1]:\nZ2:='Z2': Z2 := x[2] + I*y[2]:\nz[ 1] = Z1;\nz[2] = Z2; ` `;\nz[1]*z[2] = Z1*Z2;\nz[1]*z[2] = expand(Z1* Z2);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 263 1 "\n" }{TEXT 256 24 "Deri vation for Division," }{TEXT 291 51 " Formula (1-9), Page 9. In gener al we can derive:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "d:='d': n:= 'n': x:='x': y:='y': z:='z':\nZ1:='Z1': Z1 := x[1] + I*y[1]:\nZ2:='Z2' : Z2 := x[2] + I*y[2]:\nz[1] = Z1;\nz[2] = Z2; ` `;\nz[1]/z[2] = Z1/Z 2;\nn := expand(Z1*(x[2]-I*y[2])):\nd := expand(Z2*(x[2]-I*y[2])):\nz[ 1]/z[2] = n/d;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 26 "Definition 1.5: Real Part" }{TEXT 305 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 331 4 "T he " }{TEXT 317 12 "real part of" }{TEXT 318 1 " " }{XPPEDIT 319 0 "z; " "6#%\"zG" }{TEXT 306 9 " denoted " }{XPPEDIT 18 0 "Re(z);" "6#-%#ReG 6#%\"zG" }{TEXT 307 20 " is the real number " }{XPPEDIT 18 0 "x;" "6#% \"xG" }{TEXT 308 1 "." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 31 "Definition 1.6: Imagi nary Part" }{TEXT 309 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 332 4 "The " }{TEXT 320 18 "imaginary part of " } {XPPEDIT 321 0 "z;" "6#%\"zG" }{TEXT 310 9 " denoted " }{XPPEDIT 18 0 "Im(z);" "6#-%#ImG6#%\"zG" }{TEXT 311 20 " is the real number " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT 312 1 "." }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 26 "Definition 1.7: Conjugate" }{TEXT 313 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 333 4 "The " }{TEXT 322 13 "co njugate of " }{XPPEDIT 323 0 "z;" "6#%\"zG" }{TEXT 314 9 " denoted " } {XPPEDIT 18 0 "conjugate(z);" "6#-%*conjugateG6#%\"zG" }{TEXT 315 23 " is the complex number " }{XPPEDIT 18 0 "(x, -y) = x-i*y;" "6#/6$%\"xG ,$%\"yG!\"\",&F%\"\"\"*&%\"iGF*F'F*F(" }{TEXT 316 1 "." }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 264 1 "\n" }{TEXT 256 23 "Example 1.4a, Page 12." }{TEXT 285 10 " Find " }{XPPEDIT 18 0 "Re(z[1])" "6#-%#ReG6#&%\"zG6#\"\"\"" } {TEXT 275 9 " and " }{XPPEDIT 18 0 "Re(z[2])" "6#-%#ReG6#&%\"zG6# \"\"#" }{TEXT 276 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 106 "z:='z':\nZ1 := -3 + 7*I: z[1] = Z1;\nRe(z[1] ) = Re(Z1); ` `;\nZ2 := 9 + 4*I: z[2] = Z2;\nRe(z[2]) = Re(Z2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 265 1 "\n" }{TEXT 256 23 "Example 1.4b, Page 12." }{TEXT 286 10 " Find " }{XPPEDIT 18 0 "Im(z[1])" "6#-%#ImG6#&%\"zG6#\"\"\"" } {TEXT 273 9 " and " }{XPPEDIT 18 0 "Im(z[2])" "6#-%#ImG6#&%\"zG6# \"\"#" }{TEXT 274 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "z:='z ':\nZ1 := -3 + 7*I: z[1] = Z1;\n`Im(z1) ` = Im(Z1); ` `;\nZ2 := 9 + 4 *I: z[2] = Z2;\n`Im(z2) ` = Im(Z2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 266 1 "\n" }{TEXT 256 23 "Ex ample 1.4c, Page 12." }{TEXT 287 10 " Find " }{XPPEDIT 18 0 "conj ugate(z[1])" "6#-%*conjugateG6#&%\"zG6#\"\"\"" }{TEXT 271 9 " and \+ " }{XPPEDIT 18 0 "conjugate(z[2])" "6#-%*conjugateG6#&%\"zG6#\"\"#" } {TEXT 272 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "z:='z':\nZ1 := -3 + 7*I:\nz[1] = Z1; \nconjugate(z[1 ]) = conjugate(Z1); ` `;\nZ2 := 9 + 4*I:\nz[2] = Z2; \nconjugate(z2) \+ = conjugate(Z2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 267 1 "\n" }{TEXT 256 47 "Derivation of the Co mmutative Law for Addition," }{TEXT 292 1 " " }{TEXT 257 23 "Property \+ (P1), Page 10." }{TEXT 294 28 " In general we can derive:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "x:='x': y:='y': z:='z':\nZ1:='Z1': Z1 := x[1] + I*y[1]:\nZ2:='Z2': Z2 := x[2] + I*y[2]:\nz[1] = Z1;\nz[2] = Z2 ; ` `;\n`z1 + z2` = Z1 + Z2;\n`z2 + z1` = Z2 + Z1; ` `;\n`Does z1 + \+ z2 = z2 + z1 ?`;\nZ1+Z2 = Z2+Z1;\nevalb(Z1+Z2 = Z2+Z1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 268 1 "\n" }{TEXT 256 53 "Derivation of the Associative Law for Multiplicati on," }{TEXT 293 1 " " }{TEXT 257 23 "Property (P6), Page 10." }{TEXT 295 28 " In general we can derive:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "x:='x': y:='y':\nZ1 := x[1] + I*y[1]: `z1 ` = Z1;\nZ2 := x[2] + I*y[2]: `z2 ` = Z2;\nZ3 := x[3] + I*y[3]: `z3 ` = Z3;\nw1 := Z1*(Z2 + Z3):\nw2 := Z1*Z2 + Z1*Z3: ` `;\n`z1*(z2 + z3) ` = w1;\n`z1*z2 + z1 *z3 ` = w2;\nw1 := expand(w1):\nw2 := expand(w2): ` `;\n`z1*(z2 + z3 ) ` = w1;\n`z1*z2 + z1*z3 ` = w2; ` `;\n`Does z1*(z2 + z3) = z1*z2 + \+ z1*z3 ?`;\nevalb(w1 = w2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 256 22 "Theorem 1.1, Page 12." }{TEXT 326 3 " \+ " }{TEXT -1 8 "Suppose " }{XPPEDIT 18 0 "z[1];" "6#&%\"zG6#\"\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\"#" }{TEXT -1 6 " , and " }{XPPEDIT 18 0 "z[3];" "6#&%\"zG6#\"\"$" }{TEXT -1 38 " are ar bitrary complex numbers. Then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "(1-10) " }{XPPEDIT 18 0 "conjugate(co njugate(z)) = z" "6#/-%*conjugateG6#-F%6#%\"zGF)" }{TEXT -1 6 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(1-11 ) " }{XPPEDIT 18 0 "conjugate(z[1]+z[2]) = conjugate(z[1])+conjuga te(z[2]);" "6#/-%*conjugateG6#,&&%\"zG6#\"\"\"F+&F)6#\"\"#F+,&-F%6#&F) 6#F+F+-F%6#&F)6#F.F+" }{TEXT -1 6 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "(1-12) " }{XPPEDIT 18 0 "con jugate(z[1]*z[2]) = conjugate(z[1])*conjugate(z[2]);" "6#/-%*conjugate G6#*&&%\"zG6#\"\"\"F+&F)6#\"\"#F+*&-F%6#&F)6#F+F+-F%6#&F)6#F.F+" } {TEXT -1 6 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(1-13) " }{XPPEDIT 18 0 "conjugate(z[1]/z[2]) = conju gate(z[1])/conjugate(z[2]);" "6#/-%*conjugateG6#*&&%\"zG6#\"\"\"F+&F)6 #\"\"#!\"\"*&-F%6#&F)6#F+F+-F%6#&F)6#F.F/" }{TEXT -1 6 ", " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(1-14) \+ " }{XPPEDIT 18 0 "Re(z) = (z+conjugate(z))/2;" "6#/-%#ReG6#%\"zG*&,& F'\"\"\"-%*conjugateG6#F'F*F*\"\"#!\"\"" }{TEXT -1 6 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(1-15) " } {XPPEDIT 18 0 "Im(z) = (z-conjugate(z))/(2*i);" "6#/-%#ImG6#%\"zG*&,&F '\"\"\"-%*conjugateG6#F'!\"\"F**&\"\"#F*%\"iGF*F." }{TEXT -1 6 " , \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(1-16 ) " }{XPPEDIT 18 0 "Re(i*z) = -Im(z);" "6#/-%#ReG6#*&%\"iG\"\"\"% \"zGF),$-%#ImG6#F*!\"\"" }{TEXT -1 6 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(1-17) " }{XPPEDIT 18 0 "Im (i*z) = Re(z);" "6#/-%#ImG6#*&%\"iG\"\"\"%\"zGF)-%#ReG6#F*" }{TEXT -1 6 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 269 19 "End of Section 1.2." }}}}{MARK "0 0 0" 13 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }