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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 285 144 "\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 284 197 "\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 
281 1 "\n" }{TEXT 256 26 "CHAPTER 1  COMPLEX NUMBERS" }{TEXT 273 2 "\n
\n" }{TEXT 256 54 "Section 1.5  The Algebra of Complex Numbers, Revisi
ted" }{TEXT 274 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 123 "     The real n
umbers are deficient in the sense that not all algebraic operations on
 them produce real numbers. Thus, for " }{XPPEDIT 18 0 "sqrt(-1);" "6#
-%%sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 177 " to make sense, we must lift o
ur sights to the domain of complex numbers. Do complex numbers have th
is same deficiency? That is, if we are to make sense out of expression
s like " }{XPPEDIT 18 0 "sqrt(1+i);" "6#-%%sqrtG6#,&\"\"\"F'%\"iGF'" }
{TEXT -1 82 ", must we appeal to yet another new number system? The an
swer to this question is " }{TEXT 287 2 "no" }{TEXT -1 164 ". It turns
 out that any reasonable algebraic operation we perform on complex num
bers gives us complex numbers. In this respect, we say that the comple
x numbers are " }{TEXT 288 8 "complete" }{TEXT -1 79 ". Later we will \+
learn how to evaluate intriguing algebraic expressions such as " }
{XPPEDIT 18 0 "(-1)^i;" "6#),$\"\"\"!\"\"%\"iG" }{TEXT -1 83 ". For no
w we will be content to study integral powers and roots of complex num
bers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 286 
4 "The " }{TEXT 289 10 "n-th power" }{TEXT 290 5 " of  " }{XPPEDIT 18 
0 "z" "6#%\"zG" }{TEXT 283 6 "  is  " }{XPPEDIT 18 0 "z^n  =  r^n * e^
(i*n*theta)" "6#/)%\"zG%\"nG*&)%\"rGF&\"\"\")%\"eG*(%\"iGF*F&F*%&theta
GF*F*" }{TEXT 265 2 " ." }{TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 282 1 "\n" }{TEXT 256 22 "Example 1.15, Page 31." }{TEXT 275 14 
"   Show that  " }{XPPEDIT 18 0 "(- sqrt(3) - i)^3  =  8i" "6#/*$,&-%%
sqrtG6#\"\"$!\"\"%\"iGF*F)*&\"\")\"\"\"F+F." }{TEXT 266 15 "  in two w
ays.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 312 "z := - sqrt(3) - I: `z `
 = z;\n`Expand using Cartesian coordinates for z^3.`;\nw1 := z^3: `w1 \+
` = w1;\nw1 := evalc(z^3): `w1 ` = w1; ` `;\nr := abs(z):\nt := argume
nt(z):\n`r ` = r, theta = t;\n`Expand using polar coordinates r^3 e^(I
3t).`;\nw2 := r^3*exp(I*3*t): `w2 ` = w2;\n`Are they the same?`;\nw1 =
 w2;\nevalb(w1=w2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 256 22 "Example 1.16, Page 32." }{TEXT 
292 13 "   Evaluate  " }{XPPEDIT 18 0 "(-sqrt(3)-i)^30;" "6#*$,&-%%sqr
tG6#\"\"$!\"\"%\"iGF)\"#I" }{TEXT 293 1 "." }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Solution.  " }
{XPPEDIT 18 0 "(-sqrt(3)-i)^30;" "6#*$,&-%%sqrtG6#\"\"$!\"\"%\"iGF)\"#
I" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(2*exp(i*(-5)*pi/6))^30;" "6#*$*&
\"\"#\"\"\"-%$expG6#**%\"iGF&,$\"\"&!\"\"F&%#piGF&\"\"'F.F&\"#I" }
{TEXT -1 3 " = " }{XPPEDIT 18 0 "2^30;" "6#*$\"\"#\"#I" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "exp(-i*25);" "6#-%$expG6#,$*&%\"iG\"\"\"\"#DF)!\"\"
" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-2^30;" "6#,$*$\"\"#\"#I!\"\"" }
{TEXT -1 4 ".   " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 260 1 "\n" }{TEXT 256 22 "Extra Eample, Page 3
2." }{TEXT 276 12 "  Evaluate  " }{XPPEDIT 18 0 "(- sqrt(3) - i)^(-6) \+
= -1/64" "6#/),&-%%sqrtG6#\"\"$!\"\"%\"iGF*,$\"\"'F*,$*&\"\"\"F0\"#kF*
F*" }{TEXT 267 15 "  in two ways.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
331 "z := - sqrt(3) - I:  `z ` = z;\n`Expand using Cartesian coordinat
es for z^(-6).`;\nw1 := z^(-6):  `w1 ` = w1;\nw1 := evalc(z^(-6)):  `w
1 ` = w1; ` `;\nr := abs(z):\nt := argument(z):\n`r ` = r, theta = t;
\n`Expand using polar coordinates r^-6 e^(-I6t).`;\nw2 := r^(-6)*exp(-
I*6*t):  `w2 ` = w2;\n`Are they the same?`;\nw1 = w2;\nevalb(w1=w2);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "  An interesting applicat
ion of the laws of exponents comes from putting the equation  " }
{XPPEDIT 18 0 "exp(i*theta)^n = exp(i*n*theta);" "6#/)-%$expG6#*&%\"iG
\"\"\"%&thetaGF*%\"nG-F&6#*(F)F*F,F*F+F*" }{TEXT -1 36 "  in its polar
 form.  Doing so gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "(cos(theta)+i*sin(theta))^n;" 
"6#),&-%$cosG6#%&thetaG\"\"\"*&%\"iGF)-%$sinG6#F(F)F)%\"nG" }{TEXT -1 
5 "  =  " }{XPPEDIT 18 0 "cos(n*theta)+i*sin(n*theta);" "6#,&-%$cosG6#
*&%\"nG\"\"\"%&thetaGF)F)*&%\"iGF)-%$sinG6#*&F(F)F*F)F)F)" }{TEXT -1 
8 ".       " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 18 "which is known as " }{TEXT 294 89 "De Moivre's formula, in hono
r of the French mathematician Abraham de Moivre (1667-1754). " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 261 1 "\n" }{TEXT 256 22 "Example 1.17, Page 32." }{TEXT 277 40 
"   Use De Moivre's formula to show that\n" }{XPPEDIT 18 0 "cos(5theta
)=cos(theta)^5 -10 *cos(theta)^3*sin(theta)^2+5*cos(theta)*sin(theta)^
4" "6#/-%$cosG6#*&\"\"&\"\"\"%&thetaGF),(*$-F%6#F*F(F)*(\"#5F)*$-F%6#F
*\"\"$F)-%$sinG6#F*\"\"#!\"\"*(F(F)-F%6#F*F)-F66#F*\"\"%F)" }{TEXT 
268 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 205 "t:='t':\nz1 := cos(5*t) + I*sin(5*t);\nz2 := (cos(t)
 + I*sin(t))^5;\nz2 := evalc(z2);\nassume(t, real);\nu1 := Re(z1):\nu2
 := Re(z2):\n`Equate the real parts.`;\n`Re(z1) = Re(z2)`;\nsubs(t='t'
,u1) = subs(t='t',u2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "  \+
   A key ingredient in determining roots of complex numbers turns out \+
to be a corollary to the " }{TEXT 291 30 "fundamental theorem of algeb
ra" }{TEXT -1 157 ". We will prove the theorem in Chapter 6. Our proof
s must be independent of conclusions we derive here since we are going
 to make use of the corollary now.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 256 61 "Theorem 1.4 (Corollary to the fundamen
tal theorem of algebra)" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "P(z);" "6#-%\"
PG6#%\"zG" }{TEXT -1 28 " is a polynomial, of degree " }{XPPEDIT 18 0 
"n;" "6#%\"nG" }{TEXT -1 47 ", with complex coefficients, then the equ
ation " }{XPPEDIT 18 0 "P(z) = 0;" "6#/-%\"PG6#%\"zG\"\"!" }{TEXT -1 
15 " has precisely " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 38 " (not
 necessarily distinct) solutions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT 256 22 "Example 1.18, Page 33." }{TEXT 278 25 "   Factor the pol
ynomial\n" }{XPPEDIT 18 0 "P(z)  =   z^3  +  (2 - 2*i)*z^2  +  (-1-4*i
)*z  -  2" "6#/-%\"PG6#%\"zG,**$F'\"\"$\"\"\"*&,&\"\"#F+*&F.F+%\"iGF+!
\"\"F+*$F'F.F+F+*&,&F+F1*&\"\"%F+F0F+F1F+F'F+F+F.F1" }{TEXT 269 3 " .
\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "P :='P': z :='z':\nP := z ->
 z^3 + (2-2*I)*z^2 + (-1-4*I)*z - 2:\n`P(z) ` = P(z);\n`P(z) ` = facto
r(P(z)); ` `;\n`P(z) = 0,  The solution set is =`, \{solve(P(z)=0, z)
\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 256 36 "Definition 1.12:  Primitive nth root" }{TEXT 295 3 "  \+
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 1 "F" }
{TEXT -1 22 "or any natural number " }{XPPEDIT 18 0 "n;" "6#%\"nG" }
{TEXT -1 12 ", the value " }{XPPEDIT 18 0 "omega[n];" "6#&%&omegaG6#%
\"nG" }{TEXT -1 10 " given by " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "omega[n]" "6#&%&ome
gaG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "exp(i(2*pi/n));" "6#-%$e
xpG6#-%\"iG6#*(\"\"#\"\"\"%#piGF+%\"nG!\"\"" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "cos(2*pi/n)+i*sin(2*pi/n);" "6#,&-%$cosG6#*(\"\"#\"\"\"
%#piGF)%\"nG!\"\"F)*&%\"iGF)-%$sinG6#*(F(F)F*F)F+F,F)F)" }{TEXT -1 5 "
     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "
is called the " }{TEXT 297 18 "primitive nth root" }{TEXT -1 11 " of u
nith. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "
" 0 "" {TEXT 262 1 "\n" }{TEXT 256 22 "Example 1.19, Page 34." }{TEXT 
279 43 "   Find all the solutions of the equation  " }{XPPEDIT 18 0 "z
^8 = 1" "6#/*$%\"zG\"\")\"\"\"" }{TEXT 270 3 " .\n" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 263 "z:='z':\n`Solutions of   z^8 = 1.`; ` `;\nsolset :
= \{solve(z^8 = 1, z)\}:\n`Solutions ` = solset;\npts := map(w->[Re(w)
,Im(w)], solset):\nplot(pts,\n  style=point, symbol=circle,\n  scaling
=constrained, color=red,\n  labels=[`    x`,`y   `],\n  view=[-1.1..1.
1,-1.1..1.1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 263 1 "\n" }{TEXT 256 22 "Example 1.20, Page 3
5." }{TEXT 280 30 "  Find all the cube roots of  " }{XPPEDIT 18 0 "8i
" "6#*&\"\")\"\"\"%\"iGF%" }{TEXT 272 48 " ,\ni.e. find all the soluti
ons to the equation  " }{XPPEDIT 18 0 "z^3 = 8i" "6#/*$%\"zG\"\"$*&\"
\")\"\"\"%\"iGF)" }{TEXT 271 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 266 "z :='z':\n`Solutions of   z^3 = 8i.`; ` `;\nsolset := \{solve(z
^3 = 8*I, z)\}:\n`Solutions ` = solset;\npts := map(w->[Re(w),Im(w)], \+
solset):\nplot(pts,\n  style=point, symbol=circle,\n  scaling=constrai
ned, color=red,\n  labels=[`    x`,`y  `],\n  view=[-2.1..2.1,-2.1..2.
1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 31 "Theorem 1.5 (Quadratic
 formula)" }{TEXT 298 6 "  If  " }{XPPEDIT 18 0 "a*z^2+b*z+c = 0;" "6#
/,(*&%\"aG\"\"\"*$%\"zG\"\"#F'F'*&%\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT 
299 28 ", then the solutin set for  " }{XPPEDIT 18 0 "z;" "6#%\"zG" }
{TEXT 300 4 " is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "\{(
-b+(b^2-4*a*c)^(1/2))/(2*a)\}" "6#<#*&,&%\"bG!\"\"),&*$F&\"\"#\"\"\"*(
\"\"%F,%\"aGF,%\"cGF,F'*&F,F,F+F'F,F,*&F+F,F/F,F'" }{TEXT -1 2 " ," }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "where by \+
 " }{XPPEDIT 18 0 "(b^2-4*a*c)^(1/2)" "6#),&*$%\"bG\"\"#\"\"\"*(\"\"%F
(%\"aGF(%\"cGF(!\"\"*&F(F(F'F-" }{TEXT -1 76 "  we mean all distinct s
quare roots of the number inside the parenthesis.   " }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 22 "Exampl
e 1.21, Page 35." }{TEXT 301 38 "  Find all solutions to the equation \+
 " }{XPPEDIT 18 0 "z^2+(1+i)*z+5*i = 0;" "6#/,(*$%\"zG\"\"#\"\"\"*&,&F
(F(%\"iGF(F(F&F(F(*&\"\"&F(F+F(F(\"\"!" }{TEXT 302 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 285 "z :='z':\n`
Solutions of   z^2 +(1+i)z +5i = 0.`; ` `;\nsolset := \{solve(z^2 +(1+
I)*z +5*I, z)\}:\n`Solutions ` = solset;\npts := map(w->[Re(w),Im(w)],
 solset):\nplot(pts,\n  style=point, symbol=circle,\n  scaling=constra
ined, color=red,\n  labels=[`    x`,`y  `],\n  view=[-2.1..2.1,-2.1..2
.1]);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 264 19 "End of Section 1.5." }}}}{MARK "0 0 0
" 28 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }