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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 277 144 "\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 276 197 "\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 
274 1 "\n" }{TEXT 256 26 "CHAPTER 1  COMPLEX NUMBERS" }{TEXT 268 2 "\n
\n" }{TEXT 256 42 "Section 1.1  The Origin of Complex Numbers" }{TEXT 
269 353 "\n\n      Complex analysis can roughly be thought of as that \+
subject which applies the ideas of calculus to imaginary numbers. But \+
what exactly are imaginary numbers? Usually, students learn about them
 in high school with introductory remarks from their teachers along th
e following lines: ``We can't take the square root of a negative numbe
r. But, let's " }{TEXT 284 7 "pretend" }{TEXT 285 45 " we can---and si
nce these numbers are really " }{TEXT 286 9 "imaginary" }{TEXT 287 44 
", it will be convenient notationally to set " }{XPPEDIT 18 0 "i = sqr
t(-1);" "6#/%\"iG-%%sqrtG6#,$\"\"\"!\"\"" }{TEXT 278 93 ".'' Rules are
 then learned for doing arithmetic with these numbers. The rules make \+
sense. If " }{XPPEDIT 18 0 "i = sqrt(-1);" "6#/%\"iG-%%sqrtG6#,$\"\"\"
!\"\"" }{TEXT 279 27 ", it stands to reason that " }{XPPEDIT 18 0 "i^2
 = -1;" "6#/*$%\"iG\"\"#,$\"\"\"!\"\"" }{TEXT 280 132 ". On the other \+
hand, it is not uncommon for students to wonder all along whether they
 are really doing magic rather than mathematics" }{TEXT -1 3 ".  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 563 "     If \+
you ever felt that way, congratulate yourself! You're in the company o
f some of the great mathematicians from the sixteenth through the nine
teenth centuries. They, too, were perplexed with the notion of roots o
f negative numbers. The purpose of this section is to highlight some o
f the episodes in what turns out to be a very colorful history of how \+
imaginary numbers were introduced, investigated, avoided, mocked, and-
--eventually---accepted by the mathematical community. We intend to sh
ow you that, contrary to popular belief, there is really nothing " }
{TEXT 283 9 "imaginary" }{TEXT -1 102 " about \"imaginary numbers'' at
 all. In a metaphysical sense, they are just as real as \"real numbers
.''" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 97 "     Our story begins in 1545. In that year the Italian
 mathematician Girolamo Cardano published " }{TEXT 290 9 "Ars Magna" }
{TEXT -1 2 " (" }{TEXT 291 13 "The Great Art" }{TEXT -1 119 "), a 40-c
hapter masterpiece in which he gave for the first time an algebraic so
lution to the general cubic equation    " }}{PARA 0 "" 0 "" {TEXT -1 
7 "       " }{XPPEDIT 18 0 "x^3+a*x^2+b*x+c = 0;" "6#/,**$%\"xG\"\"$\"
\"\"*&%\"aGF(*$F&\"\"#F(F(*&%\"bGF(F&F(F(%\"cGF(\"\"!" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "     \+
His technique involved transforming this equation into what is called \+
a " }{TEXT 288 15 "depressed cubic" }{TEXT -1 83 ". This is a cubic eq
uation without the quadratic term, so that it can be written as" }}
{PARA 257 "" 0 "" {TEXT 282 5 "     " }{XPPEDIT 18 0 "x^3  + b x  +  c
  =  0" "6#/,(*$%\"xG\"\"$\"\"\"*&%\"bGF(F&F(F(%\"cGF(\"\"!" }{TEXT 
281 3 " .\n" }}{PARA 257 "" 0 "" {TEXT 289 66 "Ferro and Tartaglia sho
wed that one solution to this equation is:\n" }{XPPEDIT 18 0 "x  =  [-
c/2 + sqrt(c^2/4 + b^3/27) ]^(1/3)  -  [c/2 + sqrt(c^2/4 + b^3/27)]^(1
/3)" "6#/%\"xG,&)7#,&*&%\"cG\"\"\"\"\"#!\"\"F--%%sqrtG6#,&*&F*F,\"\"%F
-F+*&%\"bG\"\"$\"#FF-F+F+*&F+F+F6F-F+)7#,&*&F*F+F,F-F+-F/6#,&*&F*F,F3F
-F+*&F5F6F7F-F+F+*&F+F+F6F-F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT 275 1 "\n" }{TEXT 256 46 "Example in na
rrative on Page 2 of Section 1.1." }{TEXT 273 47 "\nConsider the simpl
e depressed cubic equation  " }{XPPEDIT 18 0 "x^3 - 15*x - 4  =   0" "
6#/,(*$%\"xG\"\"$\"\"\"*&\"#:F(F&F(!\"\"\"\"%F+\"\"!" }{TEXT 263 43 " \+
.\nMaple can easily find all the solutions!" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "x:='x':\neqn := x^3 - 15
 *x - 4 = 0:\n`Solve the equation  `,eqn; ` `;\nsolset := \{solve(eqn,
 x )\}:\n`Solution set `, solset;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%
5Solve~the~equation~~G/,(*$)%\"xG\"\"$\"\"\"F**&\"#:F*F(F*!\"\"\"\"%F-
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%.Solution~set~G<%\"\"%,&!\"#\"\"\"*$-%%sqrtG6#\"\"$F(F
(,&F'F(F)!\"\"" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 
"" 0 "" {TEXT 260 68 "We want to investigate the Ferro-Tartaglia formu
la and verify that  " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 271 16 "  is
 a solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 364 "b:='b': c:='c': w:='w': x:='x': X:='X':\nx^3 + b*x +
 c = 0;\nx:=(-c/2 + sqrt(c^2/4 +b^3/27))^(1/3) + \n   (-c/2 - sqrt(c^2
/4 +b^3/27))^(1/3):\n`x ` = x; ` `;\n`Substitute`;\nb := -15:\nc := -4
:\nx :='x':\n`b ` = b, `c ` = c;\neqn := x^3 + b*x + c = 0: eqn;\nx:=(
-c/2 + sqrt(c^2/4 +b^3/27))^(1/3) - \n   ( c/2 + sqrt(c^2/4 +b^3/27))^
(1/3):\n`x ` = x; ` `;\n`The result is:`;\neqn;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,(*$)%\"xG\"\"$\"\"\"F)*&%\"bGF)F'F)F)%\"cGF)\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#x~G,&*$),&%\"cG#!\"\"\"\"#*&#\"\"\"
\"#=F/-%%sqrtG6#,&*$)F)F,F/\"#\")*&\"#7F/)%\"bG\"\"$F/F/F/F/#F/F<F/F/*
$),&F)F**&#F/F0F/*$F1F/F/F+F=F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+SubstituteG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$/%#b~G!#:/%#c~G!\"%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,(*$)%\"xG\"\"$\"\"\"F)*&\"#:F)F'F)!\"\"\"\"%F,\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%#x~G,&*$)^$\"\"#\"#6#\"\"\"\"\"$F,F,*$)^$!
\"#F*F+F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%/The~result~is:G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,**$),&*$)^$\"\"#\"#6#\"\"\"\"\"$F.F.*$)^$!\"#F,F-F.!\"\"F/F.F.
*&\"#:F.F)F.F4*&F6F.F1F.F.\"\"%F4\"\"!" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 261 21 "The principal value  " }{XPPEDIT 18 0 "r[1] = (2 + 11 i)
^(1/3)" "6#/&%\"rG6#\"\"\"),&\"\"#F'*&\"#6F'%\"iGF'F'*&F'F'\"\"$!\"\"
" }{TEXT 264 97 "  is correct.  However, a different choice (or branch
) of\nthe cube root must be used to compute  " }{XPPEDIT 18 0 "(-2 + 1
1 i)^(1/3)" "6#),&\"\"#!\"\"*&\"#6\"\"\"%\"iGF)F)*&F)F)\"\"$F&" }
{TEXT 265 61 " .   Its value is obtained by multiplying\nby a \"cube r
oot of " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 272 30 " \", the special
 cube root is  " }{XPPEDIT 18 0 "1^(1/3) =   (-1)^(2/3)" "6#/)\"\"\"*&
F%F%\"\"$!\"\"),$F%F(*&\"\"#F%F'F(" }{TEXT 266 40 " .   Now compute th
e second\nroot using  " }{XPPEDIT 18 0 "r[2] = (-2 + 11 i)^(1/3) * (-1
)^(2/3)" "6#/&%\"rG6#\"\"#*&),&F'!\"\"*&\"#6\"\"\"%\"iGF.F.*&F.F.\"\"$
F+F.),$F.F+*&F'F.F1F+F." }{TEXT 267 31 "  .   And then compute using  \+
 " }{XPPEDIT 18 0 "x = r[1] - r[2]" "6#/%\"xG,&&%\"rG6#\"\"\"F)&F'6#\"
\"#!\"\"" }{TEXT 270 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 440 "r2
 := (-2 + 11*I)^(1/3)*(-1)^(2/3):\n`r2 = (-2 + 11 i)^(1/3)` = r2;\nr1 \+
:= (2 + 11*I)^(1/3):\n`r1 ` = r1;` `;\nr2 := evalc((-2 + 11*I)^(1/3)*(
-1)^(2/3)):\n`r2 ` = expand(r2);\nr1 := evalc((2 + 11*I)^(1/3)):\n`r1 \+
` = r1; ` `;\n`Use the roots to form x = r1 - r2.`;\n`Then substitute \+
x into the cubic equation:`;\nx:='x':\neqn := x^3 -15*x - 4 = 0:\neqn;
\nx := expand(r1 - r2):\n`x ` = x; ` `;\neqn;\n`Expand things on the l
eft side and obtain:`;\nevalf(eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/%7r2~=~(-2~+~11~i)^(1/3)G*&)^$!\"#\"#6#\"\"\"\"\"$F+)!\"\"#\"\"#F,F+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$r1~G*$)^$\"\"#\"#6#\"\"\"\"\"$F
+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$r2~G,&*&-%%sqrtG6#\"\"&\"\"\"-%$cosG6#,$-%'arctanG6#
#\"#6\"\"##F+\"\"$F+!\"\"*(^#F+F+F'F+-%$sinGF.F+F+" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%$r1~G,&*&-%%sqrtG6#\"\"&\"\"\"-%$cosG6#,$-%'arctanG
6##\"#6\"\"##F+\"\"$F+F+*(^#F+F+F'F+-%$sinGF.F+F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%CUse~the~root
s~to~form~x~=~r1~-~r2.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%KThen~subs
titute~x~into~the~cubic~equation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/,(*$)%\"xG\"\"$\"\"\"F)*&\"#:F)F'F)!\"\"\"\"%F,\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%#x~G,$*&-%%sqrtG6#\"\"&\"\"\"-%$cosG6#,$-%'arcta
nG6##\"#6\"\"##F+\"\"$F+F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&-%%sqrtG6#\"\"&\"\"\")-%$cosG6#,
$-%'arctanG6##\"#6\"\"##F*\"\"$F7F*\"#S*(\"#IF*F&F*F,F*!\"\"\"\"%F;\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%KExpand~things~on~the~left~sid
e~and~obtain:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"\"!F%F$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 314 "`We could use numerical app
roximations for the roots`;\n`r2 ` = evalf(r2);\n`r1 ` = evalf(r1);\n`
And use a numerical approximation for x = r1 - r2.`;\n`x ` = evalf(r1 \+
- r2); ` `;\n`Then substitute x into the cubic equation:`;\nx:='x':\ne
qn := x^3 -15*x - 4 = 0:\neqn;\nx := evalf(r1 - r2):\neqn,`  Look, it \+
is almost zero!`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%TWe~could~use~nu
merical~approximations~for~the~rootsG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/%$r2~G^$$!+++++?!\"*$\"+++++5F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/%$r1~G^$$\"+++++?!\"*$\"+++++5F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%SAnd~use~a~numerical~approximation~for~x~=~r1~-~r2.G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%#x~G^$$\"+++++S!\"*$\"\"!F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%KThen~substit
ute~x~into~the~cubic~equation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(
*$)%\"xG\"\"$\"\"\"F)*&\"#:F)F'F)!\"\"\"\"%F,\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$/^$$\"\"!F&F%F&%;~~Look,~it~is~almost~zero!G" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 262 20 "End of Section
 1.1. " }}}}{MARK "1 2 2" 85 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }