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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 272 144 "\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 271 197 "\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 
268 1 "\n" }{TEXT 256 26 "CHAPTER 1  COMPLEX NUMBERS" }{TEXT 265 2 "\n
\n" }{TEXT 256 44 "Section 1.3  The Geometry of Complex Numbers" }
{TEXT 269 1 "\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 262 "     S
ince the complex numbers are ordered pairs of real numbers, there is a
 one-to-one correspondence between them and points in the plane. In th
is section we shall see what effect algebraic operations on complex nu
mbers have on their geometric representations." }}{PARA 0 "" 0 "" 
{TEXT -1 18 "\011\n     The number " }{XPPEDIT 18 0 "z = x+i*y;" "6#/%
\"zG,&%\"xG\"\"\"*&%\"iGF'%\"yGF'F'" }{TEXT -1 198 " can be represente
d by a position vector in the xy-plane whose tail is at the origin and
 whose head is at the point (x,y). When the xy-plane is used for displ
aying complex numbers, it is called the " }{TEXT 273 13 "complex plane
" }{TEXT -1 22 ", or more simply, the " }{TEXT 274 7 "z-plane" }{TEXT 
-1 14 ". Recall that " }{XPPEDIT 18 0 "Re(z) = x;" "6#/-%#ReG6#%\"zG%
\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Im(z) = y;" "6#/-%#ImG6#%\"z
G%\"yG" }{TEXT -1 17 ". Geometrically, " }{XPPEDIT 18 0 "Re(z);" "6#-%
#ReG6#%\"zG" }{TEXT -1 22 " is the projection of " }{XPPEDIT 18 0 "z =
 (x, y);" "6#/%\"zG6$%\"xG%\"yG" }{TEXT -1 23 " onto the x- axis, and \+
" }{XPPEDIT 18 0 "Im(z);" "6#-%#ImG6#%\"zG" }{TEXT -1 22 " is the proj
ection of " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 75 " onto the y-ax
is. It makes sense, then, that the x-axis is also called the " }{TEXT 
275 9 "real axis" }{TEXT -1 31 ", and the y-axis is called the " }
{TEXT 276 14 "imaginary axis" }{TEXT -1 2 ". " }{TEXT 277 0 "" }{TEXT 
-1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT 256 24 "Definition 1.8:  Modulus" }{TEXT 292 1 " " }}{PARA 257 "
" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 293 4 "The " }{TEXT 
295 7 "modulus" }{TEXT 291 5 ", or " }{TEXT 296 14 "absolute value" }
{TEXT 297 25 ", of the complex number  " }{XPPEDIT 18 0 "z = x+i*y;" "
6#/%\"zG,&%\"xG\"\"\"*&%\"iGF'%\"yGF'F'" }{TEXT 290 40 "  is a non-neg
ative real number denoted " }{XPPEDIT 18 0 "abs(z);" "6#-%$absG6#%\"zG
" }{TEXT 294 31 " and is given by the equation  " }}{PARA 257 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "abs
(z) = sqrt(x^2+y^2)" "6#/-%$absG6#%\"zG-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*
$%\"yGF.F/" }{TEXT -1 6 " .    " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 283 13 "  The number " }{XPPEDIT 18 0 "abs(
z);" "6#-%$absG6#%\"zG" }{TEXT 284 128 " is the distance between the o
rigin and the point (x, y). The only complex number with modulus zero \+
is the number 0. The number " }{XPPEDIT 18 0 "z = 4+3*i;" "6#/%\"zG,&
\"\"%\"\"\"*&\"\"$F'%\"iGF'F'" }{TEXT 285 13 " has modulus " }
{XPPEDIT 18 0 "5;" "6#\"\"&" }{TEXT 286 14 ". The numbers " }{XPPEDIT 
18 0 "abs(x) = abs(Re(z));" "6#/-%$absG6#%\"xG-F%6#-%#ReG6#%\"zG" }
{TEXT 287 2 ", " }{XPPEDIT 18 0 "abs(y) = abs(Im(z));" "6#/-%$absG6#%
\"yG-F%6#-%#ImG6#%\"zG" }{TEXT 288 6 ", and " }{XPPEDIT 18 0 "abs(z);
" "6#-%$absG6#%\"zG" }{TEXT 289 93 " are the lengths of the sides and \+
hypotenuse of a right triangle, from which it follows that " }}{PARA 
257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }
{XPPEDIT 18 0 "abs(x) <= abs(z);" "6#1-%$absG6#%\"xG-F%6#%\"zG" }
{TEXT -1 14 "      and     " }{XPPEDIT 18 0 "abs(y) <= abs(z);" "6#1-%
$absG6#%\"yG-F%6#%\"zG" }{TEXT -1 3 " . " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 38 "Theorem 1.2  (The tria
ngle inequality)" }{TEXT 280 6 "   If " }{XPPEDIT 18 0 "z[1];" "6#&%\"
zG6#\"\"\"" }{TEXT 281 5 " and " }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"
\"#" }{TEXT 282 35 " are arbitrary comples numbers then" }}{PARA 257 "
" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 279 5 "     " }
{XPPEDIT 18 0 "abs(z[1] + z[2]) <= abs(z[1]) + abs(z[2])" "6#1-%$absG6
#,&&%\"zG6#\"\"\"F+&F)6#\"\"#F+,&-F%6#&F)6#F+F+-F%6#&F)6#F.F+" }{TEXT 
278 3 " . " }{TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 270 1 "\n
" }{TEXT 256 22 "Example  1.5, Page 19." }{TEXT 266 38 "   Verify the \+
triangle inequality for " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }
{TEXT 261 6 " and  " }{XPPEDIT 18 0 "z[2]" "6#&%\"zG6#\"\"#" }{TEXT 
262 3 " .\n" }{XPPEDIT 18 0 "abs(z[1] + z[2]) <= abs(z[1]) + abs(z[2])
" "6#1-%$absG6#,&&%\"zG6#\"\"\"F+&F)6#\"\"#F+,&-F%6#&F)6#F+F+-F%6#&F)6
#F.F+" }{TEXT 263 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 306 "z1 := 7 + I:  `z1 ` = z1;\nz2 := 3 + 5*I:  `z
2 ` = z2;\n`z1 + z2 ` = z1 + z2; ` `;\n`|z1| ` = abs(z1);\n`|z2| ` = a
bs(z2);\n`|z1 + z2| ` = abs(z1 + z2); ` `;\n`|z1 + z2| <= |z1| + |z2|`
;\nabs(z1 + z2) <= abs(z1) + abs(z2);\nevalf(abs(z1 + z2) <= abs(z1) +
 abs(z2));\nevalb(evalf(abs(z1 + z2) <= abs(z1) + abs(z2)));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 22 "Example  1.6, Page 20.
" }{TEXT 267 16 "   Verify that  " }{XPPEDIT 18 0 "abs( z[1]*z[2]) = a
bs(z[1])*abs(z[2])" "6#/-%$absG6#*&&%\"zG6#\"\"\"F+&F)6#\"\"#F+*&-F%6#
&F)6#F+F+-F%6#&F)6#F.F+" }{TEXT 264 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "z1 := 1 + 2*I;\nz2 := 3 + 2
*I;\n`z1*z2 ` = z1*z2; ` `;\n`|z1| ` = abs(z1);\n`|z2| ` = abs(z2);\n`
|z1*z2| ` = abs(z1*z2); ` `;\n`|z1*z2| = |z1|*|z2|`;\nabs(z1*z2) = abs
(z1)*abs(z2);\ncombine(abs(z1*z2) = abs(z1)*abs(z2),power);\nevalb(com
bine(abs(z1*z2) = abs(z1)*abs(z2),power));" }}}{EXCHG {PARA 257 "" 0 "
" {TEXT 260 19 "End of Section 1.3." }}}}{MARK "0 0 0" 15 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }