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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 266 144 "\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 265 197 "\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 
269 1 "\n" }{TEXT 256 30 "CHAPTER 6  COMPLEX INTEGRATION" }{TEXT 267 
2 "\n\n" }{TEXT 256 71 "Section 6.6  The Theorems of Morera and Liouvi
lle and Some Applications" }{TEXT 268 1 "\n" }{TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 246 "  In this section we investigate some of the qua
litative properties of analytic and harmonic functions. Our first resu
lt shows that the existence of an antiderivative for a continuous func
tion is equivalent to the statement that the integral of  " }{XPPEDIT 
18 0 "f;" "6#%\"fG" }{TEXT -1 138 "  is independent of the path of int
egration. This result is stated in a form that will serve as a convers
e to the Cauchy-Goursat theorem. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 30 "Theorem 6.13  (Morera Theorem)" }{TEXT -1 8 "   Let  " }
{XPPEDIT 18 0 "f(z);" "6#-%\"fG6#%\"zG" }{TEXT -1 61 "  be a continuou
s function in a simply connected domain D.   " }}{PARA 0 "" 0 "" 
{TEXT -1 5 "If   " }{XPPEDIT 18 0 "int(f(z),z = C .. ` `) = 0;" "6#/-%
$intG6$-%\"fG6#%\"zG/F*;%\"CG%\"~G\"\"!" }{TEXT -1 42 "   for every cl
osed contour in  D,  then  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }
{TEXT -1 22 "  is analytic in  D.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "  Cauchy's integral formula sho
ws how the value " }{XPPEDIT 18 0 "f(z[0])" "6#-%\"fG6#&%\"zG6#\"\"!" 
}{TEXT -1 91 " can be represented by a certain contour integral. If we
 choose the contour of integration " }{XPPEDIT 18 0 "C;" "6#%\"CG" }
{TEXT -1 28 " to be a circle with center " }{XPPEDIT 18 0 "z[0]" "6#&%
\"zG6#\"\"!" }{TEXT -1 34 ", then we can show that the value " }
{XPPEDIT 18 0 "f(z[0])" "6#-%\"fG6#&%\"zG6#\"\"!" }{TEXT -1 43 " is th
e integral average of the values of  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG
6#%\"zG" }{TEXT -1 11 " at points " }{XPPEDIT 18 0 "z;" "6#%\"zG" }
{TEXT -1 15 " on the circle " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 
3 ".  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 256 42 "Theorem 6.14  (Gauss's Mean Value Theorem)" }{TEXT -1 4 
"    " }}{PARA 0 "" 0 "" {TEXT -1 4 "If  " }{XPPEDIT 18 0 "f(z" "6#-%
\"fG6#%\"zG" }{TEXT -1 76 "  is analytic in a simply connected domain \+
 D  that contains the circle  C: " }{XPPEDIT 18 0 "abs(z-z[0]) = R" "6
#/-%$absG6#,&%\"zG\"\"\"&F(6#\"\"!!\"\"%\"RG" }{TEXT -1 9 " ,  then\n
" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "f(z[0])" "6#-%
\"fG6#&%\"zG6#\"\"!" }{TEXT -1 6 "  =   " }{XPPEDIT 18 0 "1/(2*pi);" "
6#*&\"\"\"F$*&\"\"#F$%#piGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int
(f(z[0]+R*exp(i*theta)),theta = 0 .. 2*pi)" "6#-%$intG6$-%\"fG6#,&&%\"
zG6#\"\"!\"\"\"*&%\"RGF.-%$expG6#*&%\"iGF.%&thetaGF.F.F./F6;F-*&\"\"#F
.%#piGF." }{TEXT -1 7 " .     " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 256 41 "Theorem 6.15  (Maximum Modulus Princ
iple)" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let  " }
{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 55 "  be analytic and
 nonconstant in the domain  D.  Then  " }{XPPEDIT 18 0 "abs(f(z))" "6#
-%$absG6#-%\"fG6#%\"zG" }{TEXT -1 48 "  does not attain a maximum valu
e at any point  " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 
10 "  in  D.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "We sometimes st
ate the maximum modulus principle in the following form." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 41 "Theorem \+
6.16  (Maximum Modulus Principle)" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "
" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 
58 " be analytic and nonconstant in the bounded domain D. If  " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 36 " is continuous on the close
d region " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 51 " that consists \+
of D and all of its boundary points " }{XPPEDIT 18 0 "B;" "6#%\"BG" }
{TEXT -1 8 ", then  " }{XPPEDIT 18 0 "abs(f(z))" "6#-%$absG6#-%\"fG6#%
\"zG" }{TEXT -1 59 "  assumes its maximum value, and does so only at p
oint(s)  " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 17 " on t
he boundary " }{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 37 "Theorem \+
6.17  (Cauchy's Inequalities)" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" 
{TEXT -1 5 "Let  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 
77 "  be analytic and nonconstant in the domain  D  that contains the \+
circle  C: " }{XPPEDIT 18 0 "abs(z-z[0]) = R" "6#/-%$absG6#,&%\"zG\"\"
\"&F(6#\"\"!!\"\"%\"RG" }{TEXT -1 4 " .  " }}{PARA 0 "" 0 "" {TEXT -1 
4 "If  " }{XPPEDIT 18 0 "abs(f(z)) < M" "6#2-%$absG6#-%\"fG6#%\"zG%\"M
G" }{TEXT -1 36 "  holds for all points z on C, then " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "       " }{XPPEDIT 18 
0 "abs(f^`(n)`*`(`*z[0]*`)`) <= n!*M/R^n" "6#1-%$absG6#**)%\"fG%$(n)G
\"\"\"%\"(GF+&%\"zG6#\"\"!F+%\")GF+*(-%*factorialG6#%\"nGF+%\"MGF+)%\"
RGF6!\"\"" }{TEXT -1 11 "    for    " }{XPPEDIT 18 0 "n=1,2,`...`" "6%
/%\"nG\"\"\"\"\"#%$...G" }{TEXT -1 4 " .  " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "  The next result shows t
hat a nonconstant entire function cannot be a bounded function.  " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 36 "T
heorem 6.18,  (Liouville's Theorem)" }{TEXT 274 3 "   " }}{PARA 0 "" 
0 "" {TEXT 275 4 "If  " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }
{TEXT 273 1 " " }{TEXT 276 27 " is an entire function and " }{TEXT 
277 1 " " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 270 2 "  " }
{TEXT 280 28 "is bounded for all values of" }{TEXT 281 2 "  " }
{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 271 1 " " }{TEXT 278 28 " in the co
mplex plane, then " }{TEXT 279 1 " " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#
%\"zG" }{TEXT 272 2 "  " }{TEXT 282 13 "is constant. " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 263 1 
"\n" }{TEXT 256 23 "Example 6.27, Page 256." }{TEXT 262 26 "  Show tha
t the function  " }{XPPEDIT 18 0 "sin(z" "6#-%$sinG6#%\"zG" }{TEXT 
261 5 "  is " }{TEXT 257 3 "NOT" }{TEXT 264 21 " a bounded function.\n
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 299 "i:='i': x:='x': y:='y': z:='z'
:\nf := z -> sin(z):\n`f(z) ` = f(z);\n`f(x + iy) ` = f(x + I*y);\n`f(
x + iy) ` = evalc(f(x + I*y));\n`|f(x + iy)| ` = evalc(abs(f(x + I*y))
);\nf(pi/2 + i*y) = evalc(f(Pi/2 + I*y));\nlim := limit(f(Pi/2 + I*y),
 y=infinity):\nprint(`limit(f(pi/2 + I*y), y=infinity)`,` = `,lim);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 
"We can use Liouville's theorem can be used to establish an important \+
theorem of elementary algebra.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 50 "Theorem 6.19  (The Fundamental Theorem of Algebra)" }{TEXT -1 
4 "    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 
"If  " }{XPPEDIT 18 0 "P(z)" "6#-%\"PG6#%\"zG" }{TEXT -1 39 "  is a po
lynomial of degree  n,  then  " }{XPPEDIT 18 0 "P(z)" "6#-%\"PG6#%\"zG
" }{TEXT -1 24 "  has at least one zero." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 256 13 "Corollary 6.4" }{TEXT -1 8 "   Let  " }{XPPEDIT 18 0 "
P(z)" "6#-%\"PG6#%\"zG" }{TEXT -1 39 "  be a polynomial of degree  n. \+
 Then  " }{XPPEDIT 18 0 "P(z)" "6#-%\"PG6#%\"zG" }{TEXT -1 64 "  can b
e expressed as the product of linear factors.  That is,  " }{XPPEDIT 
18 0 "P(z) = A*(z - z[1])*`...`*(z - z[2])*(z - z[n])" "6#/-%\"PG6#%\"
zG*,%\"AG\"\"\",&F'F*&F'6#F*!\"\"F*%$...GF*,&F'F*&F'6#\"\"#F.F*,&F'F*&
F'6#%\"nGF.F*" }{TEXT -1 9 "  where  " }{XPPEDIT 18 0 "z[1],z[2],`...`
,z[n]" "6&&%\"zG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT -1 20 "  ar
e the zeros of  " }{XPPEDIT 18 0 "P(z)" "6#-%\"PG6#%\"zG" }{TEXT -1 
57 "  counted according to multiplicity an  A  is a constant." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 260 19 "End of Section 6.6." }}}}{MARK "0 0 0" 11 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }