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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple
 Worksheets,  2001" }{TEXT 276 144 "\n(c) John H. Mathews          Rus
sell W. Howell\nmathews@fullerton.edu     howell@westmont.edu\n\nCompl
imentary software to accompany the textbook:" }}{PARA 258 "" 0 "" 
{TEXT 257 82 "COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed,
 2001, ISBN: 0-7637-1425-9" }{TEXT 275 197 "\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 
273 1 "\n" }{TEXT 256 30 "CHAPTER 6  COMPLEX INTEGRATION" }{TEXT 267 
2 "\n\n" }{TEXT 256 30 "Section 6.1  Complex Integrals" }{TEXT 268 6 "
\n\n    " }{TEXT -1 408 "In Chapter 3 we defined the derivative of a c
omplex function. We now turn our attention to the problem of integrati
ng complex functions. We will find that integrals of analytic function
s are well behaved and that many properties from Calculus carry over t
o the complex case. To introduce the integral of a complex function, w
e start by defining what we mean by the integral of a complex-valued f
unction of a " }{TEXT 278 4 "real" }{TEXT -1 12 " variable.  " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 33 "Definition 6.1:  Integral of f
(t)" }{TEXT 277 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 5 "Let  " }{XPPEDIT 18 0 "f(t) = u(t)+i*v(t)" "6#/-%\"fG6#
%\"tG,&-%\"uG6#F'\"\"\"*&%\"iGF,-%\"vG6#F'F,F," }{TEXT -1 9 ",  where \+
" }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v;
" "6#%\"vG" }{TEXT -1 48 " are real-valued functions of the real varia
ble " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 7 " for  a" }{XPPEDIT 
18 0 "`` <= ``;" "6#1%!GF$" }{TEXT -1 1 "t" }{XPPEDIT 18 0 "`` <= ``" 
"6#1%!GF$" }{TEXT -1 7 "b, then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 280 5 "     " }{XPPEDIT 18 0 "int(f(t),t = a .. \+
b) = int(u(t),t = a .. b)+i*int(v(t),t = a .. b);" "6#/-%$intG6$-%\"fG
6#%\"tG/F*;%\"aG%\"bG,&-F%6$-%\"uG6#F*/F*;F-F.\"\"\"*&%\"iGF7-F%6$-%\"
vG6#F*/F*;F-F.F7F7" }{TEXT 279 4 " .  " }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 83 "   We generally evaluate integrals of t
his type by finding the antiderivatives of  " }{XPPEDIT 18 0 "u;" "6#%
\"uG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 86 "
 and evaluating the definite integrals on the right side of the equati
on. That is, if " }{XPPEDIT 18 0 "`U '(t) = u(t)`;" "6#%.U~'(t)~=~u(t)
G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`V '(t) = v(t)`;" "6#%.V~'(t)~=
~v(t)G" }{TEXT -1 6 " for a" }{XPPEDIT 18 0 "`` <= ``;" "6#1%!GF$" }
{TEXT -1 1 "t" }{XPPEDIT 18 0 "`` <= ``" "6#1%!GF$" }{TEXT -1 11 "b, w
e have " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 
5 "     " }{XPPEDIT 18 0 "int(f(t),t = a .. b) = U(b)-U(a)+i*(V(b)-V(a
));" "6#/-%$intG6$-%\"fG6#%\"tG/F*;%\"aG%\"bG,(-%\"UG6#F.\"\"\"-F16#F-
!\"\"*&%\"iGF3,&-%\"VG6#F.F3-F;6#F-F6F3F3" }{TEXT 281 4 " .  " }{TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 274 1 "\n" }{TEXT 256 22 "Example 6.1, Page 202." }{TEXT 
269 15 "   Show that   " }{XPPEDIT 18 0 "int((t - i)^3, t=0..1) = -5/4
" "6#/-%$intG6$*$,&%\"tG\"\"\"%\"iG!\"\"\"\"$/F);\"\"!F*,$*&\"\"&F*\"
\"%F,F," }{TEXT -1 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "f:='
f': t:='t': u:='u': v:='v':\nf := t -> (t+I)^3:\n`f(t) ` = f(t);\n`f(t
) ` = evalc(f(t));\nu := u -> t^3 - 3*t:\nv := v -> - 3*t^2 + 1:\n`u(t
) ` = u(t);\n`v(t) ` = v(t);\nInt(u(t)+I*v(t),t) = int(u(t),t) + I*int
(v(t),t);\ndefint := int(u(t),t=0..1) + I*int(v(t),t=0..1):\nInt(f(t),
t=0..1) = defint;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 260 79 "Or we could do the integral directly, wi
th complex function for the integrand.\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 258 "f:='f': g:='g': t:='t': T:='T':\nf := t -> (t+I)^3:
\ng := t -> subs(T=t, int(f(T),T)):\n`f(t) ` = f(t);\n`g(t) = `,Int(f(
t),t) = g(t);\n`g(1) ` = g(1),`  and  `,\n`g(0) ` = g(0);\n`g(1) - g(0
) ` = g(1) - g(0);\ndefint := int(f(t), t=0..1):\nInt(f(t),t=0..1) = d
efint;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 256 22 "Example 6.2, Page 202." }{TEXT 270 15 "   Show that
   " }{XPPEDIT 18 0 "int(exp(t+i*t),t=0..pi/2) = (e^(pi/2) - 1)/2 + i/
2*(e^(pi/2) + 1)" "6#/-%$intG6$-%$expG6#,&%\"tG\"\"\"*&%\"iGF,F+F,F,/F
+;\"\"!*&%#piGF,\"\"#!\"\",&*&,&)%\"eG*&F3F,F4F5F,F,F5F,F4F5F,*(F.F,F4
F5,&)F:*&F3F,F4F5F,F,F,F,F," }{TEXT 264 3 " .\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 309 "f:='f': t:='t': u:='u': v:='v':\nf := t -> exp(t+I*t
):\n`f(t) ` = f(t);\n`f(t) ` = evalc(f(t));\nu := u -> exp(t)*cos(t):
\nv := v -> exp(t)*sin(t):\n`u(t) ` = u(t);\n`v(t) ` = v(t);\nInt(u(t)
+I*v(t),t) = int(u(t),t) + I*int(v(t),t);\ndefint := int(u(t),t=0..Pi/
2) + I*int(v(t),t=0..Pi/2):\nInt(f(t),t=0..pi/2) = defint;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 79 "
Or we could do the integral directly, with complex function for the in
tegrand.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "f:='f': g:='g': t:='
t': T:='T':\nf := t -> exp(t+I*t):\ng := t -> subs(T=t, int(f(T),T)):
\n`f(t) ` = f(t);\n`g(t) = `,Int(f(t),t) = g(t);\n`g(Pi/2) ` = g(Pi/2)
,`  and  `,\n`g(0) ` = g(0);\n`g(Pi/2) - g(0) ` = g(Pi/2) - g(0);\ndef
int := int(f(t), t=0..Pi/2):\nInt(f(t),t=0..pi/2) = defint;\nInt(f(t),
t=0..pi/2) = evalc(defint);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 1 "\n" }{TEXT 256 22 "Example 6.3, \+
Page 203." }{TEXT 271 19 "   Show that\n      " }{XPPEDIT 18 0 "int((c
 + i*d)*(u(t) + i*v(t)),t) = c*int(u(t),t) - d*int(v(t),t) + i*d*int(u
(t),t) + i*c*int(v(t),t)" "6#/-%$intG6$*&,&%\"cG\"\"\"*&%\"iGF*%\"dGF*
F*F*,&-%\"uG6#%\"tGF**&F,F*-%\"vG6#F2F*F*F*F2,**&F)F*-F%6$-F06#F2F2F*F
**&F-F*-F%6$-F56#F2F2F*!\"\"*(F,F*F-F*-F%6$-F06#F2F2F*F**(F,F*F)F*-F%6
$-F56#F2F2F*F*" }{TEXT 265 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
148 "c:='c': d:='d': S:='S': t:='t': u:='u': v:='v':\ns := int((c + I*
d)*(u(t) + I* v(t)), t): S = s;\ns := expand(s): S = s;\ns := evalc(ex
pand(s)): S = s;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 256 22 "Example 6.4, Page 204." }{TEXT 272 15 " \+
  Show that   " }{XPPEDIT 18 0 "int(exp(i*t), t=0..pi) = 2*i" "6#/-%$i
ntG6$-%$expG6#*&%\"iG\"\"\"%\"tGF,/F-;\"\"!%#piG*&\"\"#F,F+F," }{TEXT 
266 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "f:='f': g:='g': t:=
't': T:='T':\nf := t -> exp(I*t):\ng := t -> subs(T=t, int(f(T),T)):\n
`f(t) ` = f(t);\n`g(t) = `,Int(f(t),t) = g(t);\n`g(Pi) ` = g(Pi),`  an
d  `,\n`g(0) ` = g(0);\n`g(Pi) - g(0) ` = g(Pi) - g(0);\ndefint := int
(f(t), t=0..Pi):\nInt(f(t),t=0..pi) = defint;" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT 263 19 "End of Section 6.1." }}}}{MARK "0 0 0" 11 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }