Figure 7.32.
Orientation of Domain Boundary.
Theorem 7.48.
If
f
(
z
)
is analytic on a bounded domain
Ω
⊂
C
, then
contintegraldisplay
∂
Ω
f
(
z
)
dz
= 0
.
(7
.
118)
Proof
: If we apply Green’s Theorem to the two real line integrals in (7.109), we find
contintegraldisplay
∂
Ω
udx
−
vdy
=
integraldisplay integraldisplay
Ω
parenleftbigg
−
∂v
∂x
−
∂u
∂y
parenrightbigg
= 0
,
contintegraldisplay
∂
Ω
vdx
+
udy
=
integraldisplay integraldisplay
Ω
parenleftbigg
∂u
∂x
−
∂v
∂y
parenrightbigg
= 0
,
both of which vanish by virtue of the Cauchy–Riemann equations (7.18).
Q.E.D.
If the domain of definition of our complex function
f
(
z
) is simply connected, then, by
definition, the interior of any closed curve
C
⊂
Ω is contained in Ω, and hence Cauchy’s
Theorem 7.48 implies path independence of the complex integral within Ω.
Corollary 7.49.
If
f
(
z
)
is analytic on a simply connected domain
Ω
⊂
C
, then its
complex integral
integraldisplay
C
f
(
z
)
dz
for
C
⊂
Ω
is independent of path. In particular,
contintegraldisplay
C
f
(
z
)
dz
= 0
(7
.
119)
for any closed curve
C
⊂
Ω
.
Remark
: Simple connectivity of the domain is an essential hypothesis — our evalua
tion (7.114) of the integral of 1
/z
around the unit circle provides a simple counterexample
to (7.119) in the nonsimply connected domain Ω =
C
\ {
0
}
.
Interestingly, this result
also admits a converse: a continuous complexvalued function that satisfies (7.119) for
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 Fall '10
 Olver
 Differential Equations, Equations, Integrals, Partial Differential Equations, Line integral, 2k, Cauchy, closed curve, Peter J. Olver

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