Proof
: Note that the integrand
f
(
z
) = 1
/
(
z
−
a
) is analytic everywhere except at
z
=
a
, where it has a simple pole. If
a
is outside
C
, then Cauchy’s Theorem 7.48 applies,
and the integral is zero.
On the other hand, if
a
is inside
C
, then Proposition 7.50
implies that the integral is equal to the integral around a circle centered at
z
=
a
. The
latter integral can be computed directly by using the parametrization
z
(
t
) =
a
+
re
i
t
for
0
≤
t
≤
2
π
, as in (7.114).
Q.E.D.
Example 7.53.
Let
D
⊂
C
be a closed and
connected
domain. Let
a,b
∈
D
be two
points in
D
. Then
contintegraldisplay
C
parenleftbigg
1
z
−
a
−
1
z
−
b
parenrightbigg
dz
=
contintegraldisplay
C
dz
z
−
a
−
contintegraldisplay
C
dz
z
−
b
= 0
for any closed curve
C
⊂
Ω =
C
\
D
lying outside the domain
D
.
This is because,
by connectivity of
D
, either
C
contains both points in its interior, in which case both
integrals equal 2
π
i , or
C
contains neither point, in which case both integrals are 0. The
conclusion is that, while the individual logarithms are multiplyvalued, their difference
F
(
z
) = log(
z
−
a
)
−
log(
z
−
b
) = log
z
−
a
z
−
b
(7
.
124)
is a consistent, singlevalued complex function on all of Ω =
C
\
D
. The difference (7.124)
has, in fact, an infinite number of possible values, differing by integer multiples of 2
π
i ;
the ambiguity can be resolved by choosing one of its values at a single point in Ω. These
conclusions rest on the fact that
D
is connected, and are
not
valid, say, for the twice
punctured plane
C
\ {
a,b
}
.
Lift and Circulation
In fluid mechanical applications, the complex integral can be assigned an important
physical interpretation. As above, we consider the steady state flow of an incompressible,
irrotational fluid. Let
f
(
z
) =
u
(
x,y
)
−
i
v
(
x,y
) denote the complex velocity corresponding
to the real velocity vector
v
= (
u
(
x,y
)
,v
(
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Olver
 Fluid Dynamics, Differential Equations, Equations, Partial Differential Equations, Line integral, Logarithm, Peter J. Olver, Complex velocity

Click to edit the document details