148
INTRODUCTION
TO
COMPLEX
VARIABLES
I-
____
real
Figure
6.8
The Contour
C,
for
the
Cauchy Integral
Formula
Integral Theorem to argue that
12
=
0.
However, the integrand
is
analytic everywhere,
except at the point
z
=
5,
so
we can deform the contour into a infinitesimal circle of
radius
E,
centered at
b,
without
affecting
the value:
(6.76)
This deformation
is
depicted in Figure
6.9.
The integral expression of Equation
6.76
can be visualized most easily with the use
of phasors, which
are
essentially vectors in the complex plane. Consider Figure 6.10.
In Equation 6.76,
z
lies on the contour C, and is represented
by
a phasor from the
origin
to
that point. The point
5
is represented by another phasor from the origin
to
L.
In
the
same way, the quantity
z
-
z,
is
represented
by
a phasor that goes from
~0
to
z.
If the angle
<b
is
defined as shown, using polar notation we have
z
-
5
=
Ee'@.
(6.77)
On
C,,
E
is
a
constant
so
(6.78)
imag
real
Figure
6.9
The
Equivalent Circular Contour
for
the
Cauchy Integral
Formula
