More
applications of the Residue Theorem
More complicated integrals require using the following:
Lemma.
Let
be
a simple pole of f(z) with residue
.
Consider the arc
of the circle
that subtends an angle
at the center. Then
as
.
Proof. Write
where g(z) is analytic inside some
disk
. If 0<r<, then
.
Now , if
M is the bound of |g(z)| in
,
.
Moreover, with
as a parametrization on
,
.
Consequently, the proof of the lemma is complete.
To
illustrate, we consider the following example:
Let
a>0
and let us try to evaluate
.
Let C
be the upper semicircle with a large radius R
indented at –a and a by small semicircles
with small radii r and
respectively and
consider
. This integral is 0 by Cauchy’s theorem since
the integrand has no singularities inside C. Thus
.
The
last integral can be proved to tend to 0 as
using Jordan Lemma. Next we apply the previous Lemma
with
.
Since
and
,
as
and
as
.
Consequently, taking real parts the final value is. |