# 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. |