More applications of the Residue Theorem

More complicated integrals require using the following:

LemmaLet  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.