          Content Sessions  RESIDUE THEORY

1.1          Cauchys Residue Theorem

We now state a theorem of practical importance called the Cauchy Residue Theorem which allows us to compute easily many integrals some of which can only be evaluated using this theorem.

Theorem 1. (Cauchy Residue Theorem). Suppose that is an analytic function on a simply-connected domain except for a finite number of poles Suppose that is a piecewise smooth positively-oriented simple (does not intersect itself) closed curve not passing through Then ,

where the sum is over all the poles of that are inside Proof. By a previous theorem, By Cauchys Theorem, Moreover, since  1.2          Jordans Lemma

In some cases we use the following:

Lemma 1. (Jordans lemma). If represents the curve then Proof. On , and Thus  In the last integral let , then using we get, But for from which the result follows.

As an application of Jordans lemma we consider the following

Example. Show that Method : Since is an even function, it suffices to show that Let Consider the integral where However, since the singularity is on the contour of integration, we consider instead the integral In this case, has a removable singularity Thus the function is entire and is a closed contour in . Then by Cauchys Theorem we have Then So  Equating imaginary parts we get So by using Jordans lemma.
Finally, letting establishes the result.
Method : Let be large and be small. Now let represent the curve Consider the integral where Note that the function has no singularities inside . Thus So Letting in the first integral we get Combining we get or Moreover, near we have the expansion Hence  +terms that tend to as Furthermore, using Jordans lemma, we get Finally, letting and establishes the result. 