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  RESIDUE THEORY

1.1          Cauchyís 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 Cauchyís Theorem, Moreover, since

 

1.2          Jordanís Lemma

In some cases we use the following:

Lemma 1. (Jordanís 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 Jordanís 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 Cauchyís Theorem we have

Then

So

 

Equating imaginary parts we get

So

by using Jordanís 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 Jordanís lemma, we get

Finally, letting and establishes the result.