Evaluation Of Definite Integrals

Basic Example of the Indefinite Integral:

On the one hand,

where we sum only over the poles of f that lie inside These are P₁ and P₂. An easy calculation shows that

Of course the index at each point is 1. So

On the other hand,

as R → . That is good, because this last is the integral that we wish to evaluate. Better still,

Finally we get;

This solves the problem: the value of the integral is