# Residues.

Let be a function analytic on a simple Jordan curve and at all the interior points, excepted at . The residue of at , denoted Res[ ] is the complex number;

 Res

Theorem 9.1.1   The residue of at is the coefficient in the Laurent series expansion of in an annulus .

Example 9.1.2   Compute the integral , where .

We compute a Laurent series expansion for which is convergent on an annulus centered at -1.

Therefore .

How to compute quickly residues?
1. If has a pole of order 1 at , then

 Res

2. If has a pole of order 2 at , then

 Res

3. If has a pole of order n at , then

 Res

4. If has a simple pole at , then

 Res

Noah Dana-Picard 2007-12-24