# Application to integrals.

Theorem 9.2.1 (Residue Theorem)   Let be a function, analytic at all the interior points of a Jordan curve , excepted a finite number of isolated singular points . Then:

 Res

Note that:

• is oriented in the positive direction.
• Only the interior singularities are involved.

Example 9.2.2   We wish to compute the integral , where

The function has a pole of order 1 at , a pole of order 2 at1 and a pole of order 3 at 3; this last pole is irrelevant to our computation as it lies out of the Jordan curve defined by . So we compute two residues:

• Res
• Res
Thus:

Example 9.2.3   Compute the integral .

We use 2.1: . If , we have

The denominator has two complex roots and . Only the second one is a point interior to the unit-circle, i.e. it is the only pole of the function we have to deal with. Therefore:

 Res

Noah Dana-Picard 2007-12-24