ABSTRACT
Cauchy's residue theorem may then be used to evaluate the contour integral in terms of residues.
A contour integral is an integral for which the integration path is a closed contour in the complex plane. Contour integrals can often be evaluated by using Cauchy's residue theorem. One statement of the theorem is
Theorem: Let f(z) be analytic in a simply connected domain D except for finitely many points {a;} at which f may have isolated singularities. Let C be a simple closed contour, traversed in the positive sense (see Notes, below), that lies in D and does not pass through any point a; . Then
i /(z) dz = 27ri L residue of I at a; where the sum is extended over all the points {a;} that are inside the contour C (see Figure 30.1).
