Prove: Residue theorem. Let f(z) be analytic inside and on a simple closed curve C except at the singularities a, b, c, ... inside C which have residues given by ar, br, cr ... . Then
Proof. Construct circles C1, C2, C3, ... centered at a, b, c, ... , of such radii as to lie entirely within C as shown in Fig. 1.
Now by Theorem 6 in the section on Complex integration and Cauchy’s theorem we have