Prove: If f(z) has a pole of order m at z = a, then the residue of f(z) at z = a is given by

if m =1, and by

if m > 1.

Proof. We wish to prove that

If f(z) has a pole of order m at z = a, then f(z) = g(z)/(z - a)^m where g(z) is analytic inside and on C, and g(a) 0. Thus

We shall now employ Cauchy’s integral formula

which we shall rewrite as

or, changing notation,

Thus 1) above becomes

Since

5) becomes

If we define 0! = 1, the above also proves the case for m = 1, which corresponds to the case of n = 0 in Cauchy’s integral formula 2).