The ramification index can be calculated explicitly from Cauchy's integral formula.

The integral formula has broad applications.

One such formula is the following integral formula involving a product of three Bessel functions:

He used this to give the following integral formula for the zeta function:

This is really a restatement of Cauchy's integral formula.

Cauchy's integral formula suggests the following definition (purely formal, for now):

More useful integral formulas for P are necessary for any practical calculation.

Note that both of these integral formulas do not converge in any usual sense for "typical" systems .

In general, derivatives of any order can be calculated using Cauchy's integral formula:

A number of integral formulas involving the eta function can be listed.