Cauchy's Integral Formula
Cauchy's Integral Formula is a fundamental result in complex analysis. It states that if is a subset of the complex plane containing a simple counterclockwise loop and the region bounded by , and is a complex-differentiable function on , then for any in the interior of the region bounded by ,
Let denote the interior of the region bounded by . Let denote a simple counterclockwise loop about of radius . Since the interior of the region bounded by is an open set, there is some such that for all . For such values of , by application of Cauchy's Integral Theorem.
Since is differentiable at , for any we may pick an arbitarily small such that whenever . Let us parameterize as , for . Since (again by Cauchy's Integral Theorem), it follows that Since and can simultaneously become arbitrarily small, it follows that which is equivalent to the desired theorem.
By induction, we see that the th derivative of at is for . In particular, the th derivative exists at , for all . In other words, if a function is complex-differentiable on some region, then it is infinitely differentiable on the interior of that region.
Since the th derivative exists in general, it follows that the th derivative is continuous. This is not true for functions of real variables! For instance the real function is everywhere differentiable, but its derivative is mysteriously not continuous at . In complex analysis, the mystery disappears: the function has an essential singularity at , so we can't establish a derivative there in any case.
The theorem is useful for estimating a function (or its th derivative) at a point based on the behavior of the function around the point. For instance, the theorem yields an easy proof that holomorphic functions are in fact analytic.