In complex analysis, a complex-valued function ${\displaystyle f}$ of a complex variable ${\displaystyle z}$:

One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa. Among the corollaries of this theorem are

The argument, first given by Cauchy, hinges on Cauchy's integral formula and the power series expansion of the expression

Let ${\displaystyle D}$ be an open disk centered at ${\displaystyle a}$ and suppose ${\displaystyle f}$ is differentiable everywhere within an open neighborhood containing the closure of ${\displaystyle D}$. Let ${\displaystyle C}$ be the positively oriented (i.e., counterclockwise) circle which is the boundary of ${\displaystyle D}$ and let ${\displaystyle z}$ be a point in ${\displaystyle D}$. Starting with Cauchy's integral formula, we have

Interchange of the integral and infinite sum is justified by observing that ${\displaystyle f(w)/(w-a)}$ is bounded on ${\displaystyle C}$ by some positive number ${\displaystyle M}$, while for all ${\displaystyle w}$ in ${\displaystyle C}$

for some positive ${\displaystyle r}$ as well. We therefore have

on ${\displaystyle C}$, and as the Weierstrass M-test shows the series converges uniformly over ${\displaystyle C}$, the sum and the integral may be interchanged.

As the factor ${\displaystyle (z-a)^{n}}$ does not depend on the variable of integration ${\displaystyle w}$, it may be factored out to yield