Analyticity of holomorphic functions

In complex analysis, a complex-valued function f of a complex variable is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a. It is analytic at a if it can be expanded as a power series

$f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}$

whose radius of convergence is positive.

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

• the fact that two holomorphic functions that agree at every point in some infinite bounded set agree everywhere in some open set, and

• the fact that holomorphic functions are differentiable not just once, but infinitely often, and

• the fact that the radius of convergence is always the distance from the center a to the nearest singularity; if there are no singularities (i.e., f is an entire function, then the radius of convergence is infinite.

Proof

Suppose f is differentiable everywhere within open some disk centered at a. Let z be within that disk. Let C be a positively oriented circle centered at a, lying within that open disk, but farther from a than z is. Then, by Cauchy's integral formula, we have