Function complex-analytic at a point

This article defines a property that can be evaluated for a function on a (particular kind of) set, and a point in that set. A function satisfying the property at every point, it is termed a holomorphic function
View other properties of functions at points

Definition

In one dimension

Suppose $U$ is an open subset (without loss of generality, a domain, i.e. an open connected subset) of $\mathbb{C}$, $f:U\to \mathbb{C}$ be a function, and $z_0\in U$ a point. We say that $f$ is complex-analytic at $z_0$ if there exists a positive integer $r$ a sequence of complex numbers $a_n,n\ge 0$ such that:

• The ball of radius $r$ about $z_0$ lies completely inside $U$
• The power series $\sum _{n=0}^{\mathrm{\infty }}{a}_n(z-z_0{)}^n$ has radius of convergence at least $r$
• The power series converges to the function on the ball of radius $r$:

$f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n(z-z_0{)}^n\mathrm{\forall }z,|z-z_0|\le r$