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 $a_{n},n\geq 0$ such that:

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

$f(z)=\sum _{n=0}^{\infty }a_{n}(z-z_{0})^{n}\ \ \forall z,|z-z_{0}|\leq r$