# Locally analytic function

## Definition

### At a point

A function of one variable is said to be **locally analytic** (or sometimes simply **analytic**) at a point in the interior of its domain if it satisfies the following equivalent conditions:

- There exists a (unique) power series centered at that converges to on an interval of positive radius centered at .
- The Taylor series of at converges to on an interval of positive radius centered at .

Note that (1) and (2) are equivalent in the following sense: if there is a power series centered at that converges to on an interval of positive radius centered at , that power series *must* equal the Taylor series.

### On a subset of the domain

A function of one variable is said to be **locally analytic** (or sometimes simply **analytic**) on an open subset of the domain if it is locally analytic at every point of .

If the open subset is the whole domain, we may simply say that is locally analytic.