Locally analytic function

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Definition

At a point

A function f of one variable is said to be locally analytic (or sometimes simply analytic) at a point x_0 in the interior of its domain if it satisfies the following equivalent conditions:

  1. There exists a (unique) power series centered at x_0 that converges to f on an interval of positive radius centered at x_0.
  2. The Taylor series of f at x_0 converges to f on an interval of positive radius centered at x_0.

Note that (1) and (2) are equivalent in the following sense: if there is a power series centered at x_0 that converges to f on an interval of positive radius centered at x_0, that power series must equal the Taylor series.

On a subset of the domain

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

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