Locally analytic function

From Calculus
Revision as of 13:58, 8 July 2012 by Vipul (talk | contribs) (Created page with "==Definition== ===At a point=== A function <math>f</math> of one variable is said to be '''locally analytic''' (or sometimes simply '''analytic''') at a point <math>x_0</mat...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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.