Editing Analytic function (section)

{{Short description|Type of function in mathematics}}
{{Distinguish|analytic expression|analytic signal}}
{{about|both real and complex analytic functions|analytic functions in complex analysis specifically|holomorphic function|analytic functions in SQL|Window function (SQL)}}
{{Complex analysis sidebar}}
In [[mathematics]], an '''analytic function''' is a [[function (mathematics)|function]] that is locally given by a [[convergent series|convergent]] [[power series]]. There exist both '''real analytic functions''' and '''complex analytic functions'''. Functions of each type are [[smooth function|infinitely differentiable]], but complex analytic functions exhibit properties that do not generally hold for real analytic functions.

A function is analytic if and only if for every <math> x_0 </math> in its [[Domain of a function|domain]], its [[Taylor series]] about <math> x_0 </math> converges to the function in some [[neighborhood (topology)|neighborhood]] of <math> x_0 </math>. This is stronger than merely being [[smoothness|infinitely differentiable]] at <math> x_0 </math>, and therefore having a well-defined Taylor series; the [[Fabius function]] provides an example of a function that is infinitely differentiable but not analytic.