Analytic function: Difference between revisions

Revision as of 16:29, 17 June 2023

Revision as of 21:36, 19 June 2023

Line 3:

{{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 its [[Taylor series]] about <math> x_0 </math> converges to the function in some [[neighborhood (topology)|neighborhood]] for every <math> x_0 </math> in its [[Domain of a function|___domain]]. It is important to note that it is's a neighborhood and not just at some point <math> x_0 </math>, since everevery differentiable function has at least a tangent line at every point, which is hisit's [[Taylor series]] of order 1,. soSo just having a polynomial expansion at singular points is not enough, and the [[Taylor series]] must also converge to the function also on points adjacent points to <math> x_0 </math> to be considered an Analytic function,. asAs a counterexample see the [[Fabius function]].

== Definitions ==