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{{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 a neighborhood and not just at some point <math> x_0 </math>, since every differentiable function has at least a tangent line at every point, which is its [[Taylor series]] of order 1. So just having a polynomial expansion at singular points is not enough, and the [[Taylor series]] must also converge to the function on points adjacent to <math> x_0 </math> to be considered an analytic function. As a counterexample see the [[Weierstrass function]] or the [[Fabius function]]. == Definitions ==
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