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The absolute value function is elementary but not analytic. It is not correct to say that "all elementary functions" are analytic. |
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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)}}
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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 ever differentiable function has at least a tangent line at every point, which is his [[Taylor series]] of order 1, so just having a polynomial expansion at singular points is not enough, the [[Taylor series]] must converge to the function also on adjacent points to <math> x_0 </math> to be considered an Analytic function, as counterexample see the [[Fabius function]].
== Definitions ==
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