Analytic function: Difference between revisions

Revision as of 04:30, 11 October 2024

Revision as of 01:56, 20 October 2024

Line 5:

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]]. ItThis is astronger neighborhoodthan andmerely notbeing just[[smoothness|infinitely atdifferentiable]] some pointat <math> x_0 </math>,{{cn|date=October 2024}}and sincetherefore every differentiable function has at leasthaving a tangentwell-defined 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 [[TaylorFabius seriesfunction]] mustprovides alsoan convergeexample toof thea function onthat pointsis adjacentinfinitely todifferentiable <math>but x_0 </math> to be considered annot analytic function. As a counterexample see the [[Weierstrass function]] or the [[Fabius function]].

== Definitions ==