Analytic function

Mathematical analysis → Complex analysis
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In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are 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 ${\displaystyle x_{0}}$ in its domain, its Taylor series about ${\displaystyle x_{0}}$ converges to the function in some neighborhood of ${\displaystyle x_{0}}$. This is stronger than merely being infinitely differentiable at ${\displaystyle x_{0}}$, and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic.