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.

Formally, a function ${\displaystyle f}$ is real analytic on an open set ${\displaystyle D}$ in the real line if for any ${\displaystyle x_{0}\in D}$ one can write $${\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-x_{0}\right)^{n}=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+\cdots }$$

in which the coefficients ${\displaystyle a_{0},a_{1},\dots }$ are real numbers and the series is convergent to ${\displaystyle f(x)}$ for ${\displaystyle x}$ in a neighborhood of ${\displaystyle x_{0}}$.

Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point ${\displaystyle x_{0}}$ in its domain

$${\displaystyle T(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}$$

converges to ${\displaystyle f(x)}$ for ${\displaystyle x}$ in a neighborhood of ${\displaystyle x_{0}}$ pointwise. The set of all real analytic functions on a given set ${\displaystyle D}$ is often denoted by ${\displaystyle {\mathcal {C}}^{\,\omega }(D)}$, or just by ${\displaystyle {\mathcal {C}}^{\,\omega }}$ if the domain is understood.

A function ${\displaystyle f}$ defined on some subset of the real line is said to be real analytic at a point ${\displaystyle x}$ if there is a neighborhood ${\displaystyle D}$ of ${\displaystyle x}$ on which ${\displaystyle f}$ is real analytic.