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 its Taylor series about ${\displaystyle x_{0}}$ converges to the function in some neighborhood for every ${\displaystyle x_{0}}$ in its domain. It is important to note that it is a neighborhood and not just at some point ${\displaystyle x_{0}}$, 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 ${\displaystyle x_{0}}$ to be considered an analytic function. As a counterexample see the Weierstrass function or the Fabius function.