<P> 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, categories that are similar in some ways, but different in others . Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions . A function is analytic if and only if its Taylor series about x converges to the function in some neighborhood for every x in its domain . </P> <P> Formally, a function f (\ displaystyle f) is real analytic on an open set D (\ displaystyle D) in the real line if for any x 0 ∈ D (\ displaystyle x_ (0) \ in D) one can write </P>

When is a function said to be analytic