# Analytic function

In

mathematics, an

**analytic function** is one that is locally given by a convergent

power series.

Complex analysis teaches us that if a function *f* is differentiable in some open disk *D* centered at a point *c* in the complex field, then it necessarily has derivatives of all orders in that same open neighborhood, and the power series

converges to

*f*(

*z*) at every point within

*D*. That is an important respect in which complex functions are better-behaved than real functions; see

an infinitely differentiable function that is not analytic. Consequently, the term

*analytic function* becomes synonymous with

*holomorphic function*.