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

${\displaystyle \sum _{n=0}^{\infty }{f^{(n)}(c) \over n!}(z-c)^{n}}$

converges to f(z) at every point within D. Consequently, the term analytic function becomes synonymous with holomorphic function.