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.