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'\'. For a proof of this result, see proof that holomorphic functions are analytic. 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, in complex analysis, the term analytic function
is synonymous with holomorphic function''.