Definition talk:Analytic Function
Analytic vs. holomorphic
It is fairly standard terminology these days to say that an analytic function is one that can be expressed by a power series, while a holomorphic function is one that is complex differentiable (on an open set). The two definitions are equivalent, but this is not trivial - it requires the development of Cauchy's integral theorem.
I would suggest defining analytic and holomorphic separately, with a note of course that these two definitions coincide, with a reference to Analytic Functions are Holomorphic, an article that is yet to be created.
As an aside, I believe Bers used to refer to the equivalence of these definitions, and some others, as the Fundamental Theorem of Complex Analysis, as in the recent book 'Complex Analysis in the spirit of Lipman Bers', by Kra et al. lasserempe 09:13, 16 February 2009 (UTC)