Definition talk:Analytic Function/Complex Plane

I am quite sure this definition is more usually called "Holomorphic", and the one with power series "Analytic". These coincide for univariate complex analysis but in the multivariate case they differ - let alone more general Riemann surfaces. --Lord_Farin (talk) 08:10, 31 January 2013 (UTC)

This is indeed true. However, most sources on univariate complex analysis use the word "analytic" instead of holomorphic (at least, most of the sources on the net that I've read). I haven't studied multivariate complex analysis, so I can't write anything about that. Feel free to add an explanation to the page. --Anghel (talk) 16:56, 31 January 2013 (UTC)

But then what is the power series def. called? Nothing? That seems lacking. I will try to put in €0.02 later tonight. --Lord_Farin (talk) 18:35, 31 January 2013 (UTC)

We might also want to include what it means to be analytic/holomorphic in a set that is not open. (Currently there are no conditions on $D$, while the definition linked to supposes $D$ is open.) --barto (talk) 15:32, 23 January 2017 (EST)

If the differences between open and non-open that require the definitions to be fundamentally different, we need to be careful and distinguish between them -- whether as an "also defined as" section or specifically to establish separate pages to draw that difference. As I say, the most difficult part of this area is to decide upon a consistent approach, and to make sure we document the different approaches in those all-important side-notes. --prime mover (talk) 16:05, 23 January 2017 (EST)