# When does a sequence (or a series) of real-analytic functions converge to a real-analytic function?

It is well known that a sequence (or a series) of holomorphic functions converging uniformly converges to a holomorphic function. I would like to know under what condition a sequence (or a series) or real-analytic functions converges to a real-analytic function. Are there any simple criteria?

In particular, I would like to know whether a series of real-analytic functions converging absolutely and uniformly converges to a real-analytic function. I read in the proof of Lemma 1.8 in "Mock theta functions" written by S. P. Zwegers "since R is the (infinite) sum of real-analytic functions, and the series converges absolutely and uniformly, it is real-analytic." But I cannot understand why that fact holds.

• Yes, that doesn't make sense. I guess, I was being very sloppy, there... – Sander Zwegers Jul 25 '16 at 21:37
• Thank you for your comment. I will check the real-analyticity on a case-by-case basis. – user356126 Jul 26 '16 at 13:02

A uniform limit of real-analytic functions certainly need not be real-analytic. Any continuous function on $[a,b]$ is a uniform limit of polynomials (and is hence the sum of a uniformly and absolutely convergent series of polynomials).
I can't think of any "simple" criteria. Even uniform convergence of a sequence of functions together with uniform convergence of every derivative is not enough; consider the Poisson integral of a function in $C^\infty_c$.
This is true in $\mathbb{C}$ but not in $\mathbb{R}$. The reason is that uniform convergence preserves continuity, not differentiability and beyond.