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Let $\sum_{i=1}^{\infty} f_i(s)$ be a series of analytic functions and suppose it converges on some neighbourhood $V$ of $s=0$ and it converges uniformly on $V\cap\{s \;|\; |s|> \varepsilon \}$ for every $\varepsilon>0$. So it is analytic on the punctured neighbourhood $V-\{0\}$. Suppose $\lim_{s\to 0} \sum_{i=1}^{\infty} f_i(s)$ exists (and finite).

I am looking for references for conditions (other than uniform convergence in whole $V$) for which $ \lim_{s\to 0} \sum_{i=1}^{\infty} f_i(s) = \sum_{i=1}^{\infty} f_i(0) $

Thank you.

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    $\begingroup$ Dosnt Cauchys formula with contour a small circle around 0 show that the series converges uniformly in a nbh of 0? $\endgroup$ – NJK Jun 2 '14 at 15:37
  • $\begingroup$ Yes, this is correct. Thank you, NJK. $\endgroup$ – ebg Jun 3 '14 at 18:17

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