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I came across this old exam problem that I think it is a typo but I want to make sure there is not a problem with my knowledge. The problem is show

$$\sum_{n=1}^{\infty} \frac{z^2}{z^2 n^2 +1}$$

is meromorphic on $\mathbb{C}$.

This must be a typo since there is an essential singularity at $z=0$ (since the limit as $z$ goes to zero does not exist and is not $\infty$). Am I wrong? The series does seem to converge to a meromorphic function on $\mathbb{C} - \{0\}$.

glS
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Mykie
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  • The limit as $z$ approaches zero does not exist since there is a sequence of poles converging to zero. – Mykie Jun 16 '13 at 04:22
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    I think you are correct. – Potato Jun 16 '13 at 05:01
  • poles are at $z={i\over n}$ for $f_n(z)$ right? how do you conclude about es. singu? – Myshkin Jun 16 '13 at 05:03
  • @TaxiDriver A meromorphic function is a function is a function that is holomorphic except for a set of isolated points, which are poles. The poles here are not isolated. – Potato Jun 16 '13 at 05:09
  • @Potato Thank you,set of poles has limit point namely $0$, if set of poles has a limit point in the domain then $f$ has es. singu? – Myshkin Jun 16 '13 at 05:11
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    @TaxiDriver An essential singularity is a type of isolated singularity, so it is not essential either. – Potato Jun 16 '13 at 05:13
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    @TaxiDriver It depends on who is defining it, though. There is some discussion here: http://math.stackexchange.com/questions/135458/limit-point-of-poles-is-essential-singularity-am-i-speaking-nonsense – Potato Jun 16 '13 at 05:14
  • Perhaps you can compute your sum using Poisson summation. Maple issues the following guess : it may be equal to $(\pi/2) \coth(\pi/z) - z^2/2$. – Sary Feb 22 '15 at 20:24

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