I'm trying to solve a problem in Complex Analysis whose function $f$ is defined in $\mathbb{C}$, is meromorphic and have double period $(f(z)=f(z+a)=f(z+b),\ \frac{a}{b} \notin \mathbb{R})$ except for poles. Is there any relationship between such functions and elliptic functions (where the double period is for all $z \in \mathbb{C} $)?


These are elliptic functions.

The fragment "except for poles" is just an indication that the author has reservations to write $f(z_0) = f(z_0+a) = f(z_0+b)$ when $z_0$ is a pole of $f$.

If you view the functions as having values in $\mathbb{C}$, then the poles don't belong to the domain, and hence the above equation would not make sense. If you view your meromorphic functions to have values in the Riemann sphere $\widehat{\mathbb{C}}$, then the poles belong to the domain, and all difficulties nicely disappear.

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    $\begingroup$ (One might also mention that a non-constant elliptic function necessarily has poles.) $\endgroup$ – Martin R May 2 '15 at 12:02
  • $\begingroup$ I do not know these functions and I am following the book Complex Analysis- Stein and Shakarchi. In the early part of the chapter does not appear explicitly as the function behaves in singularities, that is, whether they are at infinity, what kind of singularities at infinity they are. Analysis is not my area so that the lack of such information left me lost. Thank you, for now realized that actually my problem is how I interpreted the definition. Do you have some reference point to me? $\endgroup$ – Pryscilla Silva May 2 '15 at 12:32
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    $\begingroup$ I'm not sure, @PryscillaSilva, what you'd like to have references for. Guessing that it's about meromorphic functions in general, or elliptic functions in particular, most introductory texts on complex analysis should help. The old classic by Ahlfors is always worth a read, of course. Rudin's "Real and Complex Analysis" doesn't treat elliptic functions, but is good for the general theory. However, it is very terse. Fischer/Lieb, A Course in Complex Analysis, has a good section on elliptic functions too. And there are lots of good books which I haven't read. $\endgroup$ – Daniel Fischer May 2 '15 at 12:50
  • $\begingroup$ @DanielFischer I need basic references on elliptic functions, I think the references you refer me are a good start. Thanks! $\endgroup$ – Pryscilla Silva May 2 '15 at 12:54

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