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I know that any entire function $f(z)$ can be represented in the form of an infinite product according to the Weierstrass factorization theorem:

$$ f(z) = z^m e^{g(z)}\prod_{n=1}^\infty E_{p_n}\left(\frac{z}{a_n}\right).\quad\quad (1) $$

Suppose that I am interested in a function $f(z)$ that is holomorphic at every point of a subset $\Omega\subset \mathbb{C}$ but it is not entire.

  1. Can I represent $f(z)$ as in Equation (1) in the subset $\Omega$?

  2. In the negative case, is there any generalization of the above theorem to non-entire functions?

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  • $\begingroup$ If $\Omega$ is simply connected, there are fairly nice ways of doing this. It's a very involved result though, and is much more nuanced than the usual weierstrass factorization. Can't seem to find a reference online. But a good treatment is in Theory of Complex Functions by Reinhold Remmert. $\endgroup$ – user335907 Mar 21 '18 at 18:06
  • $\begingroup$ I couldn't find anything about it in this book. Does the theorem go by another name? $\endgroup$ – Mr. K Mar 22 '18 at 8:07
  • $\begingroup$ Oh my mistake. That's the first book in the series. I meant to suggest the second one. Classical topics in Complex Function Theory by Reinhold Remmert. It should be under prescribed zeroes of holomorphic functions. $\endgroup$ – user335907 Mar 22 '18 at 19:07
  • $\begingroup$ As a heads up, reading through it again, it is very high brow. It might help to accomodate yourself to the terminology he uses in the early chapters. It doesn't quite answer question 1, it is moreso an alternative route, it's not quite a negative to 1, but the expansion is slightly different then the original factorization theorem $\endgroup$ – user335907 Mar 22 '18 at 23:42

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