# Weierstrass Factorization Theorem

Are there any generalisations of the Weierstrass Factorization Theorem, and if so where can I find information on them? I'm trying to investigate infinite products of the form

$$\prod_{k=1}^\infty f(z)^{k^a}e^{g(z)},$$

where $g\in\mathbb{Z}[z]$ and $a\in\mathbb{N}$.

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## 2 Answers

A vast generalization of Weierstraß's theorem is to Riemann surfaces.
Florack, inspired by methods due to Behnke-Stein, proved the following in 1948:

Let $X$ be a non-compact Riemann surface. Let $D$ be a closed discrete set in $X$ and to each $d\in D$ attach a complex number $a_d$.
Then there exists a holomorphic function $f\in \mathcal O(X)$ defined on all of $X$ such that $f(a_d)=c_d.$

One may think of $f$ as holomorphically interpolating some discrete data.
This result immediately implies that $X$ is a Stein manifold, a concept of fundamental importance in the theory of holomorphic manifolds: Stein manifolds are the analogues of affine varieties in algebraic geometry.

A complete proof is in Theorem 26.7 of Forster's awesome Lectures on Riemann Surfaces.
The special case where $X$ is an open subset of $\mathbb C$ is analyzed in John's answer.

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The Weierstrass factorization theorem provides a way of constructing an entire function with any prescribed set of zeros, provided the set of zeros does not have a limit point in $\mathbb{C}$. I know that this generalizes to being able to construct a function holomorphic on a region $G$ with any prescribed set of zeros in $G$, provided that the set of zeros does not have a limit point in $G$.

These are theorems VII.5.14 and VII.5.15 in Conway's Functions of One Complex Variable. They lead to the (important) corollary that every meromorphic function on an open set $\Omega$ is a ratio of functions holomorphic on $\Omega$.

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