Let $\{a_n\} \subseteq \mathbb C$ be a discrete set with no limit point . Then for every sequence $\{z_n\}$ of complex numbers , can we find an entire function $f:\mathbb C \to \mathbb C$ such that $f(a_n)=z_n , \forall n \in \mathbb N$ ? I feel I have to use Weierstrass factorization or Mittag-Leffler , but I can't quite crack it . Please help . Thanks in advance
2026-03-25 09:29:02.1774430942
$\{a_n\} \subseteq \mathbb C$ discrete set with no limit point . For any sequence $\{z_n\} $ , there is entire $f $ on $\mathbb C$ with $f(a_n)=z_n$?
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Hint: Mittag-Leffler allows you to construct a meromorphic function with prescribed principal parts. Weierstrass allows you to construct an entire function with prescribed zeros of any order, and in fact you can take such a function to be a product of elementary factors.
Suppose the zeros of the entire function $g$ coincide with the poles of the meromorphic function $h$. What would the product $g\cdot h$ behave like near these poles/zeros? Can you use this to construct the function you want?