This tag is for questions relating to Weierstrass factorization theorem, an extension of the fundamental theorem of algebra.
The Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes.
Weierstrass factorization theorem: Let $~f~$ be an entire function and let $~\{a_n\}~$ be the nonzero zeros of $~f~$ repeated according to multiplicity. Suppose $~f~$ has a zero at $~z = 0~$ of order $~m ≥ 0~$ (a zero of order $~m = 0~$ at $~0~$ means $~f(0) \ne 0~$). Then there is an entire function $~g~$ and a sequence of integers $~\{p_n\}~$ such that $$f(z) = z^m~e^{g(z)}~\prod_{n}E _{p_n}\left(\dfrac{z}{a_n}\right)~.$$
Note: A generalization of the theorem extends it to meromorphic functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's zeros and poles, and an associated non-zero holomorphic function.
References:
https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem
"Functions of One Complex Variable" by J. B. Conway
