Is there a theorem in complex analysis which says something along the lines that,
If two meromorphic function have same set of poles then they are same. I mean to say that, given a set of poles $M=\{z_1,z_2,\dots,z_n\}$, there is a unique meromorphic function having poles at $z_1,z_2,\dots,z_n$
If there is such theorem, I request you to mention books/online source where I can find the proof.
Best Answer
If $f$ is a meromorphic function and $g$ is an entire function without zeros, then $fg$ is a meromorphic function with exactly the same poles as $f$.
The relevant uniqueness theorem is the Weierstrass factorization theorem.
See also the Mittag-Leffler's theorem.