Let $(a_n{)}_{n\in \mathbb{N}}$ be a sequence of complex numbers which is pairwise distinct, and has no accumulation points. For each $a_n$, let there be a given principal part:

Then there exist a meromorphic function: $f:\mathbb{C}\to \mathbb{C}$ with 1;;poles exactly at $a_n$ with principal part given by $h_n(z)$. Moreover, $f$ is unique up to adding a holomorphic function.

PROOF

• Estimate $h_n(z)$ with a power series centered at $0$
• Pick suitable taylor polynomial $T^n(z)$ of $h_n(z)$ around this point (recall the taylor series converges), with radius of convergence $a_n$
• Use local uniform convergence