Let $G$ be an open subset of $\mathbb{C}$ let $\{a_k\}$ be a sequence of distinct points in $G$ which has no limit point in $G$ For each $k$ let $A_{1k},\dots,A_{m_kk}$ be arbitrary complex coefficients, and define $$S_k(z) = \sum_{j=1}^{m_k} \frac{A_{jk}}{(z-a_k)^j}.$$ Then there exists a meromorphic function $f$ on $G$ whose poles are exactly the points $\{a_k\}$ and such that the singular part of $f$ at $a_k$ is $S_k(z)$ for each $k$