Mittag-Leffler’s theorem

Let $G$ be an open subset of $ℂ$, let $\{a_k\}$ be a sequence of distinct points in $G$ which has no limit point in $G$. For each $k$, let$A_{1k},\mathrm{\dots },A_{m_{k}k}$ 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 $k$.