On partial fraction expansion for meromorphic functions

Анотація

The paper is a short survey of results devoted to partial fraction expansion for meromorphic functions of one complex variable. In particular, this contains new results by the author on representation of a meromorphic function $\Phi$ on $\mathbb{C}$ in the form $\Phi(z)=\displaystyle\lim_{R\to\infty}\sum_{|b_k|<R}\Phi_k(z)+\alpha(z),$ where $\{b_k\}_1^\infty$ is the sequence of all its poles arranged in the order of increase of the absolute values and tending to $\infty$, $\left\{\Phi_k(z)=\displaystyle \sum_{n=1}^{N_k}\frac{A_{k,n}}{(z-b_k)^n}, k=1,2,\ldots \right\}$ is the sequence of principal parts of the Laurent expansion of $\Phi$ near the poles, and $\alpha$ is an entire function.