Let \(\displaystyle \{ z_k \}\) be a sequence of distinct points such that \(\displaystyle |z_k| \rightarrow \infty\) and \(\displaystyle \sum_{k=1}^{\infty} |z_k|^{-m-1}< \infty\). Show that \(\displaystyle z^m \sum_{k=1}^{\infty} \frac{1}{z_k^m(z-z_k)}\) converges normally to a meromorphic function with principal part \(\displaystyle \frac{1}{z-z_k}\) at \(\displaystyle z_k\). (If \(\displaystyle z_k=0\), we replace the corresponding summand by \(\displaystyle \frac{1}{z}\).)
In this section, we covered the Mittag-Leffler Theorem. I am not sure if we need to apply that result here though. I am not sure what this sequence would converge to. I would appreciate some help with this. Thank you.
In this section, we covered the Mittag-Leffler Theorem. I am not sure if we need to apply that result here though. I am not sure what this sequence would converge to. I would appreciate some help with this. Thank you.