# Principal Part of Laurent Series Converges in Punctured Disc

I'm trying to work through the following problem:

Prove that if the holomorphic function $$f$$ has an isolated singularity at $$z_{0}$$, then the principal part of the Laurent series of $$f$$ at $$z_{0}$$ converges in $$\mathbb{C}\setminus\{z_{0}\}$$.

My Thoughts: If $$f$$ has the Laurent series expansion $$f(z)=\sum_{n=-\infty}^{\infty}a_{n}(z-z_{0})^{n}$$, then the principal part of the Laurent series is $$\sum_{n=-\infty}^{-1}a_{n}(z-z_{0})^{n}$$. I'm assuming that I need to break this up to consider all possible types of singularities that could occur at $$z_{0}$$ (i.e., a removable singularity, a pole, or an essential singularity).

• If $$z_{0}$$ is a removable singularity, the principal part of the Laurent series is trivial ($$a_{n}=0$$ for all $$n<0$$), so there is nothing to show.

• If $$z_{0}$$ is a pole (say of order $$k$$), then $$a_{-k}\neq 0$$, but $$a_{n}=0$$ for all $$n<-k$$. Then the principal part is $$\sum_{n=-k}^{-1}a_{n}(z-z_{0})^{n}$$.

• If $$z_{0}$$ is an essential singularity, then the principal part of the Laurent series has infinitely many non-vanishing terms. Then the principal part is $$\sum_{n=-\infty}^{-1}a_{n}(z-z_{0})^{n}$$.

My Questions: Am I right in saying that the convergence is trivial in the case that $$z_{0}$$ is a removable singularity? Also, how would I go about the case where $$z_{0}$$ is a pole or an essential singularity? Is it some sort of Cauchy-Hadamard argument?

Thanks in advance for any suggestions.