Poles , Radius of Convergence and Laurent Series

Question

If I have a function with $2$ poles, for example $z_1 $and $z_2$, is it true that the radius of convergence of the Laurent series near $z_1$ is always the distance between $z_1$ and $z_2? This doesn't seem very true to me but I can't find a counter-example. Thanks in advance.

Answer 1

This is true if the domain of $f$ is $\mathbb{C}\setminus\{z_1,z_2\}$. Otherwise, it is false. For instance, take an analytic function $F\colon\{z\in\mathbb{C}\,|\,\lvert z\rvert<1\}\longrightarrow\mathbb C$ which admits no analytic continuation to any domain $\Omega$ such that $\Omega\varsupsetneq\{z\in\mathbb{C}\,|\,\lvert z\rvert<1\}$. Now, let $f(z)=\dfrac{F(z)}{z\left(z-\frac34\right)}$. Then the radius of convergence of the Laurent series of $F$ at $\frac34$ is $\frac14$, which is smaller than the distance from $\frac34$ to $0$.