I need to check whether $z=\infty$ is a branch point of the function $$f(z)=\sqrt[p]{(z-a_1)(z-a_2)}$$ where $p \in \mathbb{N}$ and $a_i \in \mathbb{C}$ for $i=1,2$. I know that we could make the usual change of variable $w=1/z$ and analize the case when $w=0$ to answer this question. We would thus have that:
$$f(w)=w^{-1/p}\sqrt[p]{(1-w\cdot a_1)(1-w \cdot a_2)}$$
Under this form, the right term of $f(w)$ has no branch point at $w=0$. But what about the left term? I have read a similar question here, but I am still not quite convinced..