Suppose we have a functional element around $z_0$:
$$ f(z) = a_0 + a_1 (z-z_0) + a_2 (z -z_0)^2 + \ldots . $$
It might be the Taylor series of $\sqrt{z}$ at $z = z_0 $. The question is, how can we know that this function element will analytically continued into a single-valued function or a multi-valued function?
How can we extract the information from the coefficients $\{a_i \}$?