Let $X(s): X,s \in \mathbb{C}$ be meromorphic, with simple, isolated zeros $\{z_n\}$ and first-order, isolated poles $\{p_n\}$, and let $L_k=\lim_{s\to z_k} \frac{X(s)X^*(s)}{X(s)-X^*(s)}$.
Is $L_k=0$?
Along a curve $\alpha(t)\in s$ where $Z(\alpha(t)) \in \mathbb{R}$, the denominator is a constant zero, so there is no limit.
However, I am thinking that $X(s)=\frac{A(s)}{B(s)}$, where $A(s),B(s) \in \mathbb{C}$ are analytic. Then, $L_k = \lim_{s\to z_k} \frac{A(s)A^*(s)}{A(s)B^*(s)-A^*(s)B(s)}$. That quotient is meromorphic; the zeros, $z_k$ of $A(s)A^*(s)$ in the numerator are of second order, whereas zeros of the denominator are of first order, so $L_k=0$. Right?