Branch cut

For $a>0$, choose the branchthat satisfies $F(z)\sim z$ at infinity and whose cut is the interval $[-a,a]$. Its boundary values are $F_{\pm }(x)=\pm i\sqrt{a^2-x^2}$. Comparing the integral around the two sides of the cut with the coefficient of $z^{-1}$ ingives

Analytic continuation of an integral of a meromorphic function around a closed loop changes its value by $2\pi i$ times the winding-number-weighted sum of enclosed residues.

Two integration paths with the same endpoints differ by a closed-path period determined by the residues and winding numbers of their concatenation.

If a cut joins two poles with opposite residues, every loop avoiding the cut has equal winding number about the endpoints, so the two residue contributions cancel.

A holomorphic or meromorphic integrand has a single-valued path integral on a domain whenever its integral around every closed curve in that domain vanishes.