Semicircle integral from a slit-plane square root

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