Semicircle integral from a slit-plane square root
= Semicircle integral from a slit-plane square root
For $a>0$, choose the branch
$$
F(z)=\sqrt{z-a}\sqrt{z+a}
$$
that 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 $z^{-1}$ in
$$
F(z)=z-\frac{a^2}{2z}+O(z^{-3})
$$
gives
$$
\int_{-a}^{a}\sqrt{a^2-x^2}\,dx=\frac{\pi a^2}{2}.
$$