 Monodromy of the complex inverse sine

Let $w$ be a local value of the complex inverse sine. Since the limiting values at the two branch points are $\pi /2$ at $z=1$ and $-\pi /2$ at $z=-1$, monodromy reflection at a square-root branch point givesFollowing a counterclockwise loop that encloses both branch points in the convention where $\sqrt{1-z^2}$ is positive just above $(-1,1)$ applies the two reflections in the order that givesThe reverse orientation gives $w+2\pi$. Thus the translational part of the monodromy is generated by $2\pi$.