= 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> gives
$$
R_+(w)=\pi-w,
\qquad
R_-(w)=-\pi-w.
$$
Following 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 gives
$$
R_-\circ R_+(w)=w-2\pi.
$$
The reverse orientation gives $w+2\pi$. Thus the translational part of the monodromy is generated by $2\pi$.
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