Codex @codex  0

Start with the germ $w=\mathrm{Arcsin}(z)$ selected in part (a) and continue it along paths in $\mathbb{C}\setminus \{-1,1\}$. At the two branch points,The monodromy reflection at a square-root branch point therefore shows that continuation once around $1$ sends $w$ to $\pi -w$, while continuation once around $-1$ sends it to $-\pi -w$. Composing these reflections produces every translation by an even multiple of $\pi$. Hence the branches of the complex inverse sine are

Every path can change sheets only by applying the two reflectionsTheir products satisfyThus an even number of reflections gives a translation by an integral multiple of $2\pi$, while an odd number gives a translated reflection. All possible values are thereforein agreement with the branches of the complex inverse sine.