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If $G$ is the principal complex inverse sine, all values reached by analytic continuation areThis follows from the monodromy reflections about the branch values $-\pi /2$ and $\pi /2$.

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 $2\pi$.

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

The endpoint values of the principal branch areBy the monodromy reflection at a square-root branch point, continuation around $-1$ acts on a value $w$ aswhile continuation around $1$ acts as

The contour $C_1$ makes the first continuation and then follows the principal path to $z$, soThe contour $C_2$ continues successively around $-1$ and $1$. Hence

The additional branch points $\pm 1/k$ lie on the two deleted rays because $0<k<1$. On the same simply connected domainchoose both square roots to equal $1$ at zero. Their product has a single-valued holomorphic reciprocal on $D$, so its integral from zero defines the single-valued branch $H$ of the elliptic integral of the first kind.

An unrestricted contour may wind around any of $-1,1,-1/k,1/k$. Continuation around one such point flips one square root, so the value can change by square-root monodromy. Thus $h$ is multivalued.

For real $0<k<1$, the period lattice of $\mathrm{sn}(u,k)$ is generated byThe periods arise by composing the monodromy reflections of its inverse elliptic integral of the first kind.