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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 $2\pi$.

Let $w$ satisfy $\sin w=z$ and put $t=e^{iw}\ne 0$. The exponential definition of the complex sine givesThe quadratic formula therefore gives $t=iz\pm \sqrt{1-z^2}$. Taking the complex logarithm yields the multivalued complex inverse sine formulaThe two quadratic roots have product $-1$, so neither is zero and their logarithms exist as multivalued quantities. If $w_0$ is one local inverse value, the complete set of values is $w_0+2\pi k$ and $\pi -w_0+2\pi k$, with $k\in \mathbb{Z}$.

Branch qualification: the source's $\pm$ formula describes the branches of the complex inverse sine, not two interchangeable expressions for a single chosen inverse branch. At $z=0$, the plus sign with the usual local choices gives $0$, whereas the minus sign with $\log (-1)=i\pi$ gives $\pi$. Thus both signs cannot simultaneously equal the principal inverse at zero. A principal complex inverse sine requires compatible square-root and logarithm branches, normalized to $w_0=0$ at zero and continued on the chosen cut domain.