= Solution
Choose the branch of $\sqrt{(\zeta+1)(\zeta-1)}$ <holomorphic> in the <upper half-plane> whose boundary value at zero is $+i$. Then
$$
f'(\zeta)=\frac1{\sqrt{1-\zeta^2}},\qquad f(0)=0,
$$
and <integration> gives $\boxed{z=\arcsin\zeta,\quad\zeta=\sin z}$ on this branch. The normalization and intended upper-half-plane image fix the otherwise ambiguous square-root sign.
To see the image, put $z=x+iy$. For $-\pi/2<x<\pi/2$ and $y>0$,
$$
\operatorname{Im}\sin z=\cos x\sinh y>0.
$$
The bottom edge maps monotonically onto $[-1,1]$. The right wall maps by $\sin(\pi/2+iy)=\cosh y$ onto $[1,\infty)$, and the left by $\sin(-\pi/2+iy)=-\cosh y$ onto $(-\infty,-1]$. These boundaries describe the <conformal map of a vertical half-strip by sine>.
For completeness, the map is one-to-one in the interior: equality of two sine values forces arguments differing by $2\pi m$ or summing to $(2m+1)\pi$, and neither alternative yields two distinct points in this half-strip. It is onto the <upper half-plane> as well. Setting $w=e^{iz}$ gives $w^2-2i\zeta w-1=0$; the root product is $-1$, and no root lies on $|w|=1$ because that would make $\zeta$ real. Thus exactly one root has $|w|<1$. The identity $\operatorname{Im}\zeta=(\operatorname{Re}w)(|w|^{-2}-1)/2$ then gives $\operatorname{Re}w>0$, so $z=-i\log w$ belongs to the half-strip. The closure gives the stated boundary-inclusive region; <holomorphy> is understood in its open interior.
In this region the transformed function is $G(z)=\log(2+\sin z)$. Its only finite intrinsic <branch point> in the closure occurs where $2+\sin z=0$:
$$
\boxed{z_*=-\frac\pi2+i\operatorname{arcosh}2=-\frac\pi2+i\log(2+\sqrt3),\qquad\zeta_*=-2.}
$$
It lies on the left boundary. There are no singularities in the open half-strip. The points $\zeta=\pm1$, where $f'$ diverges, map to the bottom corners; $G$ is <holomorphic> through those corners since $2+\sin z\neq0$ there.
For the principal logarithm, the boundary segment $z=-\pi/2+iy$, $y\geq\operatorname{arcosh}2$, is the preimage of its cut $\zeta\leq-2$. Points on that cut above $z_*$ admit local <analytic continuation> after changing branch; $z_*$ itself is the genuine logarithmic <branch point>.
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