A single-valued function $\operatorname{Arcsin}(z)$ can be defined, for $0 \leqslant \arg z<2 \pi$, by means of an integral as:

$$\operatorname{Arcsin}(z)=\int_{0}^{z} \frac{d t}{\left(1-t^{2}\right)^{1 / 2}}$$

(a) Choose a suitable branch-cut with the integrand taking a value $+1$ at the origin on the upper side of the cut, i.e. at $t=0^{+}$, and describe suitable paths of integration in the two cases $0 \leqslant \arg z \leqslant \pi$ and $\pi<\arg z<2 \pi$.

(b) Construct the multivalued function $\arcsin (z)$ by analytic continuation.

(c) Express arcsin $\left(e^{2 \pi i} z\right)$ in terms of $\operatorname{Arcsin}(z)$ and deduce the periodicity property of $\sin (z)$.