Consider the function $\displaystyle f(z)=\frac{1}{\sqrt{1-z^2}}$ as a multiple-valued function of the complex plane. We can construct its Riemann surface by cutting two copies of the complex plane along the line segment from $\displaystyle -1$ to $\displaystyle 1$ and pasting them together along this cut in the usual fashion. What we obtain is a cylinder, to which we could add two points at infinity (one point in each direction of the cylinder, i.e. one point on each of the two sheets); doing that, we'd obtain a surface isomorphic to the Riemann sphere, but let's stick to the non-compactified surface we had before (the cylinder).

Now it's quite clear in view of the differential equation satisfied by $\displaystyle w=\sin z$ that $\displaystyle w$ has as inverse function the integral $\displaystyle z = \int_0^w {f(t)dt}$. However we all know $\displaystyle \arcsin w$ is far from a single valued function (unless we consider its image to be the cylinder $\displaystyle \mathbb{C}/2\pi \mathbb{Z}$, but let's not do that now). Therefore we can expect the multi-valuedness of the function $\displaystyle \arcsin w$ to be explained by the fact that the Riemann surface on which we are integrating has a non-trivial fundamental group; non homotopic paths of integration between $\displaystyle 0$ and $\displaystyle z$ should yield values of $\displaystyle w$ differing by an integral multiple of $\displaystyle 2\pi$. I've understood this to be true but I'm having trouble checking it computationally; I would like to check it.

My question arises also in the context of elliptic function. I have a book in which Jacobi's $\displaystyle \mbox{sn}$ function is constructed as the inverse of the elliptic integral $\displaystyle \int_0^w \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}$; the author states that integrating around the branch points in various ways results in paving the plane with the period lattice of $\displaystyle \mbox{sn}$, but does not show it by a computation.