Riemann Surface of $w^{2}=\sqrt{1-z^{2}}$ I'm working in the problem of finding branch points and build the Riemann Surface of the following complex function:
$$
w(z)=\sqrt{1-z^{2}} \,\, .
$$
I'm reading lots of texts about how to do this, but I'm not able to do it. How to indentify the branch points and how to build the Riemann surfaces step-by-step of this function?
Greetings!

Since the square root has two possible values, your Riemann surface will have two sheets. Branch points of the square root are at $0$ and at $\infty$, so in this case the finite branch points are at the points where $1-z^2 = 0$, i.e., at $z=\pm 1$. At both of these points you have a simple zero, so analytic continuation around them will get you to the other sheet. Now you can make a branch cut from $-1$ to $+1$ and glue the two sheets together along this cut. Visualizing this on the Riemann sphere and opening up the cuts, you will get two spheres with disks removed, glued along their respective boundaries, which will again get you a sphere.
