# Anyone a Riemann Surface guru?

by T-O7
Tags: guru, riemann, surface
 P: 55 So I'm supposed to describe the riemann surface of the following map: $$w=z-\sqrt{z^2-1}$$ I can sort of understand the basic idea and derivation behind the riemann surfaces of $$w=e^z$$ and $$w=\sqrt{z}$$, but ask me a question about another mapping, and I really don't know where to begin. How does one attack a problem like this? i.e. how do you know where exactly the map is single-valued?
 Sci Advisor HW Helper P: 2,586 MAT354? ew is single valued on those strips, and it maps those strips onto the plane - {0}, so you want to take the plane - {0} and make a cut such that log(z) maps the cut onto the lines separating those strips. wn is single valued on the sectors, and it maps each sector onto a plane, so you want to take a plane and make a cut such that z1/n maps the cut onto the lines separating those sectors. 1/2(w + 1/w) is single valued outside the unit circle, and inside the unit circle, and it maps each of those regions onto the plane, so you want to take the plane and make a cut such that $z - \sqrt{z^2-1}$ maps the cut onto the boundary separating those 1-1 regions, namely the unit circle. This function maps the interval [-1,1] to that circle, so that's where you make the cut. In the log example, there are infinitely many 1-1 regions (fundamental regions) so you have infinitely many planes that you cut and glue together. In the z1/n example, there are n fundamental regions so you have n planes. You want to make z1/n single-valued, so for points where it's many valued, you want to make a Riemann surface that makes this one piont into many points, so that instead of one point corresponding to many points, you have many points corresponding to many, in a 1-1 way. But 0 is a point that is not many-to-1, so when you have your n-separate planes, you still only want 1 point on the entire Riemann surface corresponding to 0. In this problem you're doing, -1 and 1 are the only single valued points of the function. So although you have two planes planes, you want their -1 points to be attached, and their 1 points to be attached, because you don't need to make two different copies of 1. The rest of the points are two-valued, so you need two sheets which each have a point corresponding to that one point.