The following is (a rephrasing of) problem 2, in Chapter 10 of Greene and Krantz's Function Theory of One Complex Variable:
Construct Riemann surfaces for the (local) "inverse functions" of $$k(z)=z + \frac{1}{z}.$$
I think I understand the construction for $\log$, and for $z^n$. What would be a generic method for constructing such surfaces, and how, in particular, to construct the specific one in the question above ?
There's not really a "generic method", but in this particular case, you can solve for $z$ in $w = z + 1/z$ to obtain $$z = \frac{w \pm \sqrt{w^2-4}}{2}$$ So this is similar to the Riemann surface for $\sqrt{w}$. The branch points are found where the argument of the square root is 0, i.e., at $w=\pm 2$. You can take the branch cut to be, for example, the line segment joining -2 and 2. The 2 sheets of the Riemann surface are glued along this line segment. Whenever a curve crosses this line segment, it moves from one sheet to the other. To get a compact Riemann surface, add the branch point at $\infty$.