many-valued function with a given set of branch points in addition to simple poles

Question

The question concerns what information is necessary and sufficient to define uniquely a complex function f(z). To set the stage, there is a theorem that a single-valued function with only a finite number of isolated poles (no essential singularities) must be a rational function. Upon performing a partial-fraction expansion, this rational function can be defined uniquely by giving the location of all poles and zeros and the coefficient of each singular term in the partial fraction expansion.

Is there a similar way of defining uniquely a many-valued function that may have branch points in addition to simple poles? I was hoping to do it without analytic continuation, in a way similar to the partial-fraction approach discussed above. That is, simply list the location and nature of each singularity, together with a set of coefficients describing the strength of each singular term. Are there any theorems that specify the necessary and sufficient information needed to define a function in this way?

Answer 1

You cannot do this "without analytic continuation". A multi-valued function is defined on a Riemann surface. The singularities of this function are not lying on the Riemann sphere, so you cannot "specify their position" without describing somehow the Riemann surface.

If your only singularities are finitely many ramification points and poles, the Riemann surface will be compact (but you need the additional data, for example the monodromy data, to specify this compact Riemann surface. Once the compact Riemann surface is known, and the poles position is specified on the Riemann surface, you have the Riemann-Roch theorem and which tells you (in principle) whether a function with such poles on such Riemann surface exists or not.