So far in this book, we have taken a particular surface S (usually the Riemann sphere Σ or a torus ℂ/Ω) and have considered the functions which are meromorphic on S (in these particular cases, the rational and elliptic functions). In this chapter we will reverse the process: we take a function f, and consider what is the most natural surface to take as the domain of definition of f. Two particular problems arise:

1. (i) If f is meromorphic (or analytic) on some region D ⊆ Σ, can we extend f to a function which is meromorphic (or analytic) on some larger region E ⊃ D?

2. (ii) If f is a so-called ‘many-valued function’ (such as √c or log(z)), can we represent f by a single-walued function on some suitable domain?

To solve problem (i) we introduce the concepts of meromorphic and analytic continuation, and we then show how this leads to the construction of Riemann surfaces, the ‘suitable domains’ in problem (ii). After examining in detail some of the surfaces which arise in this way, we then show how Riemann surfaces may be defined abstractly (as objects in their own right, independent of many-valued functions), and finally we investigate some of the topological properties of these surfaces.

Meromorphic and analytic continuation

Recall that a region is a non-empty open subset of Σ which is connected (or, equivalently, path-connected). We define a function element to be a pair (D, f) where D is a region and f : D → Σ is a (single-valued) meromorphic function on D.