From Complex Analysis to Riemann Surfaces

Abstract

This chapter makes a quick and targeted incursion into the world of complex analysis, with the goal of presenting the ideas that historically led to the development of the notion of a Riemann Surface. Differentiability for a function of one complex variable imposes considerably more structure than the analogous notion for functions of a real variable. Somewhat strangely, many of the remarkable properties of complex differentiable functions are natural consequences of a construction that somehow “leaves” the complex world: complex functions can be integrated along real paths, and the value of such integrals doesn't change if the path is continuously perturbed while fixing the endpoints. This phenomenon leads to Cauchy's formula, which expresses a complex differentiable function as a path integral of yet another complex function. While this may seem a slightly bizarre thing to do, Cauchy's formula has a number of remarkable consequences. In particular it gives a differentiable expression for the local inverse to a complex differentiable function at a point where the derivative does not vanish. When a differentiable f function is not injective, obviously there does not exist a global inverse function. However, at any point where f doesn't vanish, one has multiple local inverse functions (or historically one said that the inverse of f is a multivalued function) and, further, there is a natural way to view all these local inverses as part of a global function defined on a space which, around any point, “looks like” the complex numbers but globally may be different from C. Such spaces are examples of Riemann Surfaces. In this chapter, which is meant to illustrate how the concept of Riemann Surfaces was developed, we limit ourselves to exploring this picture for the power functions w = z k and their inverses (the k -th root “functions”). While this may seem unimpressive, Lemma 1.4.4 shows that the power functions, up to appropriate changes of variables, describe the behavior of any holomorphic function around a critical point – a point where the derivative vanishes. Complex analysis is a beautiful and rich subject, and there is no way that we can do it justice in a handful of pages.We have made the choice of taking a path through the subject that gives a working understanding of a small selection of ideas that are important for the development of our story.