The Riemann Surfaces

Abstract

A Riemann surface is an ingenious construct for visualizing a multivalued function. We treat all branches of a multi-valued function as a single-valued function on a domain that consists of many sheets of the zplane. These sheets are then glued together so that in moving from one sheet to another we pass continuously from one branch of the multi-valued function to another. This glued structure of sheets is called a Riemann surface for the multi-valued function. For example, in a multi-story car park, floors can be thought of as sheets of the z-plane, that are glued by the ramps on which cars can go from one level to another. Riemann surfaces have proved to be of inestimable value, especially in the study of algebraic functions. Although there is much literature on the subject, in this lecture we shall construct Riemann surfaces for some simple functions.