Introduction

The goal of this Chapter is to provide you with a very basic understanding of how Laplace's equation (in three space variables) and the scalar wave equation (in three space and one time dimension) can be solved using holomorphic methods. This chapter will build on the methods developed in Chapter 23 in a very direct way, and you are recommended to read that chapter now before proceeding further here.

The material on dimension three is, in part, a much simplified version of part of a series of lectures given by Nigel Hitchin (now Professor, F.R.S.) in Oxford in the early 1980s. The picture presented here will not give you anything like the full geometrical picture underlying the results, which we shall develop by elementary methods. Part of the theory of the intrinsic three-dimensional approach is developed in a paper published by Hitchin (1982). The four-dimensional picture is covered most comprehensively by its principle architect, Professor Sir Roger Penrose, F.R.S., in Penrose and Rindler (1984a, b).

Laplace's equation in dimension three

Perhaps the most natural place to start is with the solution of Laplace's equation in three variables. In Chapter 23 we found, in Eq. (23.85), a natural holomorphic representation of a point in (possibly complex) three-dimensional space, arising as a degenerate case of a holomorphic null curve. This representation is in terms of a special type of quadratic holomorphic function.