Throughout the nineteenth century, the attention of the mathematical world was, to a large extent, concentrated on complex function theory, that is, the study of meromorphic functions of a complex variable. Some of the greatest mathematicians of that period, including Gauss, Cauchy, Abel, Jacobi, Eisenstein, Riemann, Weierstrass, Klein and Poincaré, made substantial contributions to this theory, and their work (mainly on what we would now regard as specific, concrete problems) led to the subsequent development of more general and abstract theories throughout pure mathematics in the present century. Because of its central position, directly linked with analysis, algebra, number theory, potential theory, geometry and topology, complex function theory makes an interesting and important topic for study, especially at undergraduate level: it has a good balance between general theory and particular examples, it illustrates the development of mathematical thought, and it encourages the student to think of mathematics as a unified subject rather than (as it is often taught) as a collection of mutually disjoint topics.

Even though the subject matter of this book is classical, it has recently assumed great importance in several different areas of mathematics. For example, the recent work on W. Thurston on 3-manifolds shows the vital importance of hyperbolic geometry and Mobius transformations to this rapidly developing subject; a totally different example is given by the work of J.G. Thompson, J.H. Conway and others on the ‘monster’ simple group, where the J-function, studied in Chapter 6, seems to play an important (and, at the time of writing, rather mysterious) role.