Beijing Lectures in Harmonic Analysis

Chapter 4 is concerned with meromorphic continuation and examples of various Riemann surfaces, and Chapter 5 deals with Fuchsian groups. The chapter finishes with a proof of Hurwitz's theorem that a compact Riemann surface of genus g has at most 84(g—1) automorphisms, together with a discussion of possible equality. There are four appendices of which the first consists of a review in six pages of the basic theorems of complex analysis, mainly without proofs. The authors say that they had to ignore the connections with number theory and potential theory ' since there are excellent books on the subject'. Perhaps in future editions they might quote a book on each of these topics. They refer to books by Rankin and Schoeneberg in the index. I could not see there a reference to a book on potential theory although Farkas and Kra certainly contains something on the subject. The material is intended for undergraduates and long and detailed arguments leading to the proof of a simple theorem are avoided. There is no proof of the mapping theorem for domains or Riemann surfaces. It is a little irritating to see so many references to matters which have to be pursued outside this book, but it does enable one to learn a good deal of material, which could not otherwise have been included. The exposition is always clear. If a department with an algebraic or geometric flavour wished to introduce a second level course in complex analysis this would be a good book from which to select the material. There are many useful exercises, a very necessary index of symbols extended over two pages, an index of names and about fifty references. I enjoyed reading this book and expect others to do the same.