The mathematics literature is not rich with texts in numerical conformal mapping. The main contributions in this area are (a) the book by von Koppenfels and Stallmann [5], written in 1959; (b) the classic monograph by Gaier [1] which, although written in 1964, is still very relevant; (c) Volume III of Henrici's ``Applied and Computational Complex Analysis'' [2]; (d) the collection of papers on numerical conformal mapping which was edited in 1986 by Trefethen [9]. There are also the two more recent books by Ivanov and Trubetskov [4] and Schinzinger and Laura [7], but these concern mainly applications and do not purport to cover the whole range of numerical conformal mapping techniques. Thus, I was particularly excited to receive this latest addition to the literature of numerical conformal mapping. Unfortunately, my initial anticipation was dampened considerably on closer reading.
The book consists of 15 chapters and 5 appendixes. The topics covered include the Schwarz Christoffel method, various expansion methods such as the Bergman kernel, the Szego kernel and the Ritz methods, the mapping of nearly circular regions, the numerical evaluation of Green's function, various integral equation methods such as those of Lichtenstein, Gershgorin, and Warschawski and Stiefel, various numerical methods based on the integral equation formulation of Symm, the integral equation of Kerzman and Stein for the Szego kernel, the Theodorsen integral equation method, the method of Wegmann, the mapping of doubly and multiply connected regions, airfoils, the treatment of corner and poletype singularities, and a discussion on the use of conformal mapping for grid generation.
According to the author ``the book is intended to contribute to an effective study program at the graduate level and to serve as a reference book for scientists, engineers and mathematicians in industry.'' Unfortunately, I am not convinced that this aim has been fulfilled. In my view, the book is a welcome addition to the existing literature, but only as an auxiliary text that can serve to indicate some of the more recent developments of the subject and to direct those interested to appropriate references. In this context, the extensive bibliography and the introductory chapter (Chap. 0), which sets the historical background of the subject and describes some of the modern developments in the area, are particularly helpful.
On the other hand, I do not think that the book can be used on its own, as a self-contained text, for the detailed study of the subject. My main criticism is that the book is not well organized. As a result, the overall presentation lacks coherence. In particular, the material is often presented in a fragmented manner, by selecting particular sections directly from the original sources, without making an effort to provide the additional explanations and links needed for continuity and ease of understanding. A typical example of this is the material concerning the singularities of the mapping function, in Section 9.5 and Chapter 12. There are also some rather dubious statements. For example, in Section 9.2 and in other parts of the book, the numerical integral equation method of Symm [8], which has nothing to do with orthonormal polynomials, is called the ``orthonormal polynomial method (ONP)' ' and is, incorrectly, attributed to Rabinowitz [6].
Another criticism concerns the end-of-chapter exercises (problems). These often refer to research items, of rather technical nature, which can hardly be regarded as problems that a student (even an advanced graduate student) would be in a position to tackle. The following examples suffice to illustrate this: (i) Problems 8.10.1, 8.10.2, and 8.10.3 require the proofs of three major theorems and the development of a major algorithm, all taken from research papers. (ii) In Problem 12.7.8 the reader is asked to accomplish an impossible task, i.e., to