This unique book represents a largely successful attempt by a master craftsman to make available to the nonspecialist, but serious, reader a systematic and coherent presentation of advanced topics in complex analysis. The approach is “applied” both in the broad sense that the selection of topics is biased toward those which are applicable in other areas, and in the narrow sense that a selection of illustrative and motivating examples are drawn from such areas. The approach is “numerical” both in the broad sense that there is an underlying concern throughout with efficient and effective construction of the quantities of interest and in the narrow sense that problems of independent numerical interest are addressed using the tools at hand. The approach is rigorous, but sufficiently discursive to be readable. If the appropriate people can be persuaded to invest the considerable effort required, the effect might be to revivify a classical subject too often disregarded as “analytical” by those who are numerically inclined, and to give operational meaning to the concerns of those who are analytically inclined.

This is the third volume of a projected trilogy whose predecessors were published in 1974 and 1977 [2,3]. Periodic access to the earlier volumes is required, but this one is largely self-contained. The style, level, and format is consistent throughout, with somewhat greater emphasis on applied and numerical aspects of the subject in the new volume. The reader should beware of misprints until a revised printing is forthcoming] Due in part to developments in the field in the interim, the third volume covers only a portion of the range of topics originally contemplated, presumably in much greater depth; one or more additional volumes are now projected.

The core of the present volume is the study of potential theory in the plane and conformal mapping of both simply and multiply connected regions, comprising Chapters 15 through 18 (Chapters 1 through 12 constitute the first two volumes, and conformal mapping was originally introduced in Chapter 5). Chapter 14 reviews background material on Cauchy integrals not readily available elsewhere. The basic numerical tool used throughout is the discrete Fourier transform which is presented in detail in Chapter 13. Portions of Chapter 18, on polynomical expansions, and Chapter 19, on univalent functions, are included both for their intrinsic mathematical interest and in anticipation of subsequent discussion of approximation theory and other topics.

The treatise as a whole assumes nothing more than the standard undergraduate preparation in algebra and analysis, but quickly becomes more sophisticated. The monumental scale of the work would make it difficult to use as a textbook, though nonroutine problems are supplied throughout. The approach to the subject is nonstandard, which may make it difficult to use portions of the work as a supplementary text for parts of other courses, though standard courses would be improved thereby. There are enough references to lead the reader to both current and historical literature, without attempting to provide a definitive bibliography.