This book introduces the concept of a “prime function” as part of a framework for dealing with a large group of problems in multiply-connected domains. The “prime” terminology refers to the basic notion of a “prime form” on a Riemann surface with a history going back to Weierstrass, Schottky, Klein and others. But the author notes that its historical use in uniformization theory has only a weak connection with his current exploration of its use for problems in multiply-connected planar domains.

Function theory for the prime function is the subject of the first part, somewhat more than half of the book. The second part is exclusively about applications. It is intended to demonstrate how the theory of prime functions from the first part can be used in a variety of applied problems. The first part is written as a research monograph with exercises that could make it usable as an advanced textbook. In the second part, the author offers several essays on applications to the physical sciences of the theory developed in the first part. A pervasive aspect of his approach is to give a geometrical view of the underlying function theory. This means thinking of functions geometrically as conformal mappings between planes.

In the author’s words, his book is really about “how to extend calculus beyond the Riemann sphere, where we know it best, to spheres with ‘handles’. For applied scientists, this amounts to taking familiar results for the single-object case and extending it to any number of objects.” He notes that fields in the natural and physical sciences have well-known solutions to single object problems and other solutions for singly periodic arrays of objects, but between these two cases are important instances where one, two or three objects interact. In these cases, the available tools and literature are very sparse.

The aim of the monograph is to present a framework that makes solving problems in multiply-connected domains a natural generalization of solving them in single connected ones. The simplest prime function is \( w(z,a)=z-a \) , where \( z \) and \( a \) are complex variables; this is the prime function associated with a simply connected domain. For domains that are not simply connected the relevant prime function is no longer elementary and must often be determined from an infinite product expansion. The author begins with simply-connected domains but moves on quickly, first to multiply-connected circular domains and then to polyhedral and polycircular arc domains.

The author concludes the first part with a chapter devoted to methods of computing the prime function using infinite product expansions or representations using theta functions. The application essays of the second part consider problems that include electric transport theory and a variety of questions about fluid and thermal flow. A unifying theme of these applications is the role prime functions have for multiply-connected domains in potential theory. The second part begins with an essay on using prime functions to develop a calculus for potential theory.

The author has generally avoided proofs and rigor to make the book, as he says, more accessible to applied scientists. As it is, the book assumes a strong background in complex analysis and geometry including subjects like Schwarz-Christoffel mappings and some automorphic function theory.

Bill Satzer (

bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and material science. He did his PhD work in dynamical systems and celestial mechanics.