Joel Schiff

The Laplace Transform

The Laplace transform is an extremely versatile technique for solving differential equations, both ordinary and partial. It can also be used to solve difference equations. The present text, while mathematically rigorous, is readily accessible to students of either mathematics or engineering. Even the Dirac delta function, which is normally covered in a heuristic fashion, is given a completely justifiable treatment in the context of the Riemann-Stieltjes integral, yet at a level an undergraduate student can appreciate. When it comes to the deepest part of the theory, the complex inversion formula, knowledge of poles, residues, and contour integration of meromorphic functions is required. To this end, an entire chapter is devoted to the fundamentals of complex analysis. In addition to all the theoretical considerations, there are numerous worked examples drawn from engineering and physics.

When applying the Laplace transform, it is important to have a good understanding of the theory underlying it, rather than just a cursory knowledge of its application. This text provides that understanding.

The book is available from or Springer-Verlag.

Normal Families

This is the first book devoted solely to the subject of normal families of analytic and meromorphic functions since the 1927 treatise of Paul Montel. A considerable body of research has evolved since then, and this text provides a comprehensive treatment of the entire theory. Since its inception early this century, the notion of a normal family has played a central role in the development of complex function theory. In fact, it is a concept lying at the very heart of the subject, weaving a line of thought through Picard's theorems, Schottky's theorem, and the Riemann mapping theorem, to many modern results on meromorphic functions via the Bloch principle. It is this latter that has provided considerable impetus over the years to the study of normal families and continues to serve as a guiding hand to future work. Numerous applications of the normal family theory are discussed, particularly those found in the study of extremal problems, normal functions, harmonic functions, discontinuous groups, and complex dynamical systems.

Only a basic knowledge of complex analysis and topology is assumed. All other necessary material for the study of the subject is included in the first chapter. The scope of the book ranges from advanced undergraduate to research level.

The book is available from or Springer-Verlag.