Singapore: World Scientific Publishing. xvi, 695 p. $ 75.00/hbk; $ 38.00/pbk (1989).
This book gives an extensive introduction to the classical special functions of mathematical physics. As the authors state in the introduction, in many respects the present book is based on A course in Modern Analysis (1902) by E. T. Whittaker and G. N. Watson, although the scheme is not the same and the presentations of the subjects are different too. Besides, some topics not covered in Whittaker and Watson’s book are included here.
The organization of the material is as follows. Chapters 1 and 2 give a supplement to the basic advanced calculus and complex function theory needed for the study of special functions. For example, asymptotic expansions and the theory of linear ordinary second order differential equations are discussed here. Chapter 3 is on the Euler gamma function, the beta function, and the Riemann zeta function. Chapters 4 to 7 are concerned with hypergeometric and confluent hypergeometric functions, including their important special cases: Legendre functions and Bessel functions. Their main properties are explored from the point of view of the singularities of the differential equations satisfied by them. Chapters 8 to 10 are about elliptic functions and theta functions, including the theory of elliptic integrals. Chapters 11 and 12 give the Lamé and Mathieu functions, with a discussion on Hill’s equation. Appendices give the solutions of cubic and quartic equations, and an overview of the orthogonal curvilinear coordinate systems, which occur when separating the variables in the three dimensional wave equation. Each chapter has a collection of exercises.
This is a rather impressive piece of work, although a mentioned earlier many pages remind us to the classic Whittaker and Watson. However, the latter seems not to be attractive to all workers in this part of classical analysis, and this book may give a more convenient introduction. Many present day research in special functions is on various extensions of the class of classical orthogonal polynomials, also with discrete variables, and on the so-called q-functions. This book does not present any collection with this new interest in special functions. Also, the bibliography only contains the classical works. The now modern classic Asymptotics and special functions (1990; Zbl 0712.41006) by F. W. J. Olver is not mentioned. Another point of criticism is that no applications are given on the field that gave birth to the special functions, namely the boundary and initial value problems of mathematical physics. Apart from this, the book is a welcome addition to the classical works on special functions. The presentation is very clear. The great variety of topics makes the book for many workers in physics, engineering and applied mathematics a useful and instructive source of information.