Higher Transcendental Functions

Note: The three volumes of the Bateman manuscript are available for download from the Caltech library.

 

The book under review, published in 1953, is an excellent reference on special functions and provides a comprehensive database for researchers. Professor Harry Bateman, a California Institute of Technology scientist, made significant contributions to American applied mathematics by compiling ”special functions” solutions of mathematical and physical equations. He investigated properties, inter-relations, representations, and behavior, and constructed tables for important definite integrals. Professor Bateman’s work was continued by the California Institute of Technology and the Office of Naval Research after his death. In 1948, arrangements were made to employ four internationally renowned mathematical analysts to complete Bateman’s work: Professors Arthur Erdelyi of the University of Edinburgh; Wilhelm Magnus of the University of Gottingen; Fritz Oberhettinger of the University of Mainz; and Francesco Tricomi of the University of Torino. The three volumes on higher transcendental functions may be viewed as an updated version of Part II of Whittaker and Watson’s A Course of Modern Analysis.

 

The gamma functions, Legendre functions, and Hypergeometric functions with generalizations are the most comprehensively addressed topics in the first volume of this book. Hypergeometric functions, in particular, are covered extensively: there’s a particularly wonderful chapter that does an amazing task of inspiring them, as well as sections that cover differential equations, recurrence relations, transformations, integral representations, reduction formulas, analytic continuation, asymptotic behavior, generating func-tions, and generalizations like E-function and Meijer’s G-function.

 

The second volume includes chapters on Bessel functions and other specific confluent hypergeometric functions, orthogonal polynomials and related topics, and on elliptic functions and integrals. The theory and the numerous examples present in the chapter on orthogonal polynomials is especially notable. The chapter provides the classical orthogonal polynomial with different intervals and weight functions, which are stated in the table, and several properties are demonstrated in great detail on pages 164-166. The asymptotic behavior of Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials, Laguerre polynomials, and Hermite polynomials are all addressed in the second part of the book. Furthermore, the zeros of Jacobi polynomials, Laguerre polynomials, and Hermite polynomials are discussed.

 

The third volume includes chapters on automorphic functions, Lame and Mathieu functions, spheroidal and ellipsoidal wave functions, as well as functions that arise in number theory. The last chapter, on generating functions, has a substantial and comprehensive list of generating functions. Throughout the three volumes, a list of references is given at the end of each chapter. These lists are by no means complete but they should be sufficient to document the presentation and to enable the reader to find further information about the functions in question.

 

This work is an outstanding effort, but it is not for the general reader, nor even for those with a limited understanding of special functions and hypergeometric series unless their mathematical education has been broadened in this subject. The results are especially good, numerous, and challenging.

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Showkat Ahmad is a postdoctoral fellow at Aligarh Muslim University.