Applied Mathematical Sciences, 25. New York etc.: Springer-Verlag. XV, 411 p.; DM 88.00 (1985).
This second edition contains plenty of new material, particularly on numerical methods and on integrals involving a parameter. However, these changes do not affect the basic character of this successful introduction which was first published in 1978 [cf. Zbl 0381.44001)]. The book is intended to serve the reader an introductory and reference material for the application of integral transforms to the solution of mathematical problems in the physical, chemical, engineering and related sciences. The author has chosen an approach which is common in published research work and stressed on complex variable techniques, which provides a powerful tool.
This book is divided into four parts which contains twenty one sections. The first six sections contain standard material on Laplace transform and its application to different and integral equations. The next five sections are based on Fourier transform and the related material. The application of Fourier transform to problems in partial differential equations transforms basically to evaluating difficult integrals. Numerical methods may be necessary in general, although asymptotic and other useful information can often be obtained directly by appropriate methods. He treats simpler problems in earlier section, leaving applications involving mixed boundary problems, Green’s functions, and transforms in several variables for later sections.
The third part of this book contans the Mellin, Hankel, Kontorovich- Lebedev transform and other related transforms with their properties and applications. This part also contains the material on integrals involving a parameter, which has been rewritten to make explicit reference to the book of Bleistein and Handelsman [Asymptotic expansions of integrals (1975; Zbl 0327.41027)].
The last part is based on special techniques such as the Wiener-Hopf technique, methods based on Cauchy integrals, Laplace’s method for ordinary differential equations and numerical inversion of Laplace transforms. The Wiener-Hopf technique is mainly used in conjunction with the Fourier transform. This technique is particularly useful in boundary value problems where an integral transform may be applied but it may not lead directly to an explicit solution. Such boundary value problems frequently arise in electro-magnetic theory, hydrodynamics, elasticity and other related fields. The major difficulty is using Wiener-Hopf technique is the problem of constructing a suitable factorization. This section contains a method based on contour integration which leads by natural extensions to the use of Cauchy integrals in the solution of mixed boundary-value problems. The last section contains the numerical inversion of Laplace transforms using Gaussian quadrature, Laguerre polynomials and Piessen’s method using Chebyshev polynomials. There are several sections in each part with sufficient number of problems as exercises. Therefore the book is very much suitable for a course at the senior level or beginning graduate level.