Special Functions & Their Applications

by Nikolai Nikolaevich Lebedev, Richard R. Silverman (Translator)

Richard Silverman's new translation makes available to English readers the work of the famous contemporary Russian mathematician N. N. Lebedev. Though extensive treatises on special functions are available, these do not serve the student or the applied mathematician as well as Lebedev's introductory and practically oriented approach. His systematic treatment of the basic theory of the more important special functions and the applications of this theory to specific problems of physics and engineering results in a practical course in the use of special functions for the student and for those concerned with actual mathematical applications or uses. In consideration of the practical nature of the coverage, most space has been devoted to the application of cylinder functions and particularly of spherical harmonics. Lebedev, however, also treats in some detail: the gamma function, the probability integral and related functions, the exponential integral and related functions, orthogonal polynomials with consideration of Legendre, Hermite and Laguerre polynomials (with exceptional treatment of the technique of expanding functions in series of Hermite and Laguerre polynomials), the Airy functions, the hypergeometric functions (making this often slighted area accessible to the theoretical physicist), and parabolic cylinder functions. The arrangement of the material in the separate chapters, to a certain degree, makes the different parts of the book independent of each other. Although a familiarity with complex variable theory is needed, a serious attempt has been made to keep to a minimum the required background in this area. Various useful properties of the special functions which do not appear in the text proper will be found in the problems at the end of the appropriate chapters. This edition closely adheres to the revised Russian edition (Moscow, 1965). Richard Silverman, however, has made the book even more useful to the English reader. The bibliography and references have been slanted toward books available in English or the West European languages, and a number of additional problems have been added to this edition.

Genres: Mathematics, Reference, Physics

520 pages, Kindle Edition

First published June 1, 1972

Reviews

[Tomáš Ševček · Rating 5 out of 5]

Although the syllabi of PDE courses for general mathematicians have changed drastically over the past 70 years, with the focus shifting from special functions to qualitative analysis and applications of functional analysis, this book is still an interesting read not only for engineers. The book covers some of the most widely used special functions (such as the gamma function, orthogonal polynomials, Bessel functions and many others), deriving some of their most useful properties and presenting problems in which they are employed. The book can definitely serve as a reference when one needs to look up the most commonly used properties of the functions. In order to follow the exposition, you should know your analysis (both complex & real) and ODEs very well. It also helps to be familiar with PDEs, since most of these functions actually arise when solving certain PDEs. I believe that this part is not stressed enough in the book. The book usually starts with definitions (usually as solutions to ODEs) of said functions and sometimes readers new to this area might keep wondering why one should be interested in solutions to certain wild-looking ODEs. It becomes clear only in the applications (when these specific ODEs arise when solving certain PDE problems on specific domains through the separation of variables). Therefore, I believe it might have been better to start with motivating examples first. Nevertheless, I highly recommend this book to anyone interested in special functions or in solving PDEs on specific domains.

[Nick Black]

Getting ready for grad school! Amazon 2008-05-01