Special Functions & Their Applications

By Nikolai Nikolaevich Lebedev

Book Overview

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.

• Format: Paperback
• Language: English
• ISBN: 0486606244
• ISBN13: 9780486606248
• Release Date: June 1972
• Publisher: Dover Publications
• Length: 336 Pages
• Weight: 0.80 lbs.
• Dimensions: 0.6" x 5.4" x 8.5"

Customer Reviews

Excellent book for people who want to actually apply special functions.

As the title indicates, the book is designed with the goal of application front and center. That said, it is also important to note that the theoretical background is developed with full mathematical rigor. You can easily see this from the fact that whenever an infinite series is differentiated, its uniform convergence in the region of interest is always established beforehand. And this is just one example. Now, given the fact that special functions is a vast subject, and the fact that the book is barely 300 pages long, it is obvious that the theoretical coverage, though rigorous, has to be reined in. By this I refer to the fact that most functions are developed from the point of view of series solutions to differential equations, while solution by contour integrals in the plane is basically absent. But then again, it doesn't matter how you develop the functions, the key is to know their properties and be able to apply them. The book will show you just how to do that. HIGHLY RECOMMENDED. For a more broad-based theoretical coverage, I recommend Whittaker and Watson (but of course), and the book "Special Functions" by X. Z. Wang. These two books complement each other like lovers.

extremely useful, very concise

Well worth buying, extremely handy, tons of information very much organized for you

Excellent discussion of the functions used to solve PDE

Yet another excellect translation by Silverman. I've only been in possession of this book for a few days but it's already becoming a favorite mathematics text. Not a pure mathematics text but certainly a very thorough, lucid and most certainly enjoyable discussion of applied mathematics with a particularly engaging discussion of the solution of partial differential equations (Laplacian, Poisson etc.) by means of separation of variables and integral transforms. Along the way it develops the theoretical essentials of gamma functions, exponential integrals, orthogonal polynomials, Bessel functions, spherical harmonics among others. Clearly written with an emphasis on explaining the process of discovering solutions rather than merely presenting particular solutions (though it does have enlightening examples). IMO, well worth the price.

Excellent Source of Applications for Special Functions

This is a book which I cannot say much about except for the unusual thoroughness, accompanied by detail and depth in treatment of the underlying mathematical properties and applications of Special Functions.Lebedev is the quintessential mathematical expert in applying Special Functions to problems in Physics and Engineering, being that he can illustrate all important concepts clearly and umambiguously with carefully prepared diagrams as well as words. I was able to cite the solution of the a problem involving a propagating electromagnetic wave along a transmission line for an important Engineering course project. For such a problem, Lebedev offers a far more detailed and precise solution with given Special Functions than anything I have ever seen in other books of the same nature with the possible exception of a specialized treatise by an MIT EE faculty member on applied electromagnetism. He also comes across as meticulous in derivations of solutions to problems worked out compared to many other authors whose works I have read. This is because he hardly ever skips an important step in deriving a solution for any given problem by leaving it out for the reader's imagination. Yet we know Lebedev as perhaps a mathematician who may not be realistically expected to come up with such complete and exhaustive solutions to practical or real-world problems, worked out with clarity as well as precision and depth. There are numerous other examples which he worked out for different applications (e.g, Legendre's and Laguerre's functions) invariably after he took pains to delineate the various mathematical properties of the Special Functions utilized to obtain the closed-form solutions. He also covers various mathematical functions which may not be as familiar to many engineering practitioners but nonetheless have an important place in applied mathematical analysis. In a sense, he saves them for occasions when we as readers may need to probe further into unfamiliar territory.So if you are looking for depth and precision in analysis of physical problems in Engineering and Science, or are trying to cope with reaearch problems in Applied Mathematics, try out this book by Lebedev. It can initially come across as difficult to understand, but Lebedev expects the reader to follow along through diligence. It is almost one of a kind, being that it is very clear and lucid without noticeable loss in depth and mathematical rigor. I highly recommend it because I believe that few other books can even come close in offering good examples in solutions to real-world problems and, at the same time, demonstrate the power of Special Functions in applications. Of course, it is also very inexpensive.

Good

Bessel's, harmonics, normal CDF, gamma, hypergeometrics, orthogonal polynomials, etc. The most important functions in engineering, (except for elementary stuff on our calculators, of course).Very good coverage. No fluff. Well-organized. Conveniently sized paperback.