Special Functions & Their Applications (Dover Books on Mathematics) by N. N. Lebedev (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I was particularly interested in the development of the Bessel Function of the Second kind. This is one of the very few books that give guidance and insight in the more difficult elements of that derivation. This is an excellent text but is not for the faint of heart. You must be willing to really study the material with a pen and paper in hand. But, this is the path for anyone that wants to truly gain insight and learning.

⭐This book is a classic. It is certainly a valuable piece to any collection. This book personally is useful to me since I need to consult about the special functions constantly.

⭐good

⭐Of course this book cannot be compared to ” A course of modern analysis “, but as a book in special functions, it have served its purpose. I would also like to make a tribute to Richard Silverman for translating ( not just direct traslating, he translate in a style make it readable to English world. )

⭐There are many books on special functions, especially from around the 1960’s, and this is one of the better ones.Chapter 1: the Gamma function.Chapter 2: the error function, Gaussian integrals, Fresnel integrals.Chapter 3: the exponential integral.Chapter 4: orthogonal polynomials: the Legendre, Hermite, and Laguerre polynomials. For each of these classes of orthogonal polynomials, Lebedev works out their generating functions, first and second order differential equations satisfied by them, integral formulas, orthogonality, asymptotic formulas, and expansions of functions in terms of these orthogonal polynomials. This chapter does not cover roots of orthogonal polynomials (Sturm’s theorem).Chapters 5 and 6: Bessel functions and the Airy function.Chapters 7 and 8: spherical harmonics and the Legendre functions.Chapter 9: the hypergeometric function.Chapter 10: parabolic cylinder functions and the Hermite functions.Lebedev proves the results he gives, which makes this book useful for mathematicians who want precise statements and proofs rather than merely tables of identities and inequalities. The one big topic Lebedev doesn’t cover are the roots of orthogonal polynomials, for which the most complete reference is Szego, which could be improved.

⭐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.

⭐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.

⭐I always expect the books of good conditions.The book is very good.

⭐Muy buen libro. Excelente para curso de métodos matemáticos en la física.Tiene muchos ejemplos y una exposición clara. Es recomendable cierto manejo de matemáticas de nivel universitario (segunda mitad de la carrera en física).Leave no stones unturned. Look for gems under every stone. Even under special functions.

⭐Un ottimo testo di riferimento per le funzioni speciali con molte applicazioni fisico – ingegneristiche. Da non farsi mancare nella propria libreria!Todos los pasos y trucos utilizados son explicados con suficiente detalle.

⭐

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