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H-Transforms: Theory and Applications

Anatoly A. Kilbas and Megumi Saigo
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Analytical Methods and Special Functions
[Reviewed by
Warren Johnson
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I've been trying to estimate the size of the audience for this book. My first guess of 100 must be too low, inasmuch as the bibliography (which is very good) lists some 576 works, only a few of which seem peripheral to the subject, by 217 different authors. While Paul Halmos, here for his A Hilbert Space Problem Book, would have no interest in the book, and some of the other authors (e.g. Hardy, Littlewood, Bateman) are no longer with us, most of the rest will welcome it, and they may not be alone.

If you are wondering what an H-transform is, then this is probably not the book for you, but anyway here goes. An H-function is a contour integral with respect to s of z-s times a combination of gamma functions. Each gamma function depends on s and two parameters, either an a and an α or a b and a β, and the contour keeps all the poles of the b/β gamma functions on the left and all the poles of the a/α gamma functions on the right. If all the alphas and betas are 1, the H-function reduces to the somewhat better known Meijer G-function. Further special cases include hypergeometric functions, Bessel functions, and elementary transcendental functions such as ez, sin z, cos z, log(1+z), sinh z, cosh z, arcsin z and arctan z. An H-transform is an integral transform (e.g. Laplace, Mellin, Fourier) of an H-function.

This type of contour is usually associated with Mellin and Barnes, and a nice pre-history of the subject was given by Barnes in the first four pages of his paper in Proc. London Math. Soc. (2), vol. 6 (1908), 141-177. This fundamental paper of Barnes is curiously omitted from the bibliography of the book under review, which does include Mellin's big paper of 1910. The only overlap between the book's history and Barnes's is Pincherle's path-breaking paper of 1888.

The basic idea is this: it is not difficult to see that, after analytic continuation, the gamma function has simple poles at all the nonpositive integers, and that the residue at -n is (-1)n/n!. Thus the idea of integrating a combination of gamma functions, at first sight rather implausible, suddenly becomes promising provided that one can control the behavior on the contour. This naturally requires some care, but it is possible.

This idea of Pincherle was exploited more thoroughly by Mellin and Barnes. Barnes developed the theory of the hypergeometric function from this point of view in his 1908 paper. (He also wrote some related papers from 1904 to 1910, before leaving mathematics as he rose in the Anglican Church.) Short accounts of Barnes's theory are in Whittaker and Watson's A Course of Modern Analysis, Bailey's Generalized Hypergeometric Series, and Andrews, Askey and Roy's Special Functions. (These books also are not in the bibliography of the present book. Incidentally, I infer Halmos's lack of interest in it not just from general principles, but particularly from his comments about Whittaker and Watson in the fascinating article "Some Books of Auld Lang Syne" in A Century of Mathematics in America.) Thus, besides those who actually work on H-functions and/or integral transforms, another class of readers who might be interested in this book are those who find beauty in this idea, and want to see what it has led to. But if you haven't found the beauty already, don't expect to find it here.

The back cover of the book claims four "Key Features" for it. The second and fourth claims are correct. Technically, so is the First — that the book "provides a general, unified introduction to the theory of integral transforms" — but I don't think many people would want to teach integral transforms from this point of view. The third claim, that the book "includes complete proofs of all results", is flagrantly violated from chapter 5 on. Sixty theorems, many with up to 5 parts, are stated on pp. 135-160 of chapter 5, and chapter 6 is even more densely populated with theorems — whole pages that are just one theorem after another, with no proof or commentary. To be fair, most of the theorems in chapter 6 are special cases of those in chapter 5.

I have not found many misprints. My favorite was "milptiplier" on page 110. Magnus, Oberhettinger and Brychkov are all misspelled at least once, but they occur often and are correct the vast majority of the time. There are two misspellings in the title of Mellin's paper. Checking very many of the formulas would be a hard job, but the few I have checked have been accurate.

Another questionable omission from the bibliography is Watson's treatise on Bessel functions. The references for Bessel functions are always to the Bateman project (i.e., the 5 volume work by Erdelyi, Magnus, Oberhettinger and Tricomi) or to the 3-volume work Integrals and Series, by Prudnikov, Brychkov and Marichev (PBM). Thus, for example, the book says (2.6.4) is a known formula, with a reference to PBM, when it might have said that (2.6.4) is a special case of an integral of Hankel and Gegenbauer, with reference to pp. 384-85 of Watson if not to the original papers. Much later, they give the same reference to PBM rather than remarking that (8.12.18) is a special case of (2.6.4). In some ways this is a very good book, but it is not, in my opinion, of much general interest, and these examples show why. There is a section of bibliographical remarks at the end of each chapter, and in these sections the authors are very careful and thorough in surveying the original literature; specialists will find these sections of great value. But the authors evince little concern for ancillary results such as (2.6.4), and it is just this concern that would be most effective in engaging non-specialists. Let me conclude with a contrasting example from Watson's treatise: in section 2.32, as a means of establishing an integral of Poisson, a curious differentiation theorem of Jacobi is wanted. These authors would be content with one reference. Watson gives a thorough history, with details of three different proofs. That is how you write a classic.

Warren Johnson ( is visiting assistant professor of mathematics at Connecticut College. One of his favorite areas of mathematics is special functions. But if you read this review, then you know as much about H-transforms as he does.

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