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Hypergeometric functions and their applications. (English) Zbl 0744.33001
Texts in Applied Mathematics. 8. New York etc.: Springer-Verlag. xiv, 250 p. with 59 ill. (1991).

The book is written for undergraduates with a good background in calculus, and with basic knowledge in some areas of applied mathematics like Schrödinger’s equation and Maxwell’s equations.

The special functions considered in the book comprise ${\Gamma }$, ${}_{2}{F}_{1}$, ${}_{1}{F}_{1}$, and their particular cases, including orthogonal polynomials. While the coverage is not surprising, the presentation is less traditional. At an early stage, ${}_{2}{F}_{1}$ and ${}_{1}{F}_{1}$ are introduced as series solutions to differential equations. Then various special functions are introduced, each with a suitable problem in applied mathematics as background, and in turn the functions are seen to be in fact particular ${}_{2}{F}_{1}$’s or ${}_{1}{F}_{1}$s. For instance, Schrödinger’s equation in spherical coordinates gives us Legendre’s differential equation and the Legendre polynomials as particular ${}_{2}{F}_{1}$’s; while Maxwell’s equations for a cylindrical waveguide leads to Bessel functions as particular ${}_{1}{F}_{1}$’s.

Thus, at mid-way, the reader would be aware of the role played by the hypergeometric functions, although they act mostly via other named functions. As to the latter, the reader at this point knows only a minor part of their properties, and has seen them only as functions of real variables because of the way they were introduced. The second half of the book, therefore, begins with chapters on complex analysis, including Cauchy’s theorem, series, singularities, and contour integrals. The special functions are now in their proper setting and a deeper study of their properties can be carried out. For instance, integral representations are given. Also, asymptotics, orthogonality, and generating functions are subjects that are treated at some length.

All chapters contain exercises, and quite a few of them are, in the spirit of the book, problems taken from the applications, e.g. quantum mechanics.

In conclusion, the reviewer finds that this book helps bridging pure and applied mathematics, and it should find a natural place in university teaching.

##### MSC:
 33-02 Research monographs (special functions) 33-01 Textbooks (special functions) 81-01 Textbooks (quantum theory) 33Cxx Hypergeometric functions 33B15 Gamma, beta and polygamma functions 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 33C45 Orthogonal polynomials and functions of hypergeometric type