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Integrals and series of elementary functions
Prudnikov A., Bryčkov Y., Maričev O., Science, Moscow, Russia, 1981. Type: Book
Date Reviewed: Sep 1 1986
Comparative Review

These are two very impressive collections of formulas. The notation in both volumes is standard, and knowledge of Russian (needed only for the few short sections of text) is not essential.

The main part of the volume on elementary functions is divided into six chapters, and each chapter is divided into many sections and subsections. Chapter 1 (247 pages) deals with indefinite integrals, including integrands involving (for example) simple rational functions, trigonometric and hyperbolic functions and their inverses, and rational functions of square roots of linear and higher-order polynomials. Chapter 2 (294 pages) consists of definite integrals. The integrand expressions (often very general and sometimes complicated) contain, according to the sections, powers and algebraic functions, and exponential, hyperbolic, trigonometric, logarithmic, inverse trigonometric, and inverse hyperbolic functions. Many integrands contain mixtures of several of these functions. Chapter 3 (32 pages) deals with double, triple, and multiple integrals of an impressive variety of integrands, mostly containing several of the functions mentioned above. General formulas are given at the beginning of each section. Chapter 4 (54 pages) is a collection of finite sums, in particular sums containing products or quotients of binomial coefficients, and products of trigonometric functions. Chapter 5 (102 pages) consists of an extended collection of results for infinite series: numerical series, power series with complicated coefficients, series with exponential and hyperbolic functions, trigonometric series, and series of logarithms and arctangents. Chapter 6 (6 pages) contains results on finite and infinite products. There are three short appendices: the first two list various properties of elementary and special functions, and the third contains explicit expressions for an abbreviated notation used for the evaluation of certain integrals in the tables.

A bibliography of 107 items is included, but no direct references are given in any formulas. Two tables of definitions, and a list of cross-references between various functions and series occurring in the tables, complete the work.

The main part of the volume on special functions is divided into five chapters, with each chapter divided into many sections and subsections. Chapter 1 (54 pages) deals with indefinite integrals, including integrands involving incomplete gamma functions, the exponential integral, the error functions, Fresnel integrals, different types of Bessel and Hankel functions, and orthogonal polynomials; also included are products of these functions with powers, logarithms, and exponential functions. The very long Chapter 2 (562 pages) consists of definite integrals. This chapter is very impressive, and many of the formulas it contains seem to have been compiled for the first time. It is divided into sections for integrands containing the gamma function, the psi function, the Riemann zeta function, the exponential integral, ordinary and hyperbolic sines and cosines, error functions, Fresnel integrals, incomplete gamma functions, parabolic cylinder functions, ordinary and modified Bessel functions (very extensive), Hankel functions, and Legendre, Laguerre, Hermite, Gegenbauer, and Jacobi polynomials. Many integrands contain several of these functions, often in combination with elementary functions, and depend on a certain number of parameters. Results are frequently given as infinite series or as hypergeometric functions pFq, which may limit their practical applicability in certain cases. Chapter 3 (18 pages) contains double, triple, and some multiple integrals; in particular, many integrals involving products of Bessel functions with exponential functions or powers are discussed. Chapter 4 (11 pages) gives finite sums, presenting, in particular, sums of (ordinary and modified) Bessel functions, and of Legendre, Laguerre, Hermite, Gegenbauer, and Jacobi polynomials. Sums involving products of these polynomials are also given. Chapter 5 (73 pages) contains an extensive compilation of infinite series, including, in particular, series involving incomplete gamma functions, the Riemann zeta function, sine and cosine integrals, real and complex error functions, parabolic cylinder functions, ordinary and modified Bessel functions, and the orthogonal polynomials mentioned above.

There are two short appendices. The first (2 pages) contains formulas for the binomial coefficients and the Pochhammer symbol. The second (21 pages) contains formulas relating to some of the special functions. Formulas for the mth derivative (with respect to the argument z) of products of certain elementary functions (powers and exponentials) with orthogonal polynomials, whose argument is z or an elementary function of z (reciprocal, square root), are given particular mention. Two glossaries of notation complete the work. There is a list of 37 references, but no direct references are given in any formulas. The printing and binding of both volumes are satisfactory.

These are certainly two of the most important reference books for mathematicians, physicists, engineers, and others working in fields where such formulas are likely to occur. Especially in view of their reasonable prices, they ought, in principle, to be made available in the relevant libraries. Unfortunately, however, the number of copies printed (70,000 and 22,000, respectively) is rather small, and the books are difficult to obtain. There are reports that an English translation is in preparation.

Reviewer:  K. S. Kölbig Review #: CR110351
Comparative Review
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