Abstract
Only two years after the publication of their volume of integrals and sums of elementary functions [1], reviewed in [2], the authors now present a 749 page collection of formulas for integrals and sums of special functions. The main part of the volume is divided into five chapters, and each chapter is divided into many sections and subsections. As in [1], the notation is standard, and knowledge of Russian (required for the few short sections of text) is not essential. 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; as well as 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 involving products of Bessel functions with exponential functions or powers. Chapter 4 (11 pages) gives finite sums; in particular 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; in particular, series involving incomplete gamma functions, the Riemann zeta function,
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