##
**Integrals and series. Volume 3: More special functions. Transl. from the Russian by G. G. Gould.**
*(English)*
Zbl 0967.00503

New York: Gordon and Breach Science Publishers. 800 p. (1990).

From the preface: The solution of many problems relating to various areas of science and technology reduces to the calculation of integrals and the summation of series containing elementary and special functions. As is well known, this task is considerably simplified by means of appropriate handbook literature from which we should single out the world-famous “Bateman Manuscript Project” series “Higher transcendental functions. Vol. I, II, III.” New York: McGraw-Hill (1953; Zbl 0051.30303, Zbl 0052.29502, Zbl 0064.06302)] and “Tables of integral transforms, Vol. I, II.” New York: McGraw-Hill (1954; Zbl 0055.36401, Zbl 0058.34103)] by A. Erdélyi et al. and also the Tables of integrals, sums, series and products [English translation, corrected edition. New York: Academic Press (1980; Zbl 0521.33001)] by I. S. Gradshtein and I. M. Ryzhik.

Over several decades these handbooks have been reference manuals for theoretical and experimental physicists, research engineers, and specialists in applied mathematics and cybernetics. However, they contained only formulae up to the end of the 1940s: and this has led to the need for creating a more complete reference manual, in which new results are reflected. In this connection, the Russian originals of Volumes 1 and 2 of this series appeared in the early 1980s [Vol. I: Moscow: Nauka (1981; Zbl 0511.00044); Vol. II: Moscow: Nauka (1983; Zbl 0626.00033)]. These contain results in this area of mathematical analysis that have been published in recent years. This volume contains tables of indefinite and definite integrals, finite sums and series and includes the functions of Struve, Weber, Anger, Lommel, Kelvin, Airy, Legendre, Whittaker, the hypergeometric and elliptic functions, the Mathieu functions, the MacRobert function, the Meijer function, the Fox function and several others. The book also contains tables of representations of generalized hypergeometric functions, and tables of Mellin transforms of a wide class of elementary and special functions, combined with tables of special cases of the Meijer \(G\)-function. Sections are included devoted to the properties of the hypergeometric functions, the Meijer \(G\)-function and the Fox \(H\)-function. The appendix contains supplementary material which can be used in the calculation of integrals and the summation of series.

The main text is preceded by a fairly detailed list of contents, from which the required formulae can be found. The notation used is, by and large, the generally accepted notation of the mathematical literature and is listed in the indices at the end of the book.

For the sake of compactness of exposition, abbreviated notation is used.

The Russian original has been reviewed (Moskva: Nauka 1986) in Zbl 0606.33001.

Over several decades these handbooks have been reference manuals for theoretical and experimental physicists, research engineers, and specialists in applied mathematics and cybernetics. However, they contained only formulae up to the end of the 1940s: and this has led to the need for creating a more complete reference manual, in which new results are reflected. In this connection, the Russian originals of Volumes 1 and 2 of this series appeared in the early 1980s [Vol. I: Moscow: Nauka (1981; Zbl 0511.00044); Vol. II: Moscow: Nauka (1983; Zbl 0626.00033)]. These contain results in this area of mathematical analysis that have been published in recent years. This volume contains tables of indefinite and definite integrals, finite sums and series and includes the functions of Struve, Weber, Anger, Lommel, Kelvin, Airy, Legendre, Whittaker, the hypergeometric and elliptic functions, the Mathieu functions, the MacRobert function, the Meijer function, the Fox function and several others. The book also contains tables of representations of generalized hypergeometric functions, and tables of Mellin transforms of a wide class of elementary and special functions, combined with tables of special cases of the Meijer \(G\)-function. Sections are included devoted to the properties of the hypergeometric functions, the Meijer \(G\)-function and the Fox \(H\)-function. The appendix contains supplementary material which can be used in the calculation of integrals and the summation of series.

The main text is preceded by a fairly detailed list of contents, from which the required formulae can be found. The notation used is, by and large, the generally accepted notation of the mathematical literature and is listed in the indices at the end of the book.

For the sake of compactness of exposition, abbreviated notation is used.

The Russian original has been reviewed (Moskva: Nauka 1986) in Zbl 0606.33001.

### MSC:

00A22 | Formularies |

26-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to real functions |

33-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to special functions |

### Citations:

Zbl 0606.33001; Zbl 0626.00033; Zbl 0511.00044; Zbl 0733.00004; Zbl 0733.00005; Zbl 0781.44002; Zbl 0786.44003; Zbl 0051.30303; Zbl 0052.29502; Zbl 0064.06302; Zbl 0055.36401; Zbl 0058.34103; Zbl 0521.33001
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\textit{A. P. Prudnikov} et al., Integrals and series. Volume 3: More special functions. Transl. from the Russian by G. G. Gould. New York: Gordon and Breach Science Publishers (1990; Zbl 0967.00503)

### Digital Library of Mathematical Functions:

§11.10(x) Integrals and Sums ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions§11.10(x) Integrals and Sums ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions

§1.14(viii) Compendia ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§11.7(v) Compendia ‣ §11.7 Integrals and Sums ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions

§11.7(v) Compendia ‣ §11.7 Integrals and Sums ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions

§11.9(iv) References ‣ §11.9 Lommel Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions

§13.10(vi) Other Integrals ‣ §13.10 Integrals ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions

§13.11 Series ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions

§13.23(v) Other Integrals ‣ §13.23 Integrals ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions

§13.24(ii) Expansions in Series of Bessel Functions ‣ §13.24 Series ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions

(13.6.11_1) ‣ §13.6(iii) Modified Bessel Functions ‣ §13.6 Relations to Other Functions ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions

(13.6.11_2) ‣ §13.6(iii) Modified Bessel Functions ‣ §13.6 Relations to Other Functions ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions

§14.17(i) Indefinite Integrals ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions

§14.17(iv) Definite Integrals of Products ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions

§14.18(iv) Compendia ‣ §14.18 Sums ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions

§14.20(x) Zeros and Integrals ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions

Integration by Parts ‣ §1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

§15.14 Integrals ‣ Properties ‣ Chapter 15 Hypergeometric Function

§15.15 Sums ‣ Properties ‣ Chapter 15 Hypergeometric Function

§15.4(i) Elementary Functions ‣ §15.4 Special Cases ‣ Properties ‣ Chapter 15 Hypergeometric Function

§16.10 Expansions in Series of F q p Functions ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function

§16.20 Integrals and Series ‣ Meijer G -Function ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function

§16.20 Integrals and Series ‣ Meijer G -Function ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function

§16.5 Integral Representations and Integrals ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function

§19.13(ii) Integration with Respect to the Amplitude ‣ §19.13 Integrals of Elliptic Integrals ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

§19.13(i) Integration with Respect to the Modulus ‣ §19.13 Integrals of Elliptic Integrals ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

§19.13(i) Integration with Respect to the Modulus ‣ §19.13 Integrals of Elliptic Integrals ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

§20.10(iii) Compendia ‣ §20.10 Integrals ‣ Properties ‣ Chapter 20 Theta Functions

§23.14 Integrals ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§24.13(iii) Compendia ‣ §24.13 Integrals ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials

§24.14(iii) Compendia ‣ §24.14 Sums ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials

§25.11(ix) Integrals ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions

§25.11(xi) Sums ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions

Integral Representation ‣ §25.12(ii) Polylogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions

§28.10(iii) Further Equations ‣ §28.10 Integral Equations ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation

§28.28(v) Compendia ‣ §28.28 Integrals, Integral Representations, and Integral Equations ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation

§9.10(ix) Compendia ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions

§9.11(v) Definite Integrals ‣ §9.11 Products ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions