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**A treatise on the theory of Bessel functions.
2nd ed.**
*(English)*
Zbl 0849.33001

Cambridge Mathematical Library. Cambridge: Cambridge Univ. Press. vi, 804 p. (1995).

This is an unaltered reprinting of the second (latest) edition [Cambridge: Cambridge Univ. Press. 1944; Repr. 1952, 1958, 1962, 1966 (Zbl 0174.36202, only title); Repr. 1980] in which only some misprints and minor errors of the first edition [(1922; JFM 48.0412.02)] were corrected.

On the occasion of the present new republication (1995) by the Cambridge Mathematical Library of this well-known classical treatise, the reviewer thinks fit to give a brief additional information. Indeed, this work has been universally acknowledged as giving an excellent and systematic treatment, clearly developed, on the theory of Bessel functions. It is divided into twenty chapters accompanied by an extensive bibliography which contains an impressive list of original sources. Although obviously the book can not do account of the enormous progress in the field since 1922, it continues being an indispensable reference book warmly recommended to all those mathematicians, physicists and engineers who are interested in Bessel functions and their applications.

The summary of the chapters is as follows. I. Bessel functions before 1922; II. The Bessel coefficients; III. Bessel functions; IV. Differential equations; V. Miscellaneous properties of Bessel functions; IV. Integral representations of Bessel functions; VII. Asymptotic expansions of Bessel functions; VIII. Bessel functions of large order; IX. Polynomials associated with Bessel functions; X. Functions associated with Bessel functions; XI. Addition theorems; XII. Definite integrals; XIII. Infinite integrals; XIV. Multiple integrals; XV. The zeros of Bessel functions; XVI. Neumann series and Lommel’s functions of two variables; XVII. Kapteyn series; XVIII. Series of Fourier-Bessel and Dini; XIX. Schlömilch series; XX. The tabulation of Bessel functions.

The book ends with some tables of the studied functions, an index of symbols, a list of authors and a general index. Numerous illustrative footnotes related to the references as well as many notes and helpful comments throughout the book enhances its utility and interest.

On the occasion of the present new republication (1995) by the Cambridge Mathematical Library of this well-known classical treatise, the reviewer thinks fit to give a brief additional information. Indeed, this work has been universally acknowledged as giving an excellent and systematic treatment, clearly developed, on the theory of Bessel functions. It is divided into twenty chapters accompanied by an extensive bibliography which contains an impressive list of original sources. Although obviously the book can not do account of the enormous progress in the field since 1922, it continues being an indispensable reference book warmly recommended to all those mathematicians, physicists and engineers who are interested in Bessel functions and their applications.

The summary of the chapters is as follows. I. Bessel functions before 1922; II. The Bessel coefficients; III. Bessel functions; IV. Differential equations; V. Miscellaneous properties of Bessel functions; IV. Integral representations of Bessel functions; VII. Asymptotic expansions of Bessel functions; VIII. Bessel functions of large order; IX. Polynomials associated with Bessel functions; X. Functions associated with Bessel functions; XI. Addition theorems; XII. Definite integrals; XIII. Infinite integrals; XIV. Multiple integrals; XV. The zeros of Bessel functions; XVI. Neumann series and Lommel’s functions of two variables; XVII. Kapteyn series; XVIII. Series of Fourier-Bessel and Dini; XIX. Schlömilch series; XX. The tabulation of Bessel functions.

The book ends with some tables of the studied functions, an index of symbols, a list of authors and a general index. Numerous illustrative footnotes related to the references as well as many notes and helpful comments throughout the book enhances its utility and interest.

Reviewer: Nácere Hayek (La Laguna)

### MSC:

33-02 | Research exposition (monographs, survey articles) pertaining to special functions |

33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |

65A05 | Tables in numerical analysis |

65D20 | Computation of special functions and constants, construction of tables |

01A75 | Collected or selected works; reprintings or translations of classics |