Functional equations in several variables with applications to mathematics, information theory and to the natural and social sciences.

*(English)*Zbl 0685.39006
Encyclopedia of Mathematics and Its Applications, 31. Cambridge etc.: Cambridge University Press. xiii, 462 p. £50.00; $ 89.50 (1989).

This book deals with functional equations in several variables. In chapter 2, Cauchy’s equation is introduced providing readers with some basic techniques in handling functional equations. From this short chapter the authors lead readers to several independent area of applications. Topics encountered include combinatorics, probability and information theory, economics, mean values, harmonic analysis, operator theory, geometry, nomography, groupoids, trigonometric functions and number theory.

It is an ideal text for senior undergraduate and graduate students to get a first, but in depth, impression of what functional equation is about. Overviews on the organization and the inter relationship between various chapters are included for the convenience of the course instructors. The proofs are very well organized with sufficient details for readers to thumb through. The original paper of H. Kestelman [Proc. London Math. Soc., II. Ser. 53, 1-12 (1951; Zbl 0042.393)] can be used to perfect the proof of Theorem 6, § 5.2, on the endomorphisms of the complex field.

The exercises are at various indicated levels of difficulties. Some of the hard problems are results found in current research articles. Graduate students will find them challenging and stimulating. The book ends with a chapter on the history and includes an invaluable source of bibliographic references.

It is an ideal text for senior undergraduate and graduate students to get a first, but in depth, impression of what functional equation is about. Overviews on the organization and the inter relationship between various chapters are included for the convenience of the course instructors. The proofs are very well organized with sufficient details for readers to thumb through. The original paper of H. Kestelman [Proc. London Math. Soc., II. Ser. 53, 1-12 (1951; Zbl 0042.393)] can be used to perfect the proof of Theorem 6, § 5.2, on the endomorphisms of the complex field.

The exercises are at various indicated levels of difficulties. Some of the hard problems are results found in current research articles. Graduate students will find them challenging and stimulating. The book ends with a chapter on the history and includes an invaluable source of bibliographic references.

Reviewer: Che Tat Ng

##### MSC:

39B99 | Functional equations and inequalities |

39-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to difference and functional equations |

94A17 | Measures of information, entropy |