Kuczma M. Functional Equations in a Single Variable

Warszawa, 1968. — 385 pages.Although the greatest mathematicians, such as N. H. Abel, A. Cauchy, L. Euler, B. Riemann and others, have concerned themselves with functional equations, there existed - until recently - no exhaustive monograph of the subject. The articles by S. Pincherle in Enzyklopadie der Mathematischen Wissenschaften and the booklet by E. Picard, in which only a few particular equations were dealt with, could hardly be regarded as such. The non-existence of a general theory of functional equations was, no doubt, one of the reasons of this situation. In recent times, however, we observe great progress in this branch of mathematics. This progress is manifested also in two newly published monographs: one by M. Ghermanescu, and the other by J. Aczel. Ghermanescu's book, however, is written in the Roumanian language and thus is inaccessible to a great number of the world's mathematicians. Moreover, it concentrates on the contributions of Roumanian mathematicians and leaves out many important results. Aczel's work, on the other hand, has gained great popularity and an English translation of it, much enlarged, has just appeared. However, it deals exclusively with functional equations in several variables. It is by all means justifiable to separate the two topics: equations in a single variable and in several variables, since there is considerable difference between them, at least as great as between ordinary and partial differential equations, both in methods and in the kind of results. Nevertheless, the fact is that a satisfactory work on functional equations in a single variable has still been lacking. It is the purpose of the present book to fill this gap at least partially.