Harmonic Analysis

Abstract

THE need for a treatise on harmonic analysis, more particularly on spherical harmonics, has been recognised for a long time by mathematicians who teach the subject or who find it necessary to use spherical harmonics in their investigations. Those who had the good fortune to be introduced to the study of the subject by the author of the present work had always hoped that he would find time to undertake the preparation and publication of such a treatise. It may be said at once that the work now published fulfils the expectations they had formed. Although many books which contain information about spherical harmonics have appeared since the publication of Heine's “Kugelfune-tionen”, none of them has claims to be considered a complete treatise on the subject, and Heine's work, though it has proved of great service to students of the subject and to mathematicians who had occasion to use harmonic analysis in their investigations, is a compilation of results which contains the material for a treatise rather than a treatise. Much has been written on the subject since the appearance of Heine's two volumes. The mere perusal of the literature involves much labour, and the task of selecting from this mass of literature what is of prime importance to students of the subject and what is necessary for the mathematical physicist is one which demands a wide knowledge of the related branches of mathematics and of applied mathematics. That the author has fulfilled these requirements the present treatise is the evidence. The first four chapters give a complete account of the origin of the ordinary spherical harmonics; Legendre's coefficients of the first and second kinds and the associated functions are treated in detail, their more important properties are developed, and their expression in the form of series and of definite integrals is investigated. Solid harmonics are discussed fully, the addition theorem for ordinary spherical harmonics is established, and the connexion of the functions with potential theory is dealt with. The results given in this part of the treatise are sufficient for the solution of the problems of potential theory and of other branches of mathematical physics where the space under consideration is the space inside or outside the surface of a sphere or the space between two concentric spheres.

The Theory of Spherical and Ellipsoidal Harmonics.

By Prof. E. W. Hobson. Pp. xi + 500. (Cambridge: At the University Press, 1931.) 37s. 6d. net.