AN INTRODUCTION TO THE STUDY OF INTEGRAL EQUATIONSby
MAXIME BÔCHER, B.A., Ph.D.
Professor of Mathematics in Harvard University
We give below the Preface to Bôcher's pamphlet followed by the Introduction:
In this tract I have tried to present the main portions of the theory of integral equations in a readable and, at the same time, accurate form, following roughly the lines of historical development. I hope that it will be found to furnish the careful student with a firm foundation which will serve adequately as a point of departure for further work in this subject and its applications. At the same time it is believed that the legitimate demands of the more superficial reader, who seeks results rather than proofs, will be satisfied by the precise statement of these results as italicized, and therefore easily recognized, theorems. The index has been added to facilitate the use of the booklet as a work of reference.
In these days of rapidly multiplying voluminous treatises, I hope that the brevity of this treatment may prove attractive in spite of the lack of exhaustiveness which such brevity necessarily entails if the treatment, so far as it goes, is to be adequate.
I wish to thank Professor Max Mason of the University of Wisconsin who has helped me with some valuable criticisms; and I shall be grateful to any readers who may point out to me such errors as still remain.
AN INTRODUCTION TO THE STUDY OF INTEGRAL EQUATIONS
The theory and applications of integral equations, or, as it is often called, of the inversion of definite integrals, have come suddenly into prominence and have held during the last half dozen years a central place in the attention of mathematicians. By an integral equation [a term first suggested by du Bois-Reymond in 1888] is understood an equation in which the unknown function occurs under one or more signs of definite integration. Mathematicians have so far devoted their attention mainly to two peculiarly simple types of integral equations, - the linear equations of the first and second kinds, - and we shall not in this tract attempt to go beyond these cases. We shall also restrict ourselves to equations in which only simple (as distinguished from multiple) integrals occur. This restriction, however, is quite an unessential one made solely to avoid unprofitable complications at the start, since the results we shall obtain usually admit of an obvious extension to the case of multiple integrals without the introduction of any new difficulties. In this respect integral equations are in striking contrast to the closely related differential equations, where the passage from ordinary to partial differential equations is attended with very serious complications.
The theory of integral equations may be regarded as dating back at least as far as the discovery by Fourier of the theorem concerning integrals which bears his name; for, though this was not the point of view of Fourier, this theorem may be regarded as a statement of the solution of a certain integral equation of the first kind. Abel and Liouville, however, and after them others began the treatment of special integral equations in a perfectly conscious way, and many of them perceived clearly what an important place the theory was destined to fill.
As we shall not, except in one relatively unimportant case, take up any of the applications of the subject, it may be well to say explicitly that like so many other branches of analysis the theory was called into being by specific problems in mechanics and mathematical physics. This was true not merely in the early days of Abel and Liouville, but also more recently in the cases of Volterra and Fredholm. Such applications of the theory, together with its relations to other branches of analysis are what give the subject its great importance.