Bessel Functions

The subject of Bessel functions is an old one. The earliest mention of these functions was dated October 3, 1703, when a series now described as a Bessel function of order 1/3 appeared in a letter from Jakob Bernoulli to Leibniz [(13) p. 356]. The Bessel coefficient of order zero occurred in 1732 in Daniel Bernoulli's memoir on the oscillations of heavy chains. However, the first systematic study of the functions was made in 1824 by Bessel. Many papers have been published on the subject but most of the literature seems to be of two general types. The first type is found in text books dealing with mathematics as applied to physics. The scope of this type of publication is very limited and one cannot get sufficient background on the development of Bessel functions. The properties needed are stated and used without proof. On the other hand, there are a few publications on Bessel functions which present them in great detail. These publications are generally at a level too high for the undergraduate to read with ease. The aim of this paper is to achieve a medium somewhere between these two types of publications. The procedure to be followed here is to accept the basic physical problem without going into the development. Some of the various applications are noted but not presented in detail. Chapters II, III, and IV are devoted to introducing the subject, stating the basic definitions, and developing various properties. These will be necessary in applying the functions to physical problems. The last chapter is concerned with one such problem in which several sets of initial and boundary conditions are used. This problem is studied in some detail. Reference material for this work was obtained from Forsyth Library and through inter-library loans. Publications by most of the recognized authorities on the subject are listed in the bibliography. This material is entirely secondary in nature. Original work of this nature are difficult, if not impossible, to obtain. Several limitations are placed on this paper. The most important is the scope, which was arbitrarily set by the writer. Securing material did not present a real problem but reference books obtained through inter-library loans had to be returned in approximately two weeks with no provision for renewal. It is hoped that the reader may obtain a thorough understanding of the basic properties and uses of Bessel functions through this paper.