The solutions to this differential equation cannot be expressed in terms of simple functions. Equation (5.1) is known as Bessel the equation, and its solutions are known as Bessel functions. There are numerous books devoted to Bessel functions, but the most authoritative is the one by Watson. This 762-page book with a 36-page bibliography containing 791 references is the standard text. This book, which was published in 1944 and covers over three centuries of results, is the ultimate reference book on Bessel functions. A simpler (and shorter book) by Bowman59 is a good reference for those wanting a shorter read. In this chapter, we introduce the readers to the Bessel differential equation and its solutions in their various forms and state their properties.
These functions arose in Daniel Bernoulli’s (1700–1782) analysis of the oscillation of a hanging chain and again in Leonhard Euler’s (1707–1783) theory of the vibrations of a circular membrane and a few other problems in mechanics. In fact, Bernoulli solved an equation of the form given above and found a series solution now known as the Bessel function of the first kind. However, the German astronomer Friedrich Wilhelm Bessel (1784–1846) was the first person who carved out a systematic study of the differential equation and the properties of its solutions. Hence, we refer to this equation as the Bessel equation and the solutions as Bessel functions.
Bessel functions are ubiquitous in optical science. In addition, they appear in a great variety of other areas, such as electromagnetism, heat, hydrodynamics, elasticity, wave motion, scattering, etc. In fact, Bessel functions come into play whenever we deal with situations including cylindrical symmetry; for this reason all solutions of the Bessel equations are called cylinder functions. Because of this great use in various areas of physics and engineering, Bessel functions are arguably the most important functions beyond the elementary ones.
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