Essentials of Mathematical Methods in Science and Engineering, 2nd Edition

CHAPTER 13BESSEL'S EQUATION AND BESSEL FUNCTIONS

Bessel functions are among the most frequently encountered special functions in physics and engineering. They are very useful in quantum mechanics in WKB approximations. Since they are usually encountered in solving potential problems with cylindrical boundaries, they are also called cylinder functions. Bessel functions are used even in abstract number theory and mathematical analysis. Like the other special functions, they form a complete and an orthogonal set. Therefore, any sufficiently smooth function can be expanded in terms of Bessel functions. However, their orthogonality is not with respect to their order but with respect to a parameter in their argument, which usually assumes the values of the infinitely many roots of the Bessel function. In this chapter, we introduce the basic Bessel functions and their properties. We also discuss the modified Bessel functions and the spherical Bessel functions. There exists a wealth of literature on special functions. Like the classic treatise by Watson, some of them are solely devoted to Bessel functions and their applications [1].

13.1 BESSEL'S EQUATION AND ITS SERIES SOLUTION

Bessel's equation is defined as

(13.1)

where the range of the independent variable could be taken as the entire real axis or even the entire complex plane. At this point, we restrict to positive and real ...