BESSEL'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. This chapter introduces the basic Bessel functions and their properties. It also discusses the modified Bessel functions and the spherical Bessel functions. Bessel functions also satisfy an orthogonality relation. Unlike the Legendre, Hermite, or Laguerre polynomials, their orthogonality is not with respect to their order but with respect to a parameter in their argument. The chapter presents the Bessel equation with respect to a new argument.