The Finite Element Method in Electromagnetics, 3rd Edition

APPENDIX ESOME SPECIAL FUNCTIONS

In this appendix we describe briefly the basic definitions and properties of some special functions used in this book. These include the Bessel functions, spherical Bessel functions, associated Legendre polynomials, and Mathieu functions. A more extensive list of properties and curves can be found in a handbook by Abramowitz and Stegun [1]. The numerical evaluation of these functions is described in a book by Zhang and Jin [2].

E.1 BESSEL FUNCTIONS

The Bessel functions, denoted by J_n(z) and Y_n(z), are the two linearly independent solutions of the second-order differential equation

(E.1)

which is called Bessel’s equation. The Hankel functions of the first and second kinds are defined by

(E.2)

(E.3)

The series expressions for J_n(z) and Y_n(z) are given by

(E.4)

and

(E.5)

where γ ≈ 0.57721566490153286 denotes Euler’s constant.

The Bessel functions have many mathematical properties, which are summarized in Abramowitz and Stegun [1]. For example, ...