A Tutorial on Bessel Functions and Numerical Evaluation of Bessel Integrals

This paper aims to provide a tutorial on Bessel functions, and especially on the numerical evaluation of Bessel integrals. Bessel integrals are asymptotically evaluated using high-frequency methods, such as the stationary-phase method, steepest-descent path evaluations, and uniform asymptotics. Concepts such as saddle points and their contributions, and integration over the steepest paths are emphasized. A MATLAB simulation package, BESSEL_GUI, was prepared to compare each method with the other, and also with the MATLAB built-in function besselj (v, z). The tool also allows plotting the magnitude of the complex integrand of the Bessel integral in three dimensions. The user can thus visualize the locations of the saddle points and the steepest-decent paths.