Special Functions Evaluation Using Continued Fractions Methods. Applications in Physics and Engineering

Abstract

In this contribution we present a set of fast codes to calculate special functions (Bessel functions of integer and fractional order, Modified Bessel functions, etc.) based on the continued fractions method [1,2]. The accurate evaluation of (high order) Bessel functions (BFs) is essential in many areas of Physics and Engineering; for example, in microwave technology. Usual methods to calculate BFs take into account normalization relations. In this contribution we present algorithms to evaluate regular and irregular BFs without any recalculation through normalization relations based on a method maintaining the stability of each recurrence relation, i.e., we use forward recurrence relations for the BFs of the second kind and backward ones for the first kind functions. In fact the algorithms use forward recurrence relations to generate irregular BFs and take into account the continued fractions method to evaluate high order regular BFs. From these values we can generate BFs applying backward recurrence relations. Because of these structure (without renormalization), the algorithm is specially useful for calculating high order BFs. We also present algorithms based on a continued fractions method for other types of functions, but paying special attention to applications. These kind of codes have direct application in a wide variety of problems where (high order) BFs are necessary: in the evaluation of Lommel’s functions of two variables (very often used in wave and light guides) and in the evaluation of very high order Hankel functions or in scattering theory when one can encounter problems where large values of the impact parameter (i.e. large values of angular momentum) are involved (and where Glauber’s approach is not very appropriate).