A new class of special functions arising in plasma linear susceptibility tensor calculations

TL;DR

This paper introduces a new class of special functions related to Bessel and Weber functions, derived from plasma physics applications, which simplifies the calculation of the linear susceptibility tensor in magnetised plasma.

Contribution

The paper defines and analyzes a novel class of special functions, providing recurrence relations and series expansions that improve the computation of plasma susceptibility tensors.

Findings

Derived recurrence relations for the new functions

Presented an alternative representation involving incomplete Anger-Weber functions

Developed a simplified method for calculating the susceptibility tensor

Abstract

We investigate some fundamental properties of a peculiar class of special functions strictly related to Bessel, Anger and Weber functions, whose introduction was originally motivated by linear susceptibility tensor calculations in a hot, magnetised plasma. We show that these functions are solutions of an inhomogeneous Bessel ODE, with specified initial conditions and a distinct right-hand-side term fulfilling the Nielsen's requirement. Beside deriving recurrence relations and an alternative representation involving incomplete Anger-Weber functions, we show that these functions admit a simple series expansion in terms of Bessel functions of integer order, obtained by resorting to the Jacobi-Anger formula. In plasma applications this eventually leads to expressions involving infinite sums of products of Bessel functions, not particularly apt to numerical evaluation ought to their slow convergence rate when the particle's gyro-radius is larger than the wavelength. By exploiting the previously determined recurrence properties of the new class of functions we present a particularly simple derivation of the linear susceptibility tensor that enables to avoid this inconvenience.