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

TL;DR

This work introduces a new class of Bessel-type functions $G_\mu(z, \psi)$ arising in linear susceptibility tensor calculations for hot magnetised plasmas. It demonstrates that $G_\mu$ solves an inhomogeneous Bessel equation with a Nielsen-type right-hand side and provides multiple representations, including a Fourier expansion in cylindrical Bessel functions and a closed form involving incomplete Anger-Weber functions. The authors establish recurrence relations, Nielsen problem connections, and a governing PDE framework, and apply these results to derive a compact, efficient method for the plasma linear susceptibility tensor that avoids slow convergence of infinite Bessel sums. The approach unifies classical special-function theory with plasma physics, offering a practical pathway for numerical implementations and potential extensions to nonlinear regimes, and it clarifies links to Newberger's sum rule. $G_\mu$, its Anger-Weber connections, and the derived integrals form a cohesive toolkit for analytic and computational plasma analysis.

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.