EFFECTIVE EVALUATION OF THE EXACT RELATIVISTIC PLASMA DISPERSION FUNCTIONS

The computation of the exact relativistic plasma dispersion functions (PDFs) [1,2] is a necessary basis for both the analysis of the electron cyclotron waves in laboratory thermonuclear and hot astrophysical plasmas and the analysis of the ion cyclotron waves in extremely hot astrophysical plasmas. The full account of relativistic effects is especially important in the regimes of wave propagation in high-temperature plasma almost perpendicularly to the confining magnetic field in the vicinities of higher cyclotron harmonic resonances. These functions, as well as other PDFs (non-relativistic one and weakly relativistic ones), can be expressed in the form of Cauchy or Cauchy-type integrals defined on the real axis, provided that the densities of the corresponding integrals vanish at infinity, and hence can be computed for not very large | | z values, being z their complex argument, by means of the direct numerical calculations of these singular integrals and using their asymptotic expansions for the remaining values of | | z [2]. However in many numerical applications, PDFs must be routinely evaluated many times, therefore the efficiency of the numerical algorithm involved in their calculation is of primary importance. For the simplest case of nonrelativistic PDF 2 ( ) exp( ) erfc( ) w z z iz = − − , the use of continued fractions of Jacobi, which are the special diagonal case of Pade approximants, has been proved to provide such an efficient method for large-| | z values in combination with the Taylor expansion of special kind for the remaining values of | | z [3]. For given accuracy, these calculations are about two orders of magnitude faster than the direct computation of the Cauchy type integrals and one order of magnitude slower than the calculation of the exponential function. The same technique involving two approaches can be used for the weakly relativistic PDFs [4,5]. However, the technique [4], due to the use of recurrent relation for the weakly relativistic PDFs, lacks stability when | | z becomes large. The numerical technique [5], developed for the computation of the weakly relativistic PDFs for not very large | | z values without the use of recurrent relations, can be also used for the most complicated case of the exact relativistic PDFs. But the main purpose of the present work is to present the new and more effective method to evaluate these functions in the same region.