Algorithms and codes for the Macdonald function: recent progress and comparisons

AbstractThe modified Bessel function Kiν(x), also known as the Macdonald function, finds application in the Kontorovich–Lebedev integral transform when x and ν are real and positive. In this paper, a comparison of three codes for computing this function is made. These codes differ in algorithmic approach, timing, and regions of validity. One of them can be tested independent of the other two through Wronskian checks, and therefore is used as a standard against which the others are compared

Scattering and delay time for 1D asymmetric potentials: the step-linear and the step-exponential cases

We analyze the quantum-mechanical behavior of a system described by a one-dimensional asymmetric potential constituted by a step plus (i) a linear barrier or (ii) an exponential barrier. We solve the energy eigenvalue equation by means of the integral representation method, classifying the independent solutions as equivalence classes of homotopic paths in the complex plane. We discuss the structure of the bound states as function of the height U_0 of the step and we study the propagation of a sharp-peaked wave packet reflected by the barrier. For both the linear and the exponential barrier we provide an explicit formula for the delay time \tau(E) as a function of the peak energy E. We display the resonant behavior of \tau(E) at energies close to U_0. By analyzing the asymptotic behavior for large energies of the eigenfunctions of the continuous spectrum we also show that, as expected, \tau(E) approaches the classical value for E -> \infty, thus diverging for the step-linear case and vanishing for the step-exponential one.Comment: 14 pages, 10 figure

Basic Methods for Computing Special Functions

This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website