Solving Systems of Transcendental Equations Involving the Heun Functions

Abstract

The Heun functions have wide application in modern physics and are expected to succeed the hypergeometrical functions in the physical problems of the 21st century. The numerical work with those functions, however, is complicated and requires filling the gaps in the theory of the Heun functions and also, creating new algorithms able to work with them efficiently. We propose a new algorithm for solving a system of two nonlinear transcendental equations with two complex variables based on the Müller algorithm. The new algorithm is particularly useful in systems featuring the Heun functions and for them, the new algorithm gives distinctly better results than Newton's and Broyden's methods. As an example for its application in physics, the new algorithm was used to find the quasi-normal modes (QNM) of Schwarzschild black hole described by the Regge-Wheeler equation. The numerical results obtained by our method are compared with the already published QNM frequencies and are found to coincide to a great extent with them. Also discussed are the QNM of the Kerr black hole, described by the Teukolsky Master equation.

A Yang–Mills field on the extremal Reissner–Nordström black hole

“…An alternative (black box) method is to exploit the fact that Eq. (23) for n = 0, 1 has the form of a double confluent Heun equation and get Maple to do the rest of the job[6,15], however we prefer a more transparent approach.…”

Probability distribution function for reorientations in Maier–Saupe potential

“…Equation (12) has fundamental solutions that can be expressed via the confluent Heun's function (CHF). The latter is a known special function [47], [48], [49]. At present it is realized explicitly in the only symbolic computational software package Maple as HeunC and its derivative HeunCP rime (see [48] for detailed discussion of the merits and drawbacks of this computational tool in Maple).…”

Wave maps on a wormhole

“…In the n = 0 case (that is U 0 = 0), there are several possible methods of computing quasinormal modes semi-analytically (notably, the method of continued fractions) but the least effort way is to use the fact that Eq. ( 10) is the confluent Heun equation and get Maple to do the job [11]. For n = 0 the general solution of Eq.…”

“…The difficulty is that for ℜ(λ) < 0 the ingoing wave is exponentially small for large r and therefore very hard to be tracked down numerically. However, Maple is able to evaluate the confluent Heun function in the whole complex r-plane which allows us to impose the quantization condition along a rotated ray re iθ where the ingoing solution is dominant [11]. Since θ depends on λ, this approach in principle requires a careful examination of Stokes' lines and branch cuts but in practice (especially when we know the quasinormal frequency approximately, for example from dynamical evolution or the WKB approximation), the range of angles θ for which the procedure is convergent can be easily determined empirically.…”