Heun Functions and Some of Their Applications in Physics

Abstract

Most of the theoretical physics known today is described by using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/ or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lamé and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable.The use of this equation in physics and mathematical literature exploded in the later years, more than doubling the number of papers with these solutions in the last decade, compared to time period since this equation was introduced in 1889 up to 2008. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. Here this equation will be introduced and examples for its use, especially in general relativity literature will be given. * This paper is a revised and many times updated version of the conference talk by the same author, published in

Massless fermions on static general prolate metrics and their Heun solutions

“…This procedure applies to general static prolate metric of the form (1). For massless fermions, the general equation (6), in its concrete form (8) can be separated into its radial and angular parts. These have been solved in the general case of accelerating charged black holes, the solutions being expressed in terms of general Heun functions.…”

Klein-Gordon Oscillator in the Presence of External Fields in a Cosmic Space-Time with a Space-Like Dislocation and Aharonov-Bohm Effect

“…Equation 34is the biconfluent Heun's differential equation [28,75,76] with HðxÞ as Heun's polynomial function. Many authors studied the analytical solutions to the relativistic wave equations in terms of Heun functions (e.g., [77][78][79][80][81][82]).…”

Symbolic and numerical analysis in general relativity with open source computer algebra systems

“…Nevertheless, we will place it here in order to present a complete code structure. # Display the metric 23 show ( ' The Schwarzschild metric : ') 24 show ( g . display () ) ¦ ¥ In the Schwarzschild case, the manifold has four dimensions and the only variable is the black hole mass M .…”

“…After a formal analysis of the radial part, one can see that the radial solution can be given in terms of confluent Heun functions [20]. General and confluent forms of the Heun function are encountered in many applications in physics, especially as solutions of the wave equations [21][22][23][24][25][26].…”