Heun Functions and Their Uses in Physics

Abstract

Most of the theoretical physics known today is described 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 2010. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. An integral transform solution using simpler functions also is not obtainable. 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 few times updated version of the conference talk by the same author, published in

Wave equation for the Wu black hole

“…The angular equation, 16) has four regular singularities as a 2 , b 2 , 1 g 2 , ∞ and it can be solved in terms of the general Heun's functions [36][37][38].…”

Biconfluent Heun equation in quantum chemistry: Harmonium and related systems

“…However, this equation is known in mathematics since more than a century as the biconfluent Heun equation and its properties were studied from both purely mathematical perspective [29,30,[35][36][37][38] and in the context of its applications in different areas of physics [39][40][41]. Very recently a brief review on its physical applications has been published by Hortaçsu [42]. An analysis of its application to modeling the behavior of two interacting electrons in a uniform magnetic field and a parabolic confinement was published several years ago by Kandemir [34].…”

Quantum field theory applications of heun type functions

“…An important point in the history of this equation and its confluent forms is the "Centennial Workshop on Heun Equations: Theory and Applications, Sept. [3][4][5][6][7][8]1989, Schloss Ringberg". The presentations at this workshop were printed in a book edited by A. Ronveaux [2].…”

“…Work done before the year 2000 can be found in the book by Slavyanov and Lay [3]. For later applications, especially in general relativity, Hortaçsu gives a long list [4].…”