Heun Functions and Some of Their Applications in Physics

TL;DR

This paper reviews the Heun equation, a higher-order differential equation increasingly used in physics, especially in general relativity, highlighting its mathematical properties, solutions, and recent surge in applications over the past decade.

Contribution

It introduces the Heun equation and discusses its applications in physics, particularly in general relativity, emphasizing its mathematical complexity and recent growth in research interest.

Findings

Heun equation is increasingly used in physics, especially in relativity.

Solutions involve complex recursion relations, unlike hypergeometric equations.

Research publications on Heun functions have more than doubled in the last decade.

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 Painlev\'{e} 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, Lam\'{e} 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.