1101.0471 (M. Hortacsu)

Heun Functions and their uses in Physics    [PDF]

M. Hortacsu

Most of the theoretical physics known today is described using a small number of differential equations. If we study only 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, then one often uses one form of the Painlev\'{e} equation. There are important examples, however, where one has to use higher order equations. An example often encountered in quantum mechanics is the hydrogen atom in an external electric field, the Stark effect. One often bypasses this difficulty by studying this problem using perturbation methods. If one studies certain problems in astronomy or general relativity, encounter with Heun functions is inevitable. This is a general equation whose special forms take names as Mathieu, Lam\'{e} and Coulomb spheroidal equation. Here the coefficients in a power series expansions do not have two way recursion relations. We have a relation between three or four different coefficients. A simple integral transform solution is also not obtainable. Here I will try to introduce this equation and some examples whose solution can be expressed in terms of solutions to this equation. Although this equation was discovered more than hundred years ago, there is not a vast amount of literature on this topic and only advanced mathematical packages can identify it. Its popularity, however, increased recently, mostly among theoretical physicists, with 163 papers in SCI in the last thirty years. More than Three fourths of the papers that use these functions in physical problems were written in the last decade.

View original: http://arxiv.org/abs/1101.0471