Heun functions and their uses in physics

Abstract

Most of the theoretical physics known today is described by using a small number of differential equations. If we study only linear systems, different forms of the hypergeometric or the confluent hypergeometric equations of- ten suffice to describe this problem. These equations have power series solutions with simple relations between consecutive coefficients and can be generally represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painleve equation. There are important examples, however, where one has to use more com- plicated 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 usingperturba- tion methods. If one studies certain problems in astronomy or general relativity, encounter with Heun equation is inevitable. This is a general equation whose special forms take names as Mathieu, Lame and Coulomb spheroidal equations. Here the coefficients in a power seriesexpansions do not have two way recursion relations. We have a relation at least between three or four different coefficients. A simple integral transform solution also is not obtainable. Here I will try to introduce this equa- tion whose popularity increased recently, mostly among theoretical physi- cists,and give some examples where the result can be expressed in terms of solutions of this equation.