13th Regional Conference on Mathematical Physics, Antalya, Turkey, 27 - 31 October 2010, pp.23-39
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 often 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 complicated 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 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 series expansions 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 equation and give some examples where the result can be expressed in terms of solutions of 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 ninety four papers in SCI in the last twenty five years. More than two thirds of the papers which use these functions in physical problems were written in the last decade.