Abstract
In this thesis an attempt has been made to complete to a large extent one's knowledge of the solutions of the Heun equation. After the formulation of the equation and a discussion of the convergence of the power series solutions, a new notation is introduced, (Chapter 1). In Chapter 2 integral representations of the solutions of Heun's equation are discussed. Chapter 3 is concerned with certain orthogonality properties, while Chapter 4 deals with new non-linear integral equations satisfied by Heun functions and Heun polynomials. In Chapter 5 Lame's equation is discussed as a special case of Heim's equation. The objective of this chapter is to correct the statement that Lame functions of the second kind associated with the Lame polynomials can be expressed as a finite series of associated Legendre functions of the second kind. In Appendices A and B the work of two previous investigations is summarized.