Abstract
This thesis consists of two largely independent parts. Part I is a discussion of Campbell's work on the Associated Mathieu equation. This equation is a transformation of the spheroidal wave equation and it was to he expected that many of Campbell's results had their counterparts in the theory of spheroidal wave functions. This thesis examines Campbell's work, relates it to the latter theory, completes and corrects it at certain points. In particular, it is shown that Campbell's "finite solution" of the Associated Mathieu equation is erroneous. Part II is devoted to a study of the spheroidal wave equation in the case when the characteristic exponent is half an odd integen The standard method of construction of solutions as series of Legendre or Bessel functions, breaks down completely in this case. After a study of the form of the eigen value A , it is shown that the series of Legendre (or Bessel) functions is supplemented by a series involving derivatives of such functions with respect to the order. The corresponding second solution, and some special cases, are also considered.