Lars Onsager's Thesis

In one of Onsager's favorite mathematics books, Whittaker and Watson's Modern Analysis¹, the chapter on Mathieu functions starts with the following:

"The preceding five chapters have been occupied with the discussion of functions which belong to what may be generally described as the hypergeometric type, and many simple properties of these functions are now well known.

In the present chapter we enter upon a region of Analysis which lies beyond this, and which is, as yet, only very imperfectly understood."

This appered in the 1915 edition, and is still true.

Onsager's idea of connecting solutions of the two equations obtained from separating variables in the equation (18) has not been studied systematically. Now that much more is known about the group-theoretical nature of certain partial differential equations, this is worth studying…