384 JACKSON. 



of Liouville's irregular cases and a large class of others. The con- 

 clusion reached is not at all that suggested by the analogy of the more 

 familiar expansions. It is found that functions of the sort usually 

 developed in uniformly convergent series of characteristic functions 

 give rise here to series which are very rapidly divergent. The explana- 

 tion is that while the characteristic functions are essentially of the 

 nature of real trigonometric functions in the cases usually treated, the 

 corresponding functions here involve real exponential factors by which 

 the order of magnitude of the terms of the series is entirely changed. 



The problem will be treated, first of all, in one of its simplest par- 

 ticular cases,^ which shows clearly the essential difference between the 

 expansions now under consideration and the familiar ones (Section I). 

 Then follows a generalization (Section II) which is more than sufficient 

 to include all of Liouville's irregular boundary conditions. A subse- 

 quent further generalization (Section III) illuminates more clearly 

 the relation between these irregular cases and certain regular ones. 



I. Special Differential Equation of the Third Order. 

 Let us consider the differential equation ^ 



(1) S + ^'^ = ^ 



in the interval ^ .r ^ tt, with the boundary conditions 



(2) u (0) = 0, w' (0) = 0, u (tt) = 0. 



The parameter p is permitted to take on complex as well as real values. 

 If CO denotes the quantity 



2jri 1 1 _ 



5 Cf. Liouville, loc cit. I; also H. Laudien, Dissertation, Breslau, 1914, 

 pp. 67-90. In the latter memoir, with which the present writer became ac- 

 quainted only after reaching the main conclusions now expressed, there is an 

 extended discussion of this special case, and it is stated that the development 

 of a function satisfying the usual conditions is uniformly convergent. Laudien 

 appears to have been misled by a false analogy in thinkmg it allowable, in the 

 demonstration of an auxiliary theorem, to dispense with a careful examina- 

 tion of one of two cases, both of which are essential to the proof of the conver- 

 gence of the expansion of any one arbitrary function. 



6 The discussion in the present section has numerous points of contact with 

 the work of Liouville, loc. cit. I, and Laudien, loc. cit. 



