

FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 161 



which, for suitable values of r satisfy tin' j>air of boundary conditions 



^ -/ = at x = a, f ^ + Hv = at x = b. 



</.- ax 



Moreover, let these solutions be made definite by imposing on them the condition* 



and let the arrangement of them be such that the corresponding values of r increase 

 with n. Then, as X tends to o, 



2- -^(^9(y)^(y)F(y)dy ...... (52) 



= i (ri J ; A. Jo 



converges to the bilateral limit of(x), at each point of the open interval (a, b) where 

 this limit exists as a finite number; further, at a point where the bilateral limit has 

 one of the improper values <x>,it diverges to this value and is non-oscillatory If 

 the set of points at which F (x) is continuous include a closed interval lying tvithin 

 (a, b), then (52) converges uniformly to F (x) in this interval. At the end point 



JJ (52) converges to pfciJv when this limit ex "'* "* ^"^ number; f urther > when 

 either of these limits has one of the improper values , (52) diverges to this value 

 and is non-oscillatory at the corresponding end point. 

 From this we deduce the corollary that, when 



y-- 



exists as a finite number at a point of the open interval (a, 6), (52) converges to this 



number. 



After what was said in 25 there will be no difficulty in perceiving how the results 

 just stated must be modified when the pair of boundary conditions satisfied by the 

 functions (*) is B, B, or B. 



IV. _ THE CONVERGENCE OF STURM-LIOUVILLE SERIES. 



1. In this section it is proposed to investigate the convergence of Sturm-Liouville 

 series. It will be recalled that with the notation of III., 2, 3, V. () is a solution of 



u = 0, . ........'. (1) 



which, for /* = X,, satisfies the pair of boundary conditions B', i.e., 



*? -h'u = at s = 0, and f -^ +H' = at s = w. . (2), (3) 



ds 



VOL. rex I. A, V 



