160 PR. J- MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



a i nl that 



when the boundary condition at x = b is 



eh 



dx 



From these inequalities, together with (49), we obtain theorems of the usual kind 

 relative to the behaviour of the general Sturm-Liouville series (45) at an end point of 

 (a, b). Thus, for example, when F(a+0) exists as a finite number, and the boundary 

 condition satisfied by the functions V, (x) at x = a is not v = 0, it will be found that, 

 if the Sturm-Liouville series converges at x = a, its sum must be F(a + 0). When 

 the lioundary condition satisfied by the functions (x) at x = a is v = 0, the terms 

 of the series all vanish and no discussion of the convergence of the series at this point 

 is necessary. Similar remarks apply at the end point x = b. 



26. In conclusion, it should be observed that the theorem enunciated in 1 8 may 

 be stated in a form applicable to the general Sturm-Liouville series. Let us suppose 

 that the functions (.c) satisfy the pair of boundary conditions B, and, consequently, 

 that the normal functions / (.s) satisfy B'. By employing the transformation of 2 

 (cf. 22), the reader will be able to establish that 



v -X lp> 

 is equal to 



& (53) 



Recalling the relation (48), it follows from the inequalities of the preceding paragraph 

 that, as X tends to GO, (52) converges to SI (x) at each point of the open interval 

 (a, b), that at the end point a it converges to F(a + 0), and that at b it converges to 

 F(b-O); it l>eing assumed in each case that the limit mentioned exists as a finite 

 number. Moreover, when any one of the limits n (x), F( + 0), F(6-0) has an 

 improper value GO, it is plain that (52) diverges to this value and is non-oscillatory 

 at the corresponding point of (, b). Again, if the set of points at which F (x) is 

 continuous includes an interval (a,, 6,) lying within (a, 6), the set of points at which 

 w (*)/(*) is continuous includes the corresponding interval of (0, tr). It follows at 

 once from the theorem of 16, that (53) converges uniformly to /(*) in the latter 

 interval ; and therefore that (52) converges uniformly to F (x) in ( 1; &,). 



We have thus established the theorem : 



Let F (x) be any function which has a Lebesgue integral in the interval (a, b). Let 

 % ( r ) ^ (*'). . . (x), ...be the solutions of 



d 



