158 DR. 



Hetice 



J. MEKCEK: STURM-LIOUVILLE SEKIES OF NORMAL 



Since it can be proved in the same way that 



we see that 



_a 



=*- 1* K A (s, t) w (t)f(t) dt EEE 1 | V" [/(-< 



W(*)J. 2 -' 



that is to say 



We have hitherto supposed that s is not one of the end points of (0, TT). When 

 this is so, the reader will be able to prove in a similar manner from the formulae of 

 15, 19, that the result stated still holds. It follows that, for all values of x 

 in (a, 6), 



T 



;^lim X K A (s, t)f(t) dt~z A (x). (49) 

 Jo 



-.- . --- 



24. Let us again suppose that A is a point of the open interval (a, b). We 

 propose to show that the upper and lower bilateral limits of F (x) at x are the same 

 as the upper and lower bilateral limits of f(s) at the corresponding point s. For, 

 since 



F<*>-/K(?H. l 



it follows that, if Xi(0> Xz(^) are the functions denned in 11, 12, we have 



where y = ^~ l k l ' J g~ 1 ' J t, and the functions 3, (y), 3 9 (y) are defined by 



fx //A 13 f*+*i(y) //-A' 1 .* 



Xl (0 = f f) fa, X a(0 = d (f) dx ..... (51) 



Jl-i(y)\K/ Ji \A,/ 



Since the functions y and k are always positive and possess continuous derivatives 

 in (a, 6), it is evident that these relations define 5, (y) and 3 a (y) as functions of y 

 possessing limited second derivatives in a certain interval (0, ) (4 > 0). Further, we 

 have 



a. (o) = , (o) = o, y, (o) = y a (o) = i , ' 



