I.,,, |.|; .1 MKKCKK: STM.'M 1.IOUVILLK SK1MKS OF NORMAL 



Let f(s) be any fu**ie* /,/,-/, I,,,* a f^l^gne integral in the interval (0, TT). Let 

 *i(). MA "' *(*) - tc ' Ae <-"i/>><>>' *y*te** f normal functions which, for 

 wt,,l.l. <>f /*, satisfy 'he <////o'< //"/ equation 



' th 



<"' If " " 1 IT'i; 



_ li'u = at A = 0, -T- +n it v at * 



; !</ tin- ,-,-< t H</<'inent of these functions be such that the corresponding values 

 iiK-rcase with n. Then, as \ tends to oo, 



converges to the bilateral limit of f(s) at each point of the open interval (0, ir) where 

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

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

 tin- *ft "/' /mints at which f(s) is continuous includes a closed interval lying within 



(0, IT), then (40) converges uniformly to f(s) in this interval. At the end point , 



(40) converges to - \ Jt, when this limit exists as a finite number ; further, when 



either of these limits has one of the improper values oo, (40) diverges to this value, 

 and is non-oscillatwy at the corresjjonding end point. 



As a corollary to this theorem it should be observed that, when 



t--0 



e.mte as a finite mtmber at a point of the open interval (0, TT), (40) converges to this 

 number (vide 13). 



The reader will recall that the system of normal functions ^(s), ^ 3 (s), ..., i|/ B (s), ... 

 is unaltered when (2) is replaced by 



(vide 3). There is therefore no necessity to suppose that the values of /x referred to 

 in the enunciation of the above theorem are all positive. 



19. It will be convenient to state here how far the preceding results remain true 

 when the pair of boundary conditions for (a, b) is one of the three B, 6 B, *B, and 

 hence that for (0, ) is one of the three B', 'B', ;B'. In each case the asymptotic 



