m DR . ,. MERCER: STUKM-UOUV,U,E SERIES OF NOIiMA.. 



Consider now the integral 



and this in turn is equal to 



f\ 



^calling that ,r(.) is never zero, it is evident that x (., will be a continuous 

 function of . and t in the square < , - ir, - tS ir, if the fund 



Xi 



is everywhere continuous. The only points at which there can be any doubt as to 

 the continuity of x , (,, /) are those which lie on t = ; and it is not very difficult to 

 see that the function is continuous at these points also. For, when t* 0, we bai 



where, by the mean value theorem, < < 1. Since w' (s) is continuous we see at 

 once that 



lim 



il*. t-*-0 



which proves that X i (*, is a continuous function of and t at points on the line 

 t = 0. It is therefore clear that x (*, is continuous in the square == s ^ ir, 

 * t s v ; and hence, in virtue of the second corollary of II., 9, that (43) converges 

 uniformly to zero in (0, TT), as n increases indefinitely. It follows that 



w (s) 



converges unifonnly to zero in (0, TT). 



It may l>e shown in the same way that each of the other integrals (42) has this 

 property. As we have already seen, this is sufficient to establish that 



converges uniformly to zero in (0, TT). 



