Oscillating Dii-ichlet's Inte<;-rals. 27 



X-a'{x) 



since «7'<0, (7'>1, Hence the first part of the lemma follows im- 

 mediately. 



Now consider the second case (ii) of the lemma, in which 



p = Ax {l—i'ix)}, 



where A>0, p>0, p<l. 



Then we have 



, = 41^, 



À— o^ 



and -^ = gives 



(25) ;. = ^"(^-py +^r 



Let us Avrite p = a'y, 



so that r<0, <y'r<\. 



Then ^lQ:p^ + ,,^^l±SX^, 



P P 



and from the relation ^V-<1> we obtain ï< — ^and, by differentia- 

 tion, 



a'-r'<o". 



Hence (25) becomes 



(26) / = 



, _ a" + o-r' ^ a" 



Here we have 



since oy-l{\Jx); and, since ^ö<1, we have xp<^x and, by differentia- 

 tion, 



xp' + ~p < 1. 



