Oscillating. Diriehlet's Integrals. 9' 



include any point at which the subject of integration possesses any 

 irrelevant discontinuity or other singvilarity. We distinguish 

 as in Hardy's papers, the following three cases: 



(A) a < l(l/x), 



(B) <r^l(llx)^ 



(C) CT > l(}lx). 



II. Lemma for Case {Ä). 



6- The proofs of the theorems A and B of ''0. D. I. 1." are 

 principally carried out by means of H-lemma* 29. By examining 

 the proof of this lemma, we can easily extend the range of validity 

 of the formula given there. 

 In fact, the integrals 



./ \ U J 



there considered, are absolutely convergent also in the case 

 — I<œs0. Hence the argument of " O. D. I. 1." for the case 

 0<a<l of this lemma holds also in the case — l<a^O. Thus we 

 easily obtain the following modification of this lemma. 



Lemma 1. Let 



(13) J{X) = 1'^ a-"-'" ^P{x) (^'^^ j W, 



where a^\ and 



0(,r) 1=0 («;>'■''<*>, 

 ;r*<:6'<(l/a;)*, ip{x)<l{\lx\ 



* The work of Mr. Hardy is chiefly included in the proofs of a great number of lemmas. 

 Naturally, in my paper, these lemmas will be used freely, being referred as " H-lemma 1 ", 



" H-lemma 2 ", , in order to distinguish them from new lemmas which will be established 



here. 



