Oscillating Dirichlet's Integrals. ß5 



Now S{?.) cannot tend to zero as >^->x. For, if S(À)<^1, then, by 

 the relation (52),'^ we have 



S,(^) < 1, 



and, by (53), CiU) + 0(1) -< 1, 



contradictory to the above result Ci{?.) > 1. Thus S(À) does not 

 tend to zero as ?>->x>. Hence, by Lemma 7, we have 



(58) SiÀ) < >S(/l) 



as iî->x. Hence, h'om (53) and (55), we obtain 



which is nothing but the formula (7) in our case. Thus we may 

 state, by combining this result with theorem VI, 



Theorem VII. The integral 



Jo X 



where i{l/x) < o < {VxY and p <, a', is convergent. The behaviour of 

 S{^), as ^->30, is determined asymptotically as follows : 

 If x'< p < x^a"lo'^ then 



if x^o"la' < p < a', then 



(7) SU) r^ ^ é^''^"^' ^t: 



where ß z= Xd + a{d), 



and d is determined as a function of X by the equation 



o'{d) + ; = 0. 



* Here S'](),) is replaced for S{\). 



