Oscillatino- Dirichlot's Integrals. g.. 



The integrals SX?^) and C{1) are convergent when K^M <<y< (l/^Y 

 and pKxo'. The behaviour of the^e integrals, as X^co , is determine^, 

 asymptotically as follows : 



where ß = '^'d + (T{d) 



and 6 is determined as a function of )~ by the equation 



In the course of tlie proof of this theorem, we are led to the 

 comparison of tlie order of magnitude of the functions 



as /î->oo , when x<p<x^o" or p = Ax[l—p(x)], where A>0,p>0 and 

 ;ö<l, a and d being functions of >l determined respectively by the 

 equations 



da La[^~a'{a)]J ^ ^ 



It will be proved that 



as i^^-^Qo . The proof of this relation plays an important rôle in 

 the discussion of Case (C). 



The integral /S{À) is still convergent when xa'<p<a'. Hardy 

 did not went into the discussion of this case, his method ceasing 

 to be applicable in this case. I have succeeded in proving that 

 formula (7) holds also in this case generally, the proof being 

 left incomplete only in a few special cases. 



