Oscillating Dirichlet's Integrals. ^4.0 



If P<xy/a"lo', then 



(7(;o = o(i//), s(;o = o(i/>(); 



if x^a"la' •< ^> < X or if p = Ax{\ + ''p{x)] , where A>0, p ^0 and 

 p <1, then the formulae (45) hold, i.e., 



C{?) 



S{X) 



p[d) 



d^['la"{d)] 



f>m 



,yH')i^^ 



e<^-i''^V", 



(A-^od). 



29- It now remains to compare the order of magnitude of 



f(«) = - ''("^ 



a{k-a'{a)\ 



4.ß) = 



P{0) 



d^\:2a"{d)] 



as -^-^00, when x<p<X's/a" or p = Ax[l—p{x)], where ^>0, p>0 

 and p < 1. 



When x<p< X'y/o", we write, as before, 



(46) p(x) = x-" d{x), 



where a^ — 1 and x" <d<{\lx)^ as x-^0. We observe that0>l, 

 if a = — 1. 



Under the supposition that 1{\Ix)<cf<^(\Ix)\ we can write 



(47) a'{:x) = - x-'-'e,(x), 



where 6^0 and x''< 6,< (1/a-)* as x^O. We observe that 0,>l, if 

 = 0, since a' >■ 1/x. 



From the condition pKX'y/o", we obtain 



a<^h, 



or a = ^h, e<et 



We have to separate the discussion into several cases as follows. 



30. (i) Let a>-l. 



At first we consider the case in which 



>0 or & = 0, -l<a<0. 



