(17) 



Oscillating Dirichlets Integrals. H 



and hence 



(16) J(k)==J^'^k)-iJ^'\X), 



Avhere 



Now Ave can Avrite 



Avbere a^l and x'<e ^ {l/xY as x^O. Then, if a=^-l, 



IEi = /)-x/?' f-^ (1 + a) ic-^ö (a;) [by H-lemma 4], 



Avhere ^ is a function of the same type as ß and 6-<^d as ic->0*; 

 if a.= -l, 



B, = -x-S' = -xS,, B, = x'e, 



Avhere 6i=x6' is a function of the same type as 6 and 8i<6. 

 Hence Ave have: 



(i) Let a^ — 1 Then, applying Lemma ] to the integrals 

 (17), Ave obtain 



hence, by (15) and (16), 



S (A) = Oi^-'^'). 

 (ii) Let — l<a<0. Then, by Lemma 1, Ave obtain 



* Since 1'^(t-<Z(1/x), we easily see that «r is a function of the same type as 0. It can also 

 easily be proved that the function .t©' is a function of the same type as and a product of two 

 functions of this type is also of the same type. Hence it follows that 0(.r) = .TiT'0 is a fvinction 

 of the type ; and since xa'-^l, we have 0<0. 



