Oscillating Dirichlet's Integrals. \^ 



(i) Let a<-l. Then, by (20), 



J{X) = ^{o(l)-ioi\)} =o(lß), 



and hence, by (19), 



GW = 0(1/;). 



(ii) Let a= -1. Then 

 Hence, if xe'<l and pa'< 1, 



and CiX) = 0{\ß); 



if a;0' or pff'yl, 



G(X) ^ /(;) po -(^7:ß){ilß)e\lß) + ip{lß)aXlß)]e''(^^K 

 (iii) Let — l<a<0. Since 8<d, we have 



/(;) CO ^ /(!)(;), 



and hence 



C(k) ^ r(-a) COS (1»;:) ^(1/>1) e'-t^^*) . 



(iv) Let a=0. Since 0<é?, by H-lemma 10, we hav^e 

 T(lß)<Tilß), 

 and J^'-W < J^'\^)- 



Hence C(/) «-* T(l//l). 



We can now state 



Theorem II. The integral 



C{X) = r p{x)e''^''^ -^^^ dx, 



Ja X 



