Oscillating- Dirichlet's Integrals. ß9 



First, let 6 '-^ Aß,. 



Then = Ad/^l + elx)}, 



rv 



where e(a-) is an L-function such that e(.j;) < 1 as .r^O; and we 

 have 



sm?..x 7 , A f^.n ia sin^a? 



S(^) = Af d^é'^^^^^dx + Af eR, 



/ X / 



e^'^^^^^^dx 



X 



say. Then, performing integration ])y parts, we have 

 /,(/.) = 0(1) -U r é' cos Ix dx 



J 



^ ^^^)-'^ Vi2^"(^)] '^^''''^''^^ + ^^^)] [by Theorem VI], 



In the integral LC/'O» we have 

 Hence, by Theorem VI, 



or I^f.^) = 0(1/;,) ; 



and we have 



UX) < u?). 



Therefore 



since ^ i-' ^ 0i. 



Thns, in the case when a = and 6 r-^ Ad^, the truth of the 

 formula (7) is proved. 



Next let 8 > ßi. 



