Oscillating Dirichlet's Intt^urrals. 



67 



Sj)^) ^ f - sin Ix 



J 



^>,e' - 



dx. 



Now 



^' xa' ^ a' {a'f ' 



fK.^-(a-'2b)^ (a > 26), 



X(T 



h 



xa' 



{a = 26), 



and 



If «-—26 < 6, then 



(a-..)_^ 



-^^(a-l).- ^. 



,0, -• -^ < xa\ 

 ■ — a-ff 



and, by another apphcation of Theorem VI, we obtain 



since S,(?^) < C,iÀ) < Ci(/Î) as ?.-^-c . 



If a— 26 >6, then repeat a similar process. 



Since 6 > and b ^ a ^ l+b by (57), there exists a positive 

 integer n such tliat 



(?^ — 1) b ^ a < nb. 



Hence, after repeating n times the above process, we are led to 

 the equation 



wdiere 



Jo X 



/'. < ^^ ^ -ia-b) (a -26) •• -(a -71-16) a'-^"-"''^^ 

 ^ — xa u{' 



