Oscillatini; Dirichlet's Integrals. 93 



53. (iv) The case in which 0<q<l, (1 + g)^ < « <(3— g)-;^, 

 p<l + q. In this case 



^- + (/- < « + hj- < |~- 



Hence cos (a + h]-) < 0, 



and therefore, by (70'), we see that <f{d) tends exponentially to 

 zero as 6-^0, so that, V)y proceeding as in the case (ii), we easily 

 obtain 



(78^ J(n) = 0{l/n) . 



Thus we have established the following results, 

 (i) If q = \, a = -, j9<'2, then. 



i (_^<^<:^); 



(ii) if < g < 1 , a = (1 + g) -^, p<\^q, flwR 



J[7i) = 0{\/n) ; 

 (iii) //'0<;g<:l, a = {3-q) ^, p<:l + q, then 



Ä- =:(i+g)g''+*«^-^^ (-k<i'<i+3); 



(iv) ;fO<q<l, ;T + g -^<«<v'^-'/)-^' V<^ + (b then 



J(n) = 0(1/») . 



54- Integral J(n). The discussion is quite similar as in the 

 case of J"(?i). 

 We have 



fi^f) 





