16 



Art. 4.— M. Kimiyeda : 



where \<o<lQ.IX) and p<xa\ is convergent. If p"X'''d{x)^ ivhere 

 x'<6-<,{\lff^ so that asO, the behaviour of C{X), asX-^<x>, is deter- 

 mined asymptotically by the follounng formulae ;* 



C{X) = Oilß) 



(rt< -] or ft = 1, xS'<\, pa^l), 



(4) [ 



C{X) ^ ^{^7:ß)[{\IX)e'{\lX)^ip{\l)^a'{\ß)}e''^':'^ 



(«= — 1, a;0'or/><T'>l), 

 C{X) r^ r(-a) COS (^aTt) p(\ß)e''^''"^ (-l<a<0), 

 [ G{X)e^T(\ß) ia = 0), 



ivhere 



T{x)^ rp{t)e''^'^ 



t 



IV. Discussion of Case (B) : o (a) ^ lO-ß). 

 9. In this case, we can write 



(21) a{x) = bl(\lx) + ^yx), 



where b=hO and a<l{l/x). Then 



Hence, as Hardy remarks in the paper "0. i). /. i.", the 



treatment of tlie sine-integral Sß) in Case (B) may be done by 

 precisely the same method as in Case (A), by applying Lemma 1. 

 Thus we can easily see : 



If a^-1, 



if — <«<0, or if a=0 and /Kl, 



S{À)r^ -r(-a-bi)sm [^{a + bi)7:}p{\ß)e"^'^'\ 



Combining these results with Theorem B, Ave obtain 



* Observe that here always C(),) = o(l) as X->-00 



