18 Art. 4. -M. Kuniyeda : 



Hence, applying Tlieorem III, we obtain: 

 Jf a<-l^ or if a=-l and 0<l, 



j{)) = o{]ßy, 



if «=-1 and 6'>1. or if -l<asO, 



Therefore we have the theorem. 



Theorem IV. The integral 



Jo X 



■tvhere <^<^hl(\lx {h^O) and p-<^, is convergent. If p = x~"S(x), where 

 x''<0<(i/x/^ so that « = 0, the behaviour of C{À)^ as ?.^:c , is deter- 

 mined asymptotically by the following formulae :^ 



(6) 



(7(/) =: 0(1//) {a<-\ or « = -1, 6'<1), 



(-l<ftsOorft= -1, e>\). 



V. Examples^ of the Cases {A) and (B). 



11. As examples of the two cases (A) and (B), we shall give 

 some discussion abont the behaviour of the integral 



J{1) =re''' x'-'flocr —Y'dx 



as /^oo , where 



B(r)>0, E(s)>0. 



At first, we consider the case in which 



* Observe that, here also always t (>,) = o(l) as X-^X . 



fin the followings, I have given examples and verifications, quite similar to those given 

 in Hardy's papers, for the purpose of parallelism. 



