10 ■^''^- ^- — M. Kuniyeda : 



as x^O. Then, if am—l^ 



J(X) = (^^+0 ; 

 if — l<a<], 



(14) /(/I) «o r(-l-a-bi) sin {^[a+bi)7r]k''+'"0(lß), 



except when a=b=0, in which case the right-hand side of (14) is to be 

 replaced by è7r(Z'(l/>l) ; if a=\, b=f=0, this result (14) still holds, provid- 

 ed that the integral 



./o 



dt 



is convergent ; and if a = ^, è=0^ and T{x) is still convergent, 



III. Discussion of Case (J) : ^(x) < l{Vx). 

 7. At first we shall consider the sine-integral 



S(/l) =/ %(;c)e '■'<*> -?^5_^ dx. 



./o X 



As is given in the paper " 0. D. I. 1.", the necessary and suffici- 

 ent condition for the convergence of this integral is 



p{x) < a'{x) 



as x^Q. As in the same paper, hy perfornn'ng integration by 

 parts, we have 



(15) = 0(l//)4-J(4 



Avliere 



jK, =p — X{>', 



B., = x fja' ; 



