2 Art. 4. — M. Kuniyeda : 



convergent for |^|<1. and representing a function /(^) which has, 

 on the circle of convergence, one singular point only, at z=-l. In 

 the investigation of this problem, we are often led to consider 

 integrals of the types 



./ X ./ X 



dx. 



2- In Part I of this paper, I consider the integral? 



(1) S(X) = I ' n{x)e''^'^ ^^^^dx 



./o X 



(2) CJ) = /" ' p{x)e'''^^^ ^^^^dx 



./ X 



::e>o), 



where, f and <J denote L-functions and <t>1* as ä-^0, ^ being a 

 positive number chosen sufficiently small so as to ensure that 

 p and <J are monotonie and continuous in the interval 0<a;^6. 

 These integrals (1) and (2) will be called ''the s'me- integral and 

 " the cosine- integraV respectively. 



Hardy, following l)u Bois-Reymond, distinguishes the 

 following three cases : 



(Ä) ^{x) < l{\lx), 



(B) a{x)^l(\lx), 



iO o{x) > l{\lx). 



The results arrived at concerning the sine-integral are designated 

 as theorems A, B, C in his papers. 



As will be seen from these results, Hardy has principally 

 considered the cases in which the sine-integral oscillates as X^oo ; 

 but he did not went into a minute discussion of the cases in which 

 the integral tends to zero. It will be interesting to find asymptotic 

 formulae ior S{X) in the latter cases; and it appears quite 

 natural that the formulae obtained by him are also available to a 

 certain extent in such cases. I have succeeded in extending the 

 range of validity of his formulae considerably — roughly speaking, 

 to all cases in which the order of S{X) is greater than -r-. 



* Throughout this paper, I will entirely adopt the symbols and notations defined in 

 Section II of " 0. D. I. i." 



