JOURNAL OF THE COLLEGE OF SCIENCE, TOKYO IMPERIAL UNIVERSITY. 



VOL. XLI., AST. 4. 



Asymptotic Formulae for oscillating Dirichlefs 

 Integrals and Coefficients of Power Series* 



By 



Moloji KUNIYEDA, Eigalcashi. 



Introduction. 



1- G. H. Hardy, in his interesting papers f with the title 

 "Oscillating Dirichlefs Integrals", has discussed almost com- 

 pletely the oscillating nature of an integral of the form 



f^ Ax)^^dx (e>0), 



when X tends to infinity. Hereby /(a^) is of the form 



where p{x) and <^(aj) are logarithmico-exponential functions 

 (or L-functions) and (y{x) tends to infinity as a;->0. 



As he remarks, the problem is equivalent to that of investi- 

 gating the convergence or divergence of the Fourier's series 

 defined by a function which has a single oscillating discontinuity 

 of the type specified by 



, \ cos / ^ 

 ü[x) • <t(x). 

 ^^ ^ sin ^ ^ 



It is also closely related to that of determining asymptotic 

 formulae for the coefficients a„ of a power series 



00 



n=o 



* This paper was worked out at the suggestion of Mr. Hardy, to whom I wish to express 

 my sincere thanks for valuable advices. 



t Quarterly Journal, Vol. XLIV, pp. 1 — éO and 2 J 2— 263. We shall refer to these papers as 

 " 0. D. I. i." and " 0. D. I. 2." respectively. 



