20 -'^i'*- 4.— M. Kimiyeda : 



log— j , ^ = mloglog(l/a;). 



Applying Theorem II, we obtain 



I^'{À)^ r(-a) COS (ki tt) /"(log /)-^e"^i ^o- log x (~l<a<0). 



If we put /(^) = x--^(log|)""^-^«"^^^^(n 

 then, by performing integration by parts, we have 

 7," (/) = 0{\rÄ)~^l^f\x) sin Xx clx. 



Evidently /'(a^) has no singularity and is absolutely integrable in 

 the interval {^, ¥). Hence by a well known theorem^ 



I f (x) sin Ix dx = (1) 



as X-^oj . Hence we have 



//'(/) = 0(1//). 



Since — l<:a<:0, we have Z/>7i" as >^^x . 

 Thus we obtain 



2i(;)= r{-a) cos {\an)X''{\og /)-ie"iilt'^lûg;(l + s^) 

 = r{r) cos {\rn) A'^ (log Xy-\l + ^,), 



where lim£^=0. 



The integral I-i{^) may also be divided into the two parts 



=j/(;0+i./'(/) 



say. As in the case of -fî'(^), we easily see that 



J/(^) = 0(1/;). 



* Hobson, Theory of Functions of a Real Variable, p. 672. 



