Art. 4.— M. Kuniyeda : 



'C{X) = 0(1//) (a<-l or a= -l,xe'< 1, p^r' < 1), 



(4)^ (a= -I, x6' ov p(7' > 1), 



C(/l) r-^ / (-a) cos (ia;r) p(lß)e"^'''^ (-1 <a<0), 



dt 



In 'Case {B), 



S{?^) r^ -r{-a-hi) sill [l{a + hi)7i] ^(l/;)e''WA) 



(-l<a<0 01-^=0, (J < 1). 



Combining this result with Theorem B, we obtain the theorem: 

 If o^ hlÇilx) {b^O), /><i-, a>*tZ 



p = x-''0(x), x*<0<(}/xy, a£\, 



then, as ^-^oD , we have 



{S{l) = 0{)r'^') {a£-l\ 



^ ^ \S{X) <-> -l\-a-hi)^in {i(a + 6*>} /<!//) e^''^^^^^ {-\<a£l). 



The corresponding formulae for C{X) are 



■(7(/) = 0(1//^) (a<-lora = -1, 6'<1), 



(6) 



C{_X) ^ ri-a-bi) cos {^{a+bi)7r] p{\ß)e'^y^'> 



(-l<aéOor a=-l, 6>>1). 



In Case (C), Hardy gave formulae which were shewn to be 

 valid when x^/a"< p < xa' . I have succeeded in proving that 

 they are valid for 



x^a"\a'<^ p < xa' , 



thus extending the range of validity of the formulae considerably. 

 It will be seen that this lower limit of P (namely p^x^'y"l<^') cor- 

 responds to our natural limit y of the order of tlie integrals S{^) 



and C{X). 



Combining these results with Theorem (7, we obtain the 

 theorem : 



