78 



Art. 4. -M. Kuniyeila : 



say. Then, by writing /(.r) = rc"^"" cos ^- - ), we hav 



/,(/) 



f(ç) sin ?.ç 1 



x/, /'( 



a;) sin >^.a^ dx 



and /'W = -(1 +a.) :r-^'+'-^> cos (-^' j + ,n.T-<^+''> sin Ç^^ 



wlience 



/ /'C^) sin /ia? <^a- 

 ./ e 





since a>—l. Therefore we obtain 



/,(>() = 0(1//), 

 and evidently 



J-i(/) < />- 

 whence it follows that 



Similarly 



B(/) ^ Z, /) 



Therefore we obtain 



Ii(/) ^ -i-^?Ar4'-^;>-^cos(2;/i*/i + J;r) 



Z.,(/) '-^ J-i7/i-i-Ui<'-^sin(27;i^/i + i-) 

 C(/î) -^ ^zi,ii-^"-i/}'^-i exp [(2wi*;.è + i-)/] 



(-l<a<l), 



(-l<a<l} 



V = — a \ 

 -l<a<l/ 



(-1 <«.<!), 

 (-1 <»<!), 

 (-l<a<l), 



which agree with the results obtained from our general Theorem 

 VI, only the difference being that the lower limit of (t, is — 1 

 instead of — y, tliis limitation being introduced from the condition 

 for convergence of the integral y4(A). 



As Hardy gives in "0. D. I. .'?.", from the values of the 

 integrals 



/•=° • r ß' \ du ("^ . . ( ,9- \ du 



/ sni u cos ( -^^ —^-^ , / sm ii sin -'— - ^^ , 



we can infer that 



