Oscillatiny: Dirichlet's Inteürrals. 



63 



{K+l)d may be made as small as we please. Hence it follows that 



Si?.) < S{?) 

 as A->oo . 



37. Now we consider the integral 



S(?) = f\o{x)e''^- 



/ 



)^in^^^. 



Performing integration by parts, we have 



! a X ./ ax K(T X ) 



Since p < (t', we obtain 



(53) Si?) = 0(1) + C,X?) + i S.i?), 

 wiiere 



( c,i?:) = i?rJ^e-^^^^dx, 



J (T X 



(54) I S,il)=fj.,é''^^ 



dx, 



^ ^' dxXxa') xo' ^ a {a J ' 



Then, in the integral Ci(/), we have 



X < -i-.- <. XG , 



— a 



since a, /> satisfy the first two conditions of (51). Hence Theorem 

 VI may be applied to this integral. Thus we have 



CI?) = i? 



m 



_-^[.^-^^)^V:^ + o(l)], 



-da'id)^[^Äa"id)] 



where <y'{0) + / = 0, ß = ?d -{■ a{d). 



Hence we obtain 



(55) C,i?) 



P{0) 

 äV'-C2^o'\d)} 



[é'-'^^'^W- + o{\)}. 



