Oscillating Dirichlet's Integrals 57 



1 



where /^i = ?^ ^^^ , and y^ h are certain function î< satisfy ing the condition 



?/>f(y)>(Vuy . i 



as y-^ao . ' 



Hence we have ! 



as ?.i^cc , since 6 > 0. Thus we obtain 



as À->oo . 



34- We can now state 



Theorem VI. The integrak 



SU) = r p(x) è"^'^ ^^^^ dx, 

 Jo œ 



Ca) = f ' o(x) e''^"> _S2?^ dx, 



Jo X 



ivhere l(lfx)<(T <(i/xy' and p<x(j', are convergent. The behaviour 

 these integrals, as A->oo , is determined asymptotically as follows. 

 If x^<p< x^a"la', then 



S{k) = 0(}ß), C{}) = 0{\ll); 



if X's/a"la' < /7 < xo' , then 



where ß = }.d + a [6), 



and d is determined as a function of X by the equation 



a'{d) + X =0. 



