Oscillating Dirichlet's Integrals. Q\ 



the coefficient of sin ax m the subject of integration is absolutely 

 integrable in the range of integration $'^x^^. Hence, by a 

 well-known theorem,^ we have 



J^--')(/) = 0(1) 

 as ?^->:c . 



In the integral 



J<i)(;.) = f Kx)f{x) sin ?.x dx, 



t{x) is monotonie and, being an L-function, has a differential 

 coefficient Avith a constant sign in the range of integration. 

 Hence, by the Second Mean Value Theorem, we have 



J(i) {X) = e(,-') /' ''f{x) sin Xx dx (0 < ^^ < O 

 = 6(ç') i —J )f{x) sin ?.x dx 



say. Then 



./ ./ f 



and / f(x) sin ).x dx = o(l) 



as /'.-^oo, by the same reason as in the case of J"^-^(>^). Hence 



y(;0 = i(/i) + 0(1). 



As to the integral 



;'(A) = /' V(^) sin /a- ^a; (0 < e, < ç' < ç), 



./ 



we observe that the upper limit ç^ of integration is a function of 

 ^, and it may be inferred that 



|/(/l)i<X|J(/)| 



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



