Oscillating^ Dirichlet's Integrals. ']JJ 



Theorem III. The Inteyral 



where <y^hl{\Jx) ib=hO) and /><l/a^, is convercjent. If (>=x-''0{x)^ 

 where x^<d<(l/x)\ so fhaf rt.^l^ tlie behaviour of S(À)^ as ;^x, is 

 determined asymptoticallii by the followimj formulae : 



(5) 



I S{))^ - l\-a-U)%\\\{\Xa^-h\)-] o{\l})e''^^i'^ {-\<.a^V). 



10. We now consider the cosine-integral 

 C(;0 = /*V(;r)e'-'f-> ^^^^ dx. 



Jo X 



Since the function ^ has the form (21), tlie conchtion for the con- 

 vergence of this integral G{}^) is 



as a? ^ 0. 



As in Case (A), we have 



c(/) = o(i//) + J(;,\ 



where 



/(/) = 4- ['{-R-i B,)e'' -'-^-^ dx, 



/ ./ X 



B,=-^-p', B. = pa'. 

 X ' ' 



If we write, as before, 



p = x'^d (x), 

 wh ere a^O, x'<0<(\ jxY,* 



we have 



B,-iB.,r^ (a+l + bi)x-("-^'^e(x). 



* Observe that, when a=0, x^ -< © ^ 1. 



