Oscillating Diriohlet's Integrals. 77 



JW) =3^^°' [2,9-i:]+2.);:]: 



where I ^^1 < —5- • 



Hence, when ß is large, -J-''(2/3) + J"-'(2;5) and J\%-i) + J-\%^) 

 behave respectively like 



2 sin |v- • /o Q 1 \ 



and — 77:^^^ cos (2^9-1-). 



On the otlier liand 



tends e.xpo}icnthdlii to zero as /9->3o i.e., as ;.-^co, so that this 

 term is negligible in comparing witli the remaining terms. Hence 

 we obtain the results: — 

 The integral 



is convergent if — l<v<l, and, as l^y: , we have 



A{X) = - ^;r4,,2,^4/î-i-'-i fsin (0,^,èiè_i-) + o(l)]. 



The integral 



is convergent if — 1 < v < 2, and, as /^oo, we have 

 43. Now put V = — a and write 



