Oscillating^ DirichlctV Iiite«,'rals. 33 



d,f^ cd 

 US /->x . 



Proof. Evidently yfiu) and (jfi[y) are ultimately monotonie 

 and tei:id to infinity as .?/->x). Hence each of the equations 



yf{y) = À yf:(y) = c/ 



has, for sufficiently large values of ?•, one and only one root which 

 tends to infinity as À->:c . 



By hypothesis, we have 



Of {6)=^)., dj\{d,) = cX; 

 and, since f{y) r^ f(y), we have 



f((fô=fi(fôO+'\ 



where 6->0 as Oi^-jo or À->-x: . Hence we obtain 



Let V be a function of / sncli that 



Then we have 



(33) c.f{e) = rjf{d,){\+t), 



where e^O as I-^od . 

 We have to prove that 



Tj (^ C 



as )-^ 00 . 



Evidently ^ is positive and continuous for all sufficiently 

 large values of I, and it might tend to infinity or zero, or might 

 oscillate finitely or infinitely as X^x) . 



If we suppose that iy>l or f) oscillates in an infinite range of 

 values, then, corresponding to any prescribed positive number P, 

 however great, there will exist a sequence {E) of values of X tend- 

 ing to infinity, namely, 



