Oscillatini;' Dirichlt't's Integrals. 95 



()<7< 1, (l + (i)^<«<(3-g)-^, (unl p<l+q, fhcii 



(.S2) a,, = 0(1///.) . 



56- Wo have thus obtained asymptotic formulae for «,„ as 

 n^y^. in the three cases^ 



( i ) g = 1. 





(ii) 0<g<l, « = (H-g)|„, 



(iii) 0<g<:l, « = (3-/7)^, 



always with the condition 



wliich is introduced from the conditions that the integral Il'n) is 

 convergent and Z(/?) > 1///, as //.->oo . But, by proceeding as follows, 

 it will be seen tliat this restriction al»out the value of p niay 

 be removed. 



Now, in a certain region near the point z — 1 and interior to 

 the circle of convergence, we may put 



— ^ — .m-^f ^2a,^2-, 



and we may differentiate this equation with respect to z, since our 

 series ^cinz" is uniformly convergent in the said region. Thus 

 we ol;)tain 



Observing that a„ is a function of ii and 2^, Ave write 



«„ = a{)i, p). 

 Then we liave 



p ^ a[n, p + l)z'^ + qA ^ a(n, p + q + 1) ,~^ = };] (//• +l)a {n +l,p) z'\ 



n = « = )i = 



* Our method fails to determine the asymptotic formulae in other cases. 



