Ml;. G. N. WA'IMiN: A THKORY OF ASYMPTOTIC SKIMI.s 283 



The novelty introduced in these assumptions lies in the fact that we consider a, 

 nnd II. (or, more precisely, the moduli of these quantities) as functions of for all 

 values of n, no matter how large. 



It will be convenient to give names to the quantities A, B, k, I, p, cr; we shall call 

 k, I, p, a- characteristics of the series (1) ; k will be called a grade, I an outer grade, 

 p a radius, cr an outer radius of the expansion. The quantities A and B will be 

 called constants of the expansion. 



If an inequality of the form (!A) can exist when k ^ kg, but not when k < k , we 

 might, for greater precision, call / the princijxd <//</,/, of the series ; and the smallest 

 possible value of p associated with / might, in like manner, be called the principal 

 i-nili'ix; similarly we could define the i>ri:i,-ijial outer i/rn<fi- and the principal outer 

 radius. But it is found that this additional precision is unnecessary, and we will 

 accordingly prove all our propositions for any possible characteristics ; and, further, 

 it may be stated that the characteristics, determined for particular series (such as 

 the asymptotic expansion of the gamma function) by the ordinary processes of 

 analysis, are of such a magnitude that, although they may not be principal 

 characteristics, we could not expect to obtain any additional knowledge of the 

 functions investigated by knowing the actual values of the principal characteristics. 



Examples :- 

 (i) Let 



log o = kn log n + tn (1 + ), 



whore | | - as n -* oo^and k is real and positive. 



The principal yrade is I: 



The prinfipal radius is (by the asymptotic expansion of the gamma function) |exp {r - k(\ogk - 1)} |, 

 provided that the upper limit of R (rne n ), as n -* oo, is not + oo. 



If the upper limit of R(rn,,) is + <x, the quantity jexp {c - k (log k - 1)} | +S is a possible radius, where 

 5 is an arbitrary positive quantity, but not zero. 



(ii) Let 



log a n = kn log n + en { 7(log ) + <}, 



where k and c are real and positive. 



The quantity k + 5 t is a possible grade ; and Sj is a possible radius ; Si and &.. being arbitrary positive 

 quantities, but not zero. 



(iii) The reader may prove that, if f(x) be defined by (1), then ) {/(f) -a^-aix' 1 } Jx has the same 



rli.-iracteristics as f(x) when k ^ 1 and / i 1, where the path of integration is the straight line which, when 

 produced Iwckwards, passes through the origin. 



Theorem I. If k, I, p, tr are possible characteristics of an expansion, then, if we 

 rc;/ard I and cr as bcimj i/ii'fii, tve may always assume that the quantities k arul p arc 



such that 



(i) k == I ; (ii) if k = I, p ^ a: 



For from equation (1) we have 



+, = (U.-R. +I )x" +1 , 

 2 o 2 



