282 MR. O. N. WATSON: A THEORY OF ASYMPTOTIC SKI! IKS. 



to PoiNCARg's theory by making use of some of BOREL'S ideas, but without making 

 of any of the properties of BOREL'S associated function, $ (t), so fur as the 

 singularities of this function outside its circle of convergence are concerned. After 

 defining the characteristics of a series, we proceed to prove a number of simple 

 theorems concerning the characteristics of series derived in various manners from a 

 series with given characteristics. 



So far the analysis would not appear to have any practical importance. To justify 

 its existence we investigate, in Part II., the circumstances in 'which an analytic 

 function, known to possess an asymptotic expansion with assigned characteristics, is 

 <!!, riniiied uniquely by its asymptotic expansion* It is then possible to determine 

 circumstances in which an asymptotic expansion, with given characteristics, is 

 " .Minimal ilr " by the method of BOREL. 



Finally we investigate the characteristics of the asymptotic expansion of a function 

 derived from the " logarithmic-integral " function, as a simple example. Further 

 examples, namely, of the gamma function and of MITTAG-LEFFLER'S function, E a (x), 

 are contained in another paper by the writer. t 



It should be mentioned here that the theory described in this memoir does not 

 cover the investigation of those functions for which BOREL'S associated function, <f> (/.), 

 has a sequence of singularities at the points t l} t 2 , ..., such that L arg t n = ; such 



II = on 



an associated function would be, e.y., cot ir(t+i), which has singularities at the points 

 t = i+n (n any integer). It seems, however, that none of the ordinary functions 

 of analysis possess associated functions of this nature. 



PART I. THE CHARACTERISTICS OF ASYMPTOTIC SERIES. 



1. The type of function.f f(z), which we shall discuss, is subject to the condition 

 that, when |z| > y, a :Sarg x ^ ft (where a, ft, y are given finite quantities), it can 

 be expressed in the form 



... 



where 



o- n . . . ...... (IB) 



The number n is any integer, and the quantities A, B, /-, /, p, a- are independent of 

 n and \x\. 



'. is known that an asymptotic expansion of POINCARK'S type does not determine an analytic function 

 n^u.ly; thus (!+*)-> and (!+*)-> + <-* have the same asymptotic expansion, viz., 1 -*-'+*-'- .... 

 when |ar| is largo and |argr| < fa. 



To lx) published shortly in the 'Quarterly Journal of Mathematics.' 

 hroughout Part I. of the paper the functions are not restricted so as to 1x5 analytic. 



