MK. G. N. WATSON: A TIIKORY OF ASYMPTOTIC SKUIKS. 295 



and 



S. = R. + (-)"a,Y 1 + (-)'" 1 3 Y 3 + ... - 

 so that 



|&.|=KI +.-iC,|o| |o..,| +.-,C,|a| | a,., | + ... 

 ^ 



Now 



T{k(n-r)+l} s 

 where 



V 1 = '8856 ... if k < 1, T?, = 1 if * 1. 

 Consequently 



IM^A^r^+iKp+H}"- ....... (IOA) 



Also 



^Br(fa+l)a V A n!|a|-' L Ja 



i|(n-r)!L 



And for the values of r under consideration (since k Z), 

 Tlierefore 



ff 



Ll 



r," ........... ." . (10B) 



From (IOA) and (10B) we see that k, I, p ly o-, are characteristics of the expansion of 

 /(a;+a) in descending powers of x. 



We have now proved all the theorems which seem to be of importance concerning 

 asymptotic series in general. We proceed to discuss properties of analytic functions 

 of which asymptotic expansions are given. 



PART II. ANALYTIC FUNCTIONS DEFINED BY ASYMPTOTIC SERIES. 



7. We first propose to consider the question of the uniqueness of an analytic 

 function which is defined by means of an asymptotic expansion possessing given 

 characteristics. 



The discussion will be based on the result of the following important lemma, of 

 which we shall give a proof before proceeding further : 



Lemma. Let f(x) be a function of x which is analytic* in the sector defined by 



* Soe (iii.) on p. 280. 



