I 271* ] 



VII. A Theory of Asymptotic Series. 

 By G. N. WATSON, M.A., Fellow of Trinity College, Cambridge. 



Communicated by G. H. HARDY, F.R.S. 



Received December 5, 1910, Read March 23, 1911. 



CONTENTS. 



Page 

 Introduction and historical summary 271) 



PART I. THE CHARACTERISTICS OF ASYMPTOTIC SERIES. 



1. The definition of characteristics. Theorem 1 282 



2. The products of asymptotic series 284 



3. The exponential of an asymptotic series 288 



4. Three general theorems. Theorem IF 290 



5. Theorem III. . . ' 292 



6. Theorem IV 293 



PART II. ANALYTIC FUNCTIONS DEFINED BY ASYMITOTIC SERIES. 



7. Lemma. An extension of PHRAGMEN'S theorems 295 



8. Theorem V. The uniqueness of a function defined by an asymptotic series 300 



9. Conditions for the " summability " of a function 302 



10. Properties of a function deduced from its associated function 310 



11. The characteristics of the " li " function 311 



12. Conclusion 313 



Introduction and Historical Summary. 



IN their efforts to place mathematical analysis on the firmest possible foundations, 

 ABEL and CAUCHY found it necessary to banish non-convergent series from their 

 work ; from that time until a quarter of a century ago the theory of divergent series 

 was, in general, neglected by mathematicians. 



A consistent theory of divergent series was, however, - developed by POINCARK in 

 1886, and, ten years later, BOREL enunciated his theory of summability in connection 

 with oscillating series. So far as diverging power series are concerned, the theory 

 of BOREL is more precise than that of POINCARK. 



VOL. crxi. A 477. 19.7.11 



