MR O. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



Since the paper of POINOARE appeared, researches have been published by a host 

 of mathematicians. It is sufficient to mention the names of CESARO, LE ROY, 

 VAN VLECK, STIELTJES, MELLIN, MITTAG-LEFFLER, BARNES, HARDY, and LITTLEWOOD 

 as investigators, either of the general theory of oscillating and asymptotic series, or 

 of the asymptotic expansions of particular classes of functions. Complete biblio- 

 graphies are to be found in BROMWICH'S ' Theory of Infinite Series/ and BARNES' 

 'Memoir on Integral Functions.'* The former work contains an excellent history of 



the subject. 



It might be considered that, when the theories of POINCARE and BOREL had been 

 discussed with such vigour, there would be comparatively little room for further 

 general developments of the subject. In this memoir, however, I propose to discuss 

 an aspect of the theory which has hitherto remained unnoticed, and which promises 

 to have many useful applications. In fact, I have already found it to be of 

 importance in connection with the problem of expanding an arbitrary function in a 

 series of inverse factorials, and it is highly probable that the theory can be employed 

 with advantage in particular cases of the problem of expanding an arbitrary function 

 in a series of normal functions. A simpler application is that of determining an 

 upper limit to the number of terms that should be taken in a given asymptotic series 

 in order that, for a given value of the variable, the difference between the asymptotic 

 expansion and the analytic function represented by the expansion should be as small 

 as possible. 



It is convenient here to specify precisely the meaning of certain expressions used in the sequel : 



(i) If x be a real variable, the statement . > a means that some positive number A (independent of x) 

 exists, such that xz a + A. Thus it might be convenient to take A = 10~ 100 . 



(ii) A function /(x), of a complex variable x, is said to be analytic in the region | x < a when the 

 function has no singularities in the interior of the region, although it may have singularities on the 

 boundary x\ = a. 



We say that the function is analytic in the region | x \ S a when some positive number A exists such 

 that the function is analytic in the region or | < a + A. 



(iii) A function f(x) is said to be analytic in the sector a s arg x /J when, if o be any singularity of 

 /(x) in the finite part of the plane, / is not within the sector, and the distance of o from the boundary 

 of the sector is at least equal to A. The statement does not iman that /(*) is "regular about the point 

 x = at," i.e., that f(x) can be expanded in a convergent series of negative integer powers of ./ if \x\ be 

 sufficiently large. 



(iv) When a region is specified by means of two inequalities (e.g., |arg;| < Jir, \x\ < 1) we mean the 

 region in which both the inequalities are satisfied, unless an explicit statement is made to the contrary. 



(v) It it necessary to make use of the ideasf of " Orders of Infinity " so frequently that attention is rarely 

 directed to their use. Thus, if we have |x| < p n where p is finite, and n is any integer, no matter how 

 large, we should, if necessary, say at once that |a-,,| < K . n \ where K is finite. 



Before proceeding, it is desirable to summarize the chief points of the theories of 

 POINCARE and BOREL : 



* 'Phil. Trans.,' A, vol. 199, pp. 411-500, 1900. 



t HARDY, "Orders of Infinity" ('Cambridge Tracts in Mathematics,' No. 12). 



