Mli. G. N. WATSON: A THEORY OF ASYMPTOTIC SKIM Ms 281 



(i) A function f(x) is said to possess an asymptotic expansion (in the sense of 

 POINCAUK) for 1m ^v values of \x\ t and for a certain range of values of arg x, if, for 

 such values of arg x, the function can be expressed in the form. 



where |RJ,|r J,]*!'"' 1 when |a?|>y. 



The quantity y is finite, n is any assigned finite integer, and J is a finite quantity 

 depending on y and n, but not on x\. 



(ii) BOUEL'S theory is an attempt to associate with the series 



a quantity, S, which shall be equal to the sum of the series if the series happens to be 

 convergent, and which shall have a definite meaning if the series be divergent. 

 Putting 



BOREL defines S by the equation 



S = 



when the integral converges, the original series is said to be " summable." 



This theory requires some knowledge of the singularities of the function <j>. Such 

 a requirement is a defect of the theory, so far as many applications to asymptotic 

 expansions are concerned, for it will often happen that, when we are given a function 

 ./('') we can obtain an asymptotic expansion of POINCARE'S type with some knowledge 

 of the values (or the upper limits of the values) of a , a,, ..., a., 11,; we may be able 

 thence to deduce the radius of convergence* of the series for the function <f> (t), and 

 yet have no knowledge of the singularities of <f> (t) outside the circle of convergence of 

 the series for <f>(t). By this lack of knowledge BOKKL'S theory is robbed of a great 

 deal of its usefulness.! 



Some severe stricturesj have been passed by MITTAO-LEFFLER upon BOREL'S theory 

 for other reasons. 



After these preliminary statements, we summarize the objects of this paper. In 

 Part I. we define certain quantities called characteristic*, which l>ear much the same 

 relation to an asymptotic series as the radius of convergence bears to a convergent 

 series. The definition is a natural consequence of an attempt to impart more precision 



* Which may l>e finite, as in the case when a n ( - )" . n !. 



t See, e.g., the assumptions made in 104 of BKOMWICH'S 'Theory of Infinite Scries." 

 I At the Fourth International Congress of Mathematician* See the ' Bulletin of the American 

 Malhcmatiral Society," vol. xiv. (ser. 2), p. 



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