MR. G. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 313 



Other functions which may be investigated in a similar manner are to he found in 

 BROMWICH'S 'Theory of Infinite Series," pp. 351-352. 



The investigation of the characteristics of asymptotic series representing functions 

 defined hy integrals of types essentially different from BOREL'H type is to he found in 

 the paper by the writer, cited on p. 282. 



12. Finally, it may be stated that it is possible to conceive a function which has an 

 asymptotic expansion 



x 



wherein, as n*-oo, jet,] increases more rapidly than any expression of the form 

 Ar(n+ l) p" ; such an expansion would be obtained by taking a, = G(+l) where 

 G (n) is BAUNES' G-function. But such series have not yet occurred in analysis, and 

 they do not appear to possess the interesting properties of series with finite 

 characteristics. 



VOL. ccxi. A. 2 8 



