Ml. 1 . O. N. WATSON: A TIIKOKY OF ASYMPTOTIC M-KIES. 289 



Let A, B be possible constants of /(x) ; from the definition of f(x) we have, with 

 the usual notation, 



exp *} = exp a . exp 



Now, by the work of the preceding section, we can expand E," in an asymptotic 

 series with characteristics /, I, p, <r ; and the expansion will be of the form 



where, by (4A) and (5), 



|A|s 



A = (r < m) 



and 



|x" +1 . n ,S.|S 

 We may consequently write 



x 

 where 



and 



T. = (exp a.) f^ + ^' + + " S " 

 >; Ll! + 2! H n! 



Consequently, if n > 0, 



r A\"-I -i 



|c.|s|expa 2 - *_r(*n+l) p " 

 L - 1 m ! 



" -I* A \ / j - ^ - - , f f t * I/A.) 



I c 1 = I exp OQ I . 

 Further, 



4 "(n+l)! + (n+2)!~ K "J 



"> 



VOL. CCXI. A. 



->^ -i 



+ 2)T~ 



