288 MK. 0. N. WATSON: A THEORY OF ASYMPTOTIC SERIKS. 



where m, p are the greatest integers, such that 



mk < 1, pi < 1, 

 ,, = , if /<!, r? 2 = 1 ^ J^l- 



Also, by induction, we deduce that if f(x) have an asymptotic expansion with 

 k, I, p, a- as characteristics, {/()}' has an asymptotic expansion with the same 

 characteristics, r being any finite positive integer. 



Let us, however, examine the series for {f(x)} r in greater detail when r is a positive 

 integer. 



Changing the notation slightly, let A, B, k, I, p, a- be possible constants and 

 characteristics of f(x) ; we may therefore take A r , B r , k, I, p, a as constants and 

 characteristics of {/()}', where 



A r = XA, A r _,, B r = ( A^,., + A^B.) X'+ B^^y-'. 

 This follows from (3A) and (SB) combined with the identity 



we have written Aj, Bj for A, .B, and we have also written X, X' for the quantities 

 L'//i + 2 + />"', l+i} a (2p + l+r 1 ) of (3A) and (SB). 

 We see at once that 



A r = X'-'A', B r = (AX'+By- 1 ) B r _ t + A r - 1 BX'- 2 X'. . . . (4 A , B) 



Dividing through (4B) by (AX'+By" 1 ) 1 ", putting r = 2,3, ... and adding the results, 

 we get without difficulty 



B r = 



From equation (4s) we see that we should get a rather larger value of B r than is 

 given by that equation if we put /* for both X and X', where 



It = 

 so that /uiX, /iSrX', since m>p, k~ l szl~\ 



The result is so much simpler that we shall do so ; and the result we get is that 

 H' r is a possible outer constant of {f(x)} r where 



B' r = y{(A / i + By-')'-(A / x)"}, . . . ..... (5) 



We shall now prove that, if f(x) has an asymptotic expansion with k, I, p, cr as 

 characteristics, then exp {f(x)} possesses an asymptotic expansion with the same 

 characteristics. 



