Mil. G. N. WATSON: A THEORY OF ASYMPTOTIC SERII > 293 



Consequently, if n > 0, 



|4,|< 2 IA.IA-X ^(foi+ijp- 



: 



< i \- I H(AX/t-)-r(A-n+l) /} " ....... (9A) 



m = 1 



If AX ^ c, we have 



From this result, combined with (9x), we see that, if AX ^ c, we take p a , the inner 

 radius of*, {f(x)}, to be the smaller of the quantities p, pA\/c. 

 If AX = c, we have 



and if S > 0, we have up* < K, (p + 8)*, where K, is a finite quantity depending on p 

 and 8, but not on n. That is to say, if AX = c, p + 8 is a possible radius of <I>, 

 and k is a possible grade. 

 Further, 



HI = 1 10 = -H 



_ _ _ ,11+1 



= +i i|ar| 



I! 



Hfi-^l" 1 . 



Using the inequalities B<c|x|, (A/i + By'^c"^ 1, and (Be" 1 )'* 1 < K,r(/'*+ l)a", 

 (where K 3 is independent of n), we get a formula of the form 



where K 3 is independent of n. That is to say, / and <T O , defined as above, are a 

 possible outer grade and a possible outer radius of <I>, \J (*")} 



6. Theorem IV. Suppose that for a certain range of values of arg (x+a), f(x+a) 

 possesses an asymptotic expansion in negative powers of x+a valid when |x+a|>y 

 with coiixttiitt.* iiinl characteristics A, B, k, I, p, <r. Then for the same range of values 

 of&vg (.c + a),f(x + a) possesses an asymptotic expansion in negative powers of x valid 

 when |x|iy+ |a| with cltaract<-ri*t;,-x k, I, p lt o-,, where 



Pi =p-t||, 



