MR. G. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 301 



we have 



for all values of n, provided x lie in the region C. 

 Now choose n to depend on x in such a way that 



so that we may put 



-!->{ !*->! }"_*, 



where : < 1. 



Now let y he the greater of the two quantities (1 +/)' <r and y ; and let the region 

 in which \x\ > y', a S arg j- :S /8 l>e called the region C 7 . 



When x lies in the region C', we have 



ln= {\x<r- l \} rl -W>(l+l)-W> 1. 



* 



But when / > 1, hy the asymptotic expansion of the gamma function, 

 logr(Jn+l) = 



where J does not exceed a finite quantity independent of n. Consequently, when x 

 lies in the region C', 



I/I CO-/. (<)! <B|p[(fcn.|)log(lm-W)-fa--{n+i)log Ixcr-'i], 



where B, is a finite quantity independent of n and x. 

 Substituting for In + Id in terms of x, we get 



| /, (x) -/, (x) |< B, | *r" | - 1+ "<> exp [- | *r-' 1 -*]. 



Putting /i (*') ./() = ./((>), we want to show that if yi (x) is analytic in the 

 region C' and subject to the inequality 



| /, (x) |< B! | r- 1 - exp [- | o 

 j-) = 0. 



Let us put* x = ory 1 and/ 3 (.r) =J\ (y). 

 Then/ 4 (y) is analytic in the region, C", in which 



\'J\> *-"(/)"> sUrgy 

 dyi (y) is subject to the inequality (when y lies in C") 



If 1 2: J, we see at once that in C" 



|/,(y)|<B,exp{-|y|} . . ...... (14) 



where B, is a finite constant depending mi B, and y'. 



* y = a. is a singularity of this transformation ; but see (iii.) on p. 280. 



