306 



MK. (!. X. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



converges when |arg|s X 6; and by putting z = tu, we find that the integral is an 

 analytic function* of t in the region |arg t\ < A 6. 



That is to say, BOREL'S associated function $ (t) is analytic in the region \t \ < p~ l 

 and also in the region |arg t\ < X 0. 



We shall require an upper limit for 



dr 



when |arg 't\ < X 6. 



To ohtain this upper limit we consider the integral 



We may show that it can be expanded in the form 



1! 



when [t\<tr \ by replacing F(z/) in the work immediately preceding by 



Therefore, by the theory of analytic continuation, we have 



for all values of |f | in the interior of the region |arg t\ < K-B. 

 Now 



< A+Bo- 



so that 



1-1 



Making the substitutions made in obtaining (16 A ), we find that 

 <**] 



(n = 0), 



( > 0) 



(n = 0). 



n > 



n = 



,... (17) 



where K, K' are finite quantities independent of t and n. 



from an obvious modification of the thcorem 8tate<i 



