Ml: <:. N. WATSON: A THEORY OF ASYMPTOTIC SKIMKS. 



307 



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We iiw tin- funiml:e (17) when \t\ is ^re.-iter than, say, 



When y[r=}tr~', since the series for (f>(() converges when |/|<p~' (and p ^ cr), 

 we have \<f>(t)\ < K" when- K" is finite and independent of t. 



Also when 1 1 \ S (i+ 3) o-"' we have | <j> (t) \ < K"' where K"' is finite and independent 

 of t. 



Accordingly, when !/|^^or~', we use the formula 



the contour of integration heing a circle of radius 

 Hence we get, when |<|s {tr~\ 



de 



Combining these results with (17), we see that for all values of t such that either 

 |arg t\ < X 6 or \t\^ <r~ l we have 



|+()|<K,ezp|yt|, 



dt 



. . (17x) 



where K 1( K,, are finite and independent of t. 



Consequently, if y, > y, and /<. is any assigned integer, 



= 0, 



if y, > y ; and the function <j>(t) is analytic in the region in which either of the 

 inequalities 



(\.)\t\<p-\ (i\.)\argt\<\-0, 

 is satisfied. 



Now let us study the function 



The function F, (2) is an analytic function of z when R (:) > y, where y, > y. 

 If also It {2 exp ( //*)} > yi, we can see that 



f 



(-!) 



(18) 



where /i is any ciu;mtity such that < p. < X 0, p. < J-jr. 



Ecjuation (18) gives the analytic continuation of F, (z) over the whole of the area 

 fur which R {2 exp ( t/n)} > y,. 



Let ff be a small quantity such that < ff < /n. 



2 K 2 



