MU. O. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 

 Consequently, for the integral round the arc, 



305 



On the ray arg z = \n + 0, we put 



where r varies from 1 to oo; so that for the integral along this ray, 







cosec 0exp { |y|sin 0}; 



and we get the same inequality for the integral along the other ray of L. 

 Combining our results we get 



(16A). 



|s ^ and |/ 1 < o-" 1 , tve may expand the integral 



Remembering that p^cr, we see from this formula that, if \t \ < cr~ l , then 



uniformly as n * oo. 



That is to say, when |arg 



)/* the f'm-m 



where 11, * as n*-oo. 



In other words, when | arg t\^\0 and t \ < a" 1 , the integral 



represents r."iti:i.s associated function defined by tlie series 



converging when ||<p"'; so that BOREL'S associated function is analytic when 



I'Kp- 1 - 



Now the integral -i 



VOL. CCXI. 



2 R 



