BOG 



MR <;. X. WATSON: A THEORY OP ASYMPTOTIC SERIES. 



Then F, (z) is certainly analytic in the region (see figure) in which both 



inequalities 



I 2 | > y\ cosec (fjitr), 



are satisfied. 



In like manner, when 



<argz 



we have 



K(z)>yi,R{zexp(i/i)}> yi , 



L- 



and we can deduce that F, (2) is analytic in the region in 

 which both the inequalities 



|z| > yi cosec (/i-^), -^n-ff < argz < ^ + 6'-^ 



are satisfied. 



That is to say, F, (2) is analytic in the region in which 



^-^), | a rgz| 



the inequalities 



|2| 

 are satisfied. 



Now consider the function F, (z) in the region in which 



l*l>(ri+l) cosec (/n-fl'), jargj 

 We may define F, (2) in this region by the equation 



F >( 2 )=f 



J ( 



where the upper sign may be taken if arg z - 0, and the lower sign if arg 2 ^ 

 By repeated integration by parts we get 



,. 



From 

 have 



where 



(1?A) a " thu hlte ^ ted Parts vanish at the upper limit; and we 



-if e 



JJfTM) 



