302 MB. 0. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



Also |j,rp{-*|y|} Ceases when Jyj > l- if I<* 



T^t v be the greater of the quantities (<r y ) , 1 



SlL the e^ ion in which |, | > y - * * arg ^ A ^e region C we see from 

 <H)t Ahis result that in the region C, whether I be greater than or less than | we 

 havlan inequality of the form \f t (y)\ < B 3 exp {^\y\ }, and y is analytic in the 



wh.-re A > ; and let /, (y) =/ : 



Then /(z) is certainly analytic in the sector jarg^-^+X, and at all points at a 

 distance not greater than 2A from the boundary of the sector ; for when z hes m the 

 region just specified, y certainly lies in the region C 2 . 

 Also, in the region specified for z, 



|/( 2 )|<B 3 exp{-]2+(|y a + 2A)secX|} 



< B 3 exp {(y a + 2A) sec X} exp { - | z I }. 



Therefore, by the lemma, /(z) = 0. 



But /(z) = fi(x)-f,(x)', and therefore we have proved that/^ae) = /, (x), when 

 /,(*) and /,(*) are subject to the conditions stated at the beginning of the section. 



The reader might be inclined to think, at first sight, that if /3-a>27r, we could infer that f,(x) is 

 identically zero on account of the theorem that " a non-convergent series cannot represent asymptotically 

 the same one-valued analytic function for all arguments of /."t 



Tliis theorem w not applicable, because /, (4 />(.) may not be analytic inside the circle |*j-yj a 

 multiform function may have an asymptotic expansion valid for a range of values of arg x greater than 2ir. 



Thus, the generalised hypergeometric function formula J 



(a) V(l - P ) ,F, (a ; /; /) + I 1 (+ 1 - P) r (p ' l)* a+ '" P iF, (a - p+ 1 ; 2 - p ; ) 



is valid when |argz| <Jir. 



9. We can now show that if an analytic function, f(x), possesses an asymptotic 

 expansion for large values of | x with a grade and an outer grade equal to unity, the 

 range of values of arg x over which the expansion is valid being greater than TT, the 

 function f(x) is absolutely summable by the method of BOREL for a range of values 

 of arg x just less than the range over which the expansion is valid, provided that 

 MOREL'S integral be taken along an appropriate path from to oo, not necessarily the 

 real axis. 



* The definition is possible since P - a.>vL 



t HROMWICH, ' Theory of Infinite Series,' p. 335. 



J See BARNES, ' Cambridge Philosophical Transactions,' vol. 20, p. 260. 



The range of validity being taken less than 2ir. 



