Ml; i;. N. WATSON: A THEORY OF ASYMPTOTIC SKKIKS. 

 Now y, 1 1 1 - H (zt) : - 1 1 1 ; and hence from (17x) 



309 



: 



; so that, applying the formula just obtained to 



and K a is independent of n. 



But we have | S, | = 

 S B+1 , we get 



But we can find a finite quantity K independent of 7i such that ^ /Ji\\ < ^ 



when n is a positive integer ; and a fortiori < K ; therefore, since p s <r, 



< Ap" +1 (/< + !)! + K, (f o-)"* 1 . (n + 2) ! . 



B, . 



. n ! , say. 



That is to say, if |z| > (y, + 1) cosec (/A ff), larg ej < %ir + ff, we have asymptotic 

 expansions of the form 



(1'J) 



Taking 1=1, and writing 2cr for <r in Tlicon-ni V., \vc conclude, since the 

 expansions (19) are valid when |arg z|S ^ir + ^0' (i.e., lor a range of values of arg z 



greater than n), that 



K (z) = F, (z). 



That is to say, in the region |z|> (yi + 1 ) cosec (p. ff), |urgz| < ^n + ff, we have 

 proved that 



F(z)=f z<l>(t)c-",/t. 



.'(T.) 



But z<f> (t) e~" dt is an analytic function of 2 when R {2 exp (/*)} > y,, where 



J(-) 



y, is any quantity greater than y ; hence, by the theory of analytic continuation, we 

 have 



.iiid more generally, if (X 0) < v < X 6, we have* 



P| )=| z<f>(()e- :l <lt, provided R {2 exp (-iV)} > y,. 

 ia analytic along the ray arg t = - v. 



