Hyper geometric Function with n parameters. 177 



no sense in which any one set of values of f r (n) corre- 

 sponding to such values of n z r gives the asymptotic behaviour 

 of F n _i(2) when \z\ is large rather than any other. It 

 would therefore be misleading to leave (1) as it stands, 

 where certain functions of n are substituted as the values 

 of f r (n) ; for the asymptotic equivalence persists, what- 

 ever value may be given to those of the set of f r (n) 

 functions corresponding to nth roots of z with negative 

 real parts. There is, indeed, no sense in which one can 

 talk, for example in the case of n even, of a " sudden 

 jump"* in the value of the function of n multiplying 



the exponential term e~ n I" 1 ''* as arg (z) changes from being 



positive to being negative, since the product of e~ n \ z{l> 

 and any function of n whatever is asymptotically equivalent 

 to zero, whatever the value of arg [z). We therefore 

 proceed to find the unique set of functions of n, whose 

 existence is already plain, which makes the asymptotic 

 expansion of F n _i(^) equal to 



\z\ n Z fr (n) e n ' ( 1 + Z — . — s~ J 

 R(,,~ r )iO V d=i n\ n Zrf) 



n-l n ]_ 



when \z\ is large, s n being 2a s -f -. 



s=l ^ 



If C is a closed contour containing the origin 

 F„ (l + «i, l + « 2? ••• l-t-a»-i ; z) 



— 2-ni | oF«_ i (l + a 1 ...l + a n „ 1 ; t) 



I ~f «n 



OA) 2 1 ^ 



l+a w .2 -f « ? 



..]*.(:') 



Further, if the contour is so chosen that \t\ and [ z/t\ tend 

 to infinity as \z\ tends to infinity (and it is plain that such 

 a choice is possible), it will be possible to obtain the leading- 

 terms of the asymptotic expansion of F n by approximating 

 to the value, as \z\ tends to infinity, of the integral in 

 which the leading terms of the asymptotic expansion of 

 F n _x(0 and of 



1 + «« 1 + ct n . 2 + u„ 



have been put in the place of these series. 



* Cp. E. W. Barnes, Proc. Oamb. Phil. Soc. L906. 

 Phil. Man. S. G. Vol. 41. No. 242. Feb. 1921. X 



