Hyper geometric function with n parameters. 18 1 



From the equations (3) it is plain itbat every value of f is 

 such that 



\z\ e b 



is an (;i-fl)th root of z, and the summation is taken over 

 those values of f alone which are less than or equal to tt/2 

 in absolute value. Thus the asymptotic equivalent for ^I x 

 is the sum of certain terms of the form 



s/ 



n -f 1 



or since 



J »+i = *» + *» + * 



of the terms of the form 



VidJjUi*) 





The question remains as to whether, when all the integrals 

 of the term t I x are taken into account, any of the (n+l)th 

 roots of z with a non-negative real part are excluded (it is 

 of no importance whether those with negative real parts are 

 excluded or not) and as to how many times terms corre- 

 sponding to each of these roots occur. A slight consideration 

 of the equations (3) shows that the first possibility is cut out. 

 As to the second point, it is clear that only values of f less 

 than or equal to 7r/2 in absolute value need be considered, 

 In this restricted class of cases, it is plain that the same 

 value of £ cannot occur in the case of integrals corresponding 

 to two different values of t. Hence 



it 



nr(i + a s ) nr(i+«.,) 



(2ttj 2 ~ \/n (2tt) 2 y/n + 1 



the summation being taken over values of H+l z r satisfying 

 the conditions 



( * \»+l . 



IT 7T 



- 2 <arg K + 1 c r < 2 , 



( n+l z r )~ s n+i being interpreted to mean the complex number 

 whose argument is 



-*«+!>< ai 'gn+rV 

 The integral I 3 has no "critical points" in the Kelvin 



