Hyper geometric function with n parameters. 179 



Both of these results fall under the hypothesis that 

 F n _i(l + a 1? 1 -\-a 2 , ... 1 .+.«»_! ; s) 



Iir(l+«J 



H— 1 



2 



(2tt) 2 ^n E(, r ~,)-0 





where (»^ r ) ~ s " means the complex number whose argument 

 is — 5 ft Xarg(n^), and the summation is taken over those 

 values of n z r , the »th root of z whose real parts are not 

 negative. 



We may now proceed to discuss the value as ( z ] — > co of 

 the integral in which the integrand of the integral (1) has 

 been replaced by the leading terms of the asymptotic 

 expansions of the series. 



We have 



nr(i+ as ) r 



2^i Q F n (z)^ +-^- - \te**Hntry 



- — " ,— JVr 



(2tt) 2 v'n 



x (/ !l (zit)-*«- i *— \* 



X V W'J * l tV{a.-l-+v))t' 



The only limitations on C, which is a contour in the 

 i-plane, are, that it shall enclose the origin and be such, that 

 on it \t\ and | z/t | are large when | z | is large. We may 

 therefore choose as the contour a circle with its centre at 

 the origin and with radius 8, provided 8 and \z \/8 are large 

 when \z\ is large. Then 



nr(i +«,)/», ^ Mn'— -5 -i e -±^ s „ 



(2tt) 2 V» 



the 5-summation being taken over integer values of s, such 

 that 



7T # ■+- 2.97T 7T 



<£ — being the argument of z/t (requiring sometimes a 

 different value of </> in different parts of the range), which 

 lies between — tt and +tt. [The special case when 



N2 



