Hypergeometric Function with n parameters. 183 



then 



nr(i+«,) 



F„(1 + «!, . . . 1 + a n ; z) ^ J^ r ^^l 



It is worth while to notice the forms which result for 



oFn-^z) and for F„_i( — z) when z is real and positive. If 



n ~ 1 n — 1 



x is real, positive, and large, and s a — X a r -\ — 



i * 



1+— S=i +— i=I +••■ 



nn(i4*r) 2in(i + « r )(2 + « r ) 

 i i 



nr(i + « r ) _ 



(2tt) 2 ^n 



+ 2 2 e" ,r cos « —VcosU# 1/n sin- (5„J-i') • 



t=1 n v x v/n L w n J J 



As a first approximation we may record the result 



n nr(i + * r ) 

 nnci+^j (27r) i a/ - 



Vn . » * «J/»«n.** / „_ • 2-rrt 2irts r 



or 



f nx l l n »** war 1 /* oos— / i/„ ■ 2? ™ 27rtS„\ } 

 ) g«* + 2 Z e ,2a cos n cos ( n^ J/ " sm~ | J > , 



/ ■ / n n r(i + «,.) 



nn(i + «r) (27r ) * ^ 



2* . 1 



giving the familiar result 



[* + 2 S *- os »cos(usm- - .,-)[. 



lit 



