INFINITE SERIES OP ANALYTIC PUNCTION& 491 



0i, 6,, n being any uniform analytic functions of 2 without singularities in the finite 

 part of the z-plane, and at, /3 two unequal constants. The integral 



vanishes if L is a suitably chosen path. 



For the integral 



1 



and vanishes if the expression in square brackets has the same value at the beginning 

 and the end of the path. A suitable path is, therefore, a " Doppelumlauf " round two 

 of the roots of 6\(z) = if these roots are not all equal ; the path may in special cases 

 reduce to a straight lina 



11. The special case of the hypergeometric function was considered by JACOBI, 

 who proved in the paper already quoted that if the real parts of y and p+ 1 y are 

 positive, 



f'F(-m, p+m,y, z)F(-n, p+n t y, z) z^-^l -)'- rfz = if m^n . (22). 

 Jo 



JACOBI proved further that, under the same conditions, 



(23). 



An important special case of (22) is given by m = 0, in which case, under the same 

 conditions, 



f F(-n t p + n,y,z)z>-*(l-zy-' f dz = 0, unless n = . . . (24). 



Jo 



We are now in a position to investigate the expansion of an arbitrary uniform 

 analytic function in a series of hypergeometric functions. 



SECTION II. HYPERGEOMETRIC FITNOTIONS. 



12. The hypergeometric function 



F(a,/8,y,z) 

 is an analytic function of x for all values of x, with branch-points at 



x = 0, 1, and oo. 



If |x| < 1, one of the branches of the function can be represented by the series 



^ , . ,. 



3 R 2 



