1904] 



Infinite Series of Analytic Functions. 



\v) 7 . 



^ . at, 



317 



taken along a suitable path, satisfies the equation, and there are in 

 general (n - \ ) such integrals linearly independent, 

 (ii) If u and v be integrals of the equations 



and 



Tz 



where <i, fa, fa are any holomorphic functions of z, and a, /3 are two 

 unequal constants, the integral 



vanishes if L is a suitably chosen path. 



We deduce the following result : 



Let <j> (z) be any function of z which is regular at all points in the 

 interior of an ellipse, C, whose foci are at the points z = and z = \. 

 The ellipse passes through one (or more) of the singularities of <f> (z). 

 The curve is thus completely defined when <f> (z) is given. Further, let 

 p y q, y be any constant quantities whatever, real or complex, save that 

 neither (p+ 1) nor y is a negative integer. Then < (s) can be expanded 

 in the infinite series of hypergeometric functions. 



where 



= 



{n< r -i)}*.n(-ff).n(#- 



r(0 + , 1 + ,0-, I-) 



fr-i (1 - ^ + ?-Y. F(p + n,q-n,y,t).<f>(t). dt. 



The series is convergent if z is inside C, and divergent if z is outside 

 C ; if z is on C the series is in general oscillatory and the expansion 

 fails. 



2 B 2 



