INFINITE SERIES OF ANALYTIC FUNCTIONa 495 



The series is convergent if i is inside C and divergent if z is outside C. If 2 is on C, 

 the series is in general oscillatory and the expansion fails. 



Moreover, the expansion holds for unrestricted values of p and y, save that neither 

 p+ 1 nor y may be a negative integer, if we write 



= (~ I 





' ' ' ' (33)t 



a form equivalent to (32) when the real parts of y and /> y+ 1 are jMjeitive ; C w being 

 a simple closed contour containing the points t = i) .-mil (=[, hut no singularity 

 of ^(l). 



16. The following is a generalisation of (21). 



The integral 



r.i+,o-,i-) 2y -i /i _~\rr 

 _F(- N ,p+n,y,2)<fc 

 2~*r 



is equal to 



) . (34), 



the equality holding for unrestricted values of n, p, y, and t, save that neither p+ 1 

 nor y may be a negative integer. 



We deduce 



Tfieorem VIII. Let <f>(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 2=0 and 2= 1. The ellipse passes 

 through one (or more) of the singularities of ^(2). The curve is thus completely 

 defined when < (z) is given. 



Further, let p, q, and y be any constant quantities whatever, real or complex, save 

 that neither p+l nor y is a negative integer. Then 4>(z) can be expanded in the 

 infinite series of hypergeometric functions 



i ?, y, z)+<*iF(p+l, q- l,y, z) + ...+<* F (p+n, <?-n, y, 2) + ... . (35), 

 where 



_ 



y-n- y/ >-y+ 



?(p+n,q-n, y, 



The series is convergent if 2 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. 



17. Expansions in Legendre functions can be deduced from expansions in 

 hypergeometric functions by an appropriate transformation. On account of the 

 special interest of Legendre functions, we give a list of the formulae obtained. 



