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XIV. On the Conveiyence of Infinite. Series of Analytic Functions. 



By H. A. WEBB, B.A., Fellow of Trinity College, Cambridge. 

 Communicated ly Professor A. R. FOUSYTH, Sc. D., LL.D., F.tt.S. 



Received Novemlwr 10, Read November 24, 1904. 



IN the first section of the following work an attempt is made to deal with the 

 convergence of infinite series of functions defined by linear differential equations of 

 the second order from the most general point of view. Functions of LAME, BESSEL 

 and LEGENDRE are considered as examples. In the second section the results 

 obtained are applied to the expansion of an arbitrary uniform analytic function of 2 in 

 a series of hypergeometric functions, and the expansion is shown to be valid if the 

 function is regular within a certain ellipse in the z-plane. An expansion in a series 

 of LEG EN ORE'S associated functions is deduced by a transformation. The method has 

 been applied by the writer to other cases, but the foregoing offer adequate illustration 

 of the general theory. 



SECTION I. GENERAL THEOREMS. 

 1. Jltcoi'cm I. Consider the differential equation 



where 



n _ n j.Q]j.Q. 



k P 



k is a large constant, v/Q,,, Qi, Q 2 ... are analytic functions of :, independent of /, 

 without singularities or branch-points so long as z lies within a given simply - 

 connected region S in the 2 -plane, though v/Qo, Qi, Q a ... may have singularities on the 

 Ixnmdary of S ; and the series defining Q converges if \k\ >R, so long as z is in the 

 region S. 



Let z = a lie a point within S. 



VOL. CCIV. A 385. 3 Q G.C.05 



