E SERIES OF ANALYTIC FUNCTIONS. 4811 



If, then, we integrate the equations 



D(n.) + E(u..,) = (m = 1, 2...), 

 with the initial conditions 



i'i m = 0, u' m = (m = 1, 2...), 



we shall have 



y 



provided this series converges. 

 We find that 



'- 



, (n.-,)<) 



the integrals being taken along any finite path witiiin the region S. 

 It can he proved by induction that either le"*"^! or | <""** | 



(2r)~' ml 



according as )<"* | is greater or less than unity; where M is a finite real positive 

 (juantity, independent of'/-. 

 Hence the series 



y = +, + ,+ ..., 



when multiplied by either ** or <"**", accoitling to the value of arg(oi), is alwolutely 

 and Uniformly convergent for all values of such that || >K. 



2. A slight change of notation is convenient. Consider the equation 



!f-0 ....... . . (3), 



where n is a jx)sitive integer, /*, (2) is exjMinsible in the series 



arranged in descending powers of n and convergent if ; is confined to a finite simply- 



3 Q 2 



