388 JACKSON. 



Theorem I. If a series of the form (8) converges uniformly throughout 

 any interval 



(9) ^ .r < a-o, 



where < .To ^ tt, its sum must have continuous derivatives of all orders 

 throughout the interval (9). 



Let xi and X2 be any two numbers subject to the inequalities < Xi 

 < xi < xq. Since the series converges uniformly throughout the 

 interval (9), it converges uniformly for 



(10) Xi^X< .To, 



and there must be a number G such that ^^ 



(11) \anUn(x)\<G 



for all values of x in (10) and for all values of n. 



Now if n is sufficiently large, pn will be real, and the interval from 

 I V3pn.ri to I VSpn.To will be of length greater than 2t, and so will 

 contain at least one value of | V3p«.T whose sine is 1. That is, there 

 will be one or more values of x in (10) for which the parenthesis in (7) 

 takes on the value V3. Let x'n be such a value. Then un{x'n) is 

 positive, and 



Since (11) must be satisfied when .r = x'n, 



\an\< GV'-^""^'. 

 On the other hand, if k is any integer, 



f^Unix) = (-p„)^-(c-P"^ + co^+ie-"P"^ + co2*^+2e— ^'"'^), 

 dx^ 



and 



dx' 



k ^n (.V) 



^ 3p/ ei"""", 



for any value of x in (0, tt); we are assuming still that pn is real. If 

 ^ .r ^ xo, then 



d"" 



<3Gp/ci''"^^^-^'), 



13 Of the assumed uniform convergence, only this consequence will be used; 

 and the statement of the theorem might have been made correspondingly 

 more general. 



