EXPANSION PROBLEMS. 401 



approaches zero, uniformly for ^ .t ^ tt, when n becomes infinite. 

 When 71 is sufficiently large, hi + (/xo + n)x will vary by more than 2x 

 as X increases from .I'l to .To, and there will surely be an intermediate 

 value X = X7i' for which the expression (26) is equal to 1. If this 

 value is substituted for x in (23), for the successive values of n, the 

 last factor remains from a certain point on numerically greater than a 

 positive constant. Independently of this particular choice of x, 

 the quantity 



gPnX COS (ir/f) I g(/io +n) X cot (tt/j') _- a^nX COt (jr/y) 



remains numerically greater than a positive constant beyond a certain 

 point. Consequently the whole expression (23) remains for .r = a-/ nu- 

 merically greater than a constant positive multiple of g(w+n)a;/cot(ir/:')^ 

 and the same is true of Un{xn'). The inequality is strengthened if xn 

 in the exponent is replaced by .I'l; it follows from (24) that 



(27) \an\< c'e -('«+'») ^^ cot (^/.)^ 



where c' is independent of 7i. 



We have now to consider the order of magnitude of the derivatives of 

 un{x). Suppose X restricted to the interval ^ .r ^ X2, where X2 < xi. 

 If k is one of the numbers 0, 1,. . ., v — I, it is at once deduced 

 from (17) and (18) that 



(28) 





<C. c"'n^6 ('*'+'') ^- cot {ir/n) 



where c" is independent of n. This inequality may be established 

 step by step for larger values of k, by combining the inequalities 

 already found with the given differential equation (15) and the equa- 

 tions obtained from it by successive differentiation. Different values 

 of the constant c" will be found for different values of /.-, but this is 

 immaterial. 



Because of (27) and (28), the series 



Y, an TJ Un (x) 



converges uniformly for ^ .r ^ X2, and consequently represents a 

 continuous function, inasmuch as the individual terms are continu- 

 ous. The statement of the hypothesis for any particular value of xi 

 implies its validity for any larger value of Xi which still remains less 

 than xo, and hence Xi may be any positive number less than .ro. With 



