EXPANSION PROBLEMS. 389 



or, in consequence of the inequalities 



n< pn< 2n 

 which hold for all values of n from a certain point on, 



< 3-2^' G n^ (>\n{x2— x\) ^ 





dx' 



However large k may be, and however small numerically the negative 

 quantity X2 — xi, the series which has this last expression for its 

 7ith term is convergent. It follows that the series obtained by differ- 

 entiating (8) term by term k times is uniformly convergent ^* for 

 ^ .T ^ xo; and as .^2 may be any number between and Xo, the 

 theorem is proved. -^^ 



The question is now inevitable: Is it possible to develop every 

 analytic function in a uniformly convergent series of the form (8), 

 with the restriction, perhaps, that the function shall vanish at the ends 

 of the interval? As in other problems of this sort, it is easy to deter- 

 mine what the values of the coefficients must be, if such an expansion 

 exists. The formal series being given, an answer to the question may 

 be expressed as follows : 



Theorem II. // N is any positive integer, however large, it is possible 

 to define a function f{x), analytic throughout the interval ^ a; ^ ir, 

 and vajiishing with its first N derivatives both for x = and for x = r, 

 such that its formal expansion in a series of the form (8) diverges at every 

 interior point of the interval. 



The system adjoint to (1), (2) consists of the differential equation 



dx"^ 

 and the boundary conditions 



«(0) = 0, (j(7r) = 0, »'(7r) = 0. 



Its characteristic numbers are the numbers p„, and its characteristic 

 solutions have the form 



d2) Cr,{x) = ^P"(^--'-f a;c"''»(^-'^) + w-f?"'""'^-'^'. 



14 Of course the terms of the series are continuous and in fact analytic func- 

 tions of X. 



15 Professor Birkhoff points out that it may be shown by a slight modifica- 

 tion of this argument that the series converges uniformly throughout a region 

 of the complex x-plane including the segment (0, Xo) of the axis of reals in its 

 interior, and so represents a function analytic for ^ x < Xq. 



