86 BIRKHOFF AND LANGER. 



whence it follows that for X sufficiently large, M is less that 2 and 

 y//(x, X) is uniformly bounded, for X in the sector in question and further 

 | X | >iV. Accordingly we have 



i.e. Y& -(*&)+!*£$) E(z). 



Since the matrices Y (x) and D are analytic in X the same is true also 

 of Y(x). It is seen, moreover, that | Y | =(= 0. Hence, when the 

 functions yt(x) satisfy the conditions of the theorem on page S3 there 

 exists in every sector of the type described above a matrix solution 

 Y(x) whose elements are continuous in x and analytic in X, and are 



the same as those of S(x) to terms in— zi . But by construction the 



A 



elements of S(x) are the same as those of S(x) to terms in — . Hence 



A 



we have the 



Theorem: Given any formal solution S(x) of equation (26) in which 



R(x) and B(x) both possess derivatives up to and including those of 



order k, then in each sector of the complex plane within which none of 



the quantities ft{\(7 A (.r) — y r (x)} } change sign there exists an actual 



matrix solution Y(x) which is continuous with its first derivative in x 



and analytic in X, and is throughout the sector identical with S(x) to 



. 1 

 terms in —r . 

 X i 



In virtue of this there exists in any sector of the type described a 



pair of actual associated matrix solutions of the form 



(45) 



Y(x) = (*!>(.)«/.) [3«|m) 

 Z(x) = ( e -xr,-(x)-B,-(x) [d .. ]k _j t 



provided only that the matrices R(x) and B(x) are differentiable 

 k terms. In all cases the functions [5*y]fc_i which occur in the expres- 

 sions for Y(x) and Z(.r) are, of course, respectively identical, to terms 

 in l/X " with the series [5 W ] in the formulas (37) and (38). 



