BIRKHOFF. — THE GENERALIZED RIEIIAXX PROBLEM. 529 



The matrices F~(.v), F+{.v), G~(.v), 6'+(.r) thus obtained will satisfy 

 (11), as appears from (12) by letting n become infinite. 



§ 4. Application to the Solution of a Single Matrix Equation. 



Multiply the second matrix equation (11) on the right by 

 T^(x)A(x) and subtract it, member for member, from the first equation 

 (11). There results 



(16) F+ (x) = rP (x) [I + G~ (x)] A (x) along C. 



Inasmuch as 6'~(.r) reduces to a matrix of zero elements at x = oo , 

 the determinant of / + G~(x) and also of F+(x) is not identically zero. 

 The matrix equation (16) admits of further simplification. In fact 

 the function log t(x) is analytic along C and increases by 2t V — 1 

 as X makes a positive circuit of C. If c lies within C the function 



4) (x) = log r (x) — log (x-c) 



is accordingly single-valued as well as analytic along C. But (/)(.i-) is 

 of the form 



T^ (x) <7+ (x) a (x) (p = 0, g (x) = 1, a (x) = 4> (.r)), 

 so that by (7) we can find ^+ {x) and d' {x) such that 



6+ (.r) — d' {x) = {x) along C; 

 this gives us 



tP{x) = (a-c)PePe^WeP»-(.T). 

 Now let us write 



$ (x) = e-P«"W F+ (x), ^(x) = {x-cYeP^~^'^ [I + G- (.r)]. 



By these equations we define $(.r) as a matrix of regular inner func- 

 tions, and ^(.r) as a matrix of functions analytic outside of C except 

 for a pole of order p at .i- = =0 , and continuous along C ; the determi- 

 nant of neither 4>(.r) nor ^(.r) vanishes identically. Between $(.r) and 

 ^(.r), by (16), we have the matrix relation 



(17) $ (.v) = ^ (.r) A (.r) along C. 



It is this type of matrix equation which is important for the present 

 paper. 



