BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 561 



result may be expressed as follows : There exist in general two matrix 

 solutions 



{ Fo(aO =ix''jaij(x)) 

 (58) \ M(,_. 



where each function Oy (.r) is analytic at x = 0, and each function &y (.r) 

 is analytic at x = oo ; and where furthermore the determinants of the 

 leading coefficients of Oy (.v) and 5y- (x) at x = and x = oo respec- 

 tively are not zero. It is only the case when such series exist that 

 will be here considered. 



It follows at once from (54) that }"o G^') is a matrix of functions ana- 

 lytic for X 5^ 0, CO , and that Y~ (x) is a matrix of functions analytic in 

 the finite plane except for poles when x 9^ 0. Further, if we write 



Yo{x)= Y^{x)P{x), 



then P(x) is a matrix of functions analytic for x 5^ 0, 00 , and possessing 

 the property that P (qx) = P(x).^^ These properties are in close 

 analogy with the properties for a linear difference system, to which 

 indeed (54) reduces formally by the substitution x = qK 



I propose now to determine completely the nature of P(x), as I 

 have done for the analogous functions P{x) associated with the linear 

 difference system; it is the doubly periodic functions which enter here 

 instead of the simply periodic functions. By means of this determina- 

 tion it will be possible for us to state the problem which, for this field, 

 is analogous to the problems above treated for linear differential and 

 linear difference equations. 



§ 19. On the Matrix P(x). 

 Let us make the transformation x = q' and write 



p(.v) = m. 



The function P{t) is a single-valued function of t analytic save for 

 poles; for, this transformation takes the Riemann surface of infinitely 

 many leaves, with logarithmic branch points at x = and x == go in 

 a one-to-one and conformal manner into the f-plane. 



Let us conceive of the f-plane as divided into parallelograms which 



33 Cf. Carmichael, loc. cit., p. 1.59. 



