BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 557 



Let US now apply the preliminary theorem a second time, taking 

 r = 1, and for Ci a circle with center at the origin and radius so large 

 as to include within it all those points of the axis of imaginaries at 

 which an element of Ai{x) as chosen above is not analytic, and also 

 so as not to pass through a zero of | ^"(.r) |. 



In this second application of the theorem we choose Ai{x) to be 

 [}'"(.r)]~\ and in this way satisfy the restrictions of the theorem. 

 Furthermore let us take a = oo . 



Along the circle Ci we have for the solution $(.r) 



(52) ,1!^. $ (x) = [,!!^- $ (x)] [Y- (a-i)]-', 



where the approach to the point .vi of C is from without and within C 

 respectively. Now write 



Y- (x) = $ (.1-) Y- (x), 7+ (.r) = $ (x) 7+ (x) 



for X outside of Ci. It follows that Y~(x) is composed of elements 

 analytic in this region; also along Ci, Y~{x) coincides with the inner 

 determination of $(.r) by (52), so that the elements of Y-(x) are also 

 analytic within and on Ci. Hence F~(.r) is a matrix of entire functions. 

 Similar considerations show that the elements of Y+{x) are analytic 

 in the right half plane like the elements of Y+(x). 



At X = GO, Y~(x) and Y-^(x) are asymptotically represented by a 

 matrix S{x) in which however /i, ...,/•„ are not necessarily the same 

 as in S(x), and in which the determinant of the leading coefficients 

 may be zero. This results from the fact that the elements of ^(x) 

 are rational in character at x = cc and in consequence can be ex- 

 panded in convergent series in descending integral powers of x. In- 

 asmuch as we have | S (.r) | = | $ (.r) | • | S (.r) |, it is also true that 

 I S (x) I cannot reduce formally to zero. 



Finally from (51) and (52) we infer that 



(53) Y-(x)= Y^(x)P(x). 



A first conclusion to be derived is that Y~ (x) <^ S (x) in any left half 

 plane, and that also F+ (x) <^ S (x) in any right half plane. In fact 

 we have already determined the asymptotic form of P (x), and this 

 known form combined with the known asymptotic form of F+(.r) in 

 the right half plane gives us the form of Y~(x) in the part of the plane 

 to the left of any line parallel to the axis of imaginaries; a similar 

 remark applies to the asymptotic form of F+ (x) in any right half 

 plane. 



