BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 549 



where x in A^ (•»") is taken along arg x = r'„, and the determinations 

 of T(x) chosen are obtained from one another by allowing arg x to 

 increase from ti to ry. The matrix A^ (x) thus defined is analytic 

 along its line save at x = 0. Also A^ (x) — / is a matrix whose only 

 non-zero element is 



Now we have 



Pkm i'V) — Pirn (-0 = {o-km — 0.jj ' -^ + {^km — ^Jm)~^ 



+ . . . + {\km — ^jm)x, 



which quantity by definition of j,„, /■•,„ has a tiegativc real part for x 

 along the ray arg .r = t',„, at least when | x \ is sufficiently large. In 

 consequence of the form of this non-zero element it is certain that 

 Afn (x) is unlimitedly difTerentiable along the ray at x = Qo and be- 

 haves there like the matrix /. As this is true along each of the rays, 

 the matrices A,n (■''-') clearly satisfy the conditions of the theorem in the 

 vicinity of .c = c^. Choose .4i(.r),. . ., Aiffix) along the straight lines 

 Ci,..., C,v/2 as equal to the corresponding matrix Ajn(x) along either 

 of its component rays outside of some circle | .r | = r, within which 

 they are chosen so as to satisfy the conditions of the preliminary 

 theorem. Since Ai (x), . . . , A}^ (x) are of the nature above described 

 at .r = CO, the matrices Ai (.r),. . .,Aj^/o (x) are unlimitedly differen- 

 tiable, and satisfy the permutability condition (see (21)) of the pre- 

 liminarv theorem. It is therefore possible to make such a choice 

 'oiA,(x),...At;/2{x).^' 



It follows by the theorem that there exists a matrix $(.i-) of determi- 

 nant nowhere zero save possiljly at .r = a = 0, analytic except along 

 the rays arg x = tJ and such that 



(42) ■ JZ^ * (.r) = UZ~ * (-^O] Am (.r„0, 



where .r„i is a point of arg .t = r^' for which [ x„i |^ /•. It also follows 

 from the theorem that each element of ^(x) is represented as^miptoti- 

 cally by a series in negative powers of x in each sector (t'^, t'^+i) ', 

 since Aj^ (x) o^ I, these series are the same in all the sectors and we may 

 write 



(43) $(.r) c. (sij{x)), 



where Sij(x) is a series of negative powers of x in which the determinant 

 of the constant terms is not zero. 



22 Cf. § 9. 



