BIRKHOFF. — THE GENERALIZED RIEMANX PROBLEM. 547 



and of determinant not zero. At x = these matrices are properly 

 equivalent to a Cauchy matrix. These further facts come at once 

 from the form of (34). 



Conversely let us prove that if Zi (x), . . . , Z^ (x) exist possessing 

 such properties, they are all matrix solutions of one and the same 

 canonical system (34).-^^ 



In the first place it is clear that the matrix 



P (.r) = X ^^ Zn-' (.r) {m=\,..., N) 



is defined in each sector r„j ^ arg x < r^+i {i=l, . . .,N) ancl hence 

 is defined in the entire plane. Now Z^+i (x) is obtained from Z^ (-i-) 

 by multiplication on the right Ijy tlie matrix 



/ + Cm, 



where Cm is a matrix of zero elements except for the single element r^ 

 in the /I'^th row and j,„th column. It appears therefore that P (.r) is 

 single- valued and analytic along each ray arg x = r^; for we have 

 along this ray 



^^^Z.,r^ (.r) = '^^^ 1/ + C4 [I + Cr.rZr,r' (x) 



_ dZrrAjc) nt _] ( \ 

 — -j Zjm \X). 



ax 



Moreover, on account of the equivalence of Zi(.v), . . . ,Zm(x) to a Cauchy 

 matrix at x = 0, the elements of xP (x) are analytic at x = 0. 



Secondly, in the neighborhood of .v = oo , tlie matrix P(x) is asymp- 

 totic to 



(39) X ^^ S-i (x), Tm^ arg.r <Tmu, 



where the meaning of the notation is manifest. It is to be recalled 

 that the relation (37) holds (by definition) in a small sector including 

 arg X = Tra as an interior ray. This enables us to write 



dZmix) dS(x) ^ _^_ 20 



dx dx 



Tm ^ arg X < Tmi-l- 



19 I stated this fact without proof earlier; see I, p. 468. 



20 Ford, loc. cit. 



