P(.r) = 



BIRKHOPF. — THE GENERALIZED RIEMANN PROBLEAr. 543 



1, 0, 



^- 1 



[x — am)'- 



"""" — 7, 0, 1 



(x — a„, 



Inasmuch as | P(.v) | = 1, and the elements of P[.v) are analytic save 

 at .V = a,n, this modification cannot affect any of the properties already 

 secured for Y(x). However by choosing co, . . ., c^ properly we can 

 clearly make all the coefficients cioi, . . . o„j vanish at x = a^ pro- 

 vided fi 9^ 0. In this manner we can increase an exponent hi by 1 

 without altering the determinant of Y{x) except by a constant factor. 



Inasmuch as ] Y{x) \ does not vanish identically, a succession of 

 steps of this type will finally bring to light a solution Y {x) for which 

 Y{x) C has the form (33) and in addition | «,_,• | ^ 0. When this 

 stage is reached Y(x) will be properly equivalent to a Cauchy matrix 

 belonging to T^ at the point 0^- 



When Y{x) has thus been given a normal form, it is the solution 

 of a linear differential system (24) with regular singular points at 

 X = ai,...,an and having no other singular points, as may be at once 

 proved. ^^ The elements of R{x) therefore have the form of rational 

 functions whose numerators are polynomials in x of degree at most m — 2, 

 and whose denominators are the product of {x — Oi), . . ., {x — Um). 



§ 12. Irregular Singular Points and Canonical Systems. 



The Cauchy matrix is the simplest possible matrix of fvmctions to 

 which a matrix solution of a given linear differential system is properly 

 equi\alent at a regular singular point. Let us determine the simplest 

 possible matrix Z{x) to which the matrix solution Y(x) of a given linear 

 differential system (24), in which the elements of R(x) need not be 

 rational, is properly equivalent at a prescribed irregular singular point. 

 It is convenient to take this point to lie at infinity. If the highest 

 order of any pole of an element of xR{x) at x = go is ;:> + 1 (p~ 0), 

 then ;>+ 1 is said to be the rank of the singular point x = oo. 



According to the results of § 10 we can find a matrix Z{x) which at 

 X = CO is properly equivalent to Y (x) and at another point .r = 



13 Schlesinger, loc. cit. pp. 21.5-221. 



