On a Result of G. D. Birkhoff on Linear Differential Systems

We give a simple example to show that a result on the equivalent singular points of systems of ordinary linear differential equations due to G. D. Birkhoff [3; 5, pp. 252-257] needs amendment. In matrix notation, in which Y, P, etc. denote nXn matrix-valued functions of a complex variable z, this result is as follows. A. Result. Every linear differential system (1) Y'(z) = P(z)Y(z) with a singular point of rank q + l at z= oo (q^—l) is equivalent at z= oo to a canonical system (!) Y'(z) = P(z)Y(z) in which zP(z) is a polynomial of degree less than or equal to q + l. B. Definitions.1 (a) The equation (1) is said to have a singular point of rank q +1 at «°, if and only if the function P has a pole of order q at oo, i.e.