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

Abstract

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, 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.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.(2) P(z) = £ PkZ~k P-q A 0. *=-8In case q = -1, i.e. the rank is 0, we say that (1) has a regular singular point at oo.(b) We call equations ( 1) and (1) equivalent at oo, if and only if a matrix-valued function A, holomorphic at <*> and with det ^4(oo)^0, can be found such that the substitutioncarries ( 1) into (1).C. Example.Consider the 2X2 matrix equation with a regular singular point at «, i.e.

Locations

• Proceedings of the American Mathematical Society - View - PDF