Birkhoff invariants and effective calculations for meromorphic linear differential equations. I
type: Item
instance of: scholarly article
title: Birkhoff invariants and effective calculations for meromorphic linear differential equations. I
zbMATH Open document ID: 0358.34007
DOI: 10.1016/0022-247X(76)90122-0
author: W. B. Jurkat
author: A. Peyerimhoff
author: Donald A. Lutz
published in: Journal of Mathematical Analysis and Applications
publication date: 1976-01-01T00:00:00Z
review text: The authors consider systems of meromorphic linear differential equations of first order, namely (1) \(x' =A(z)x\) with \[ A(z)=z^{r-1}\sum_{0}^{\infty}A_{\nu}z^{-\nu},\; r\geq 0, A_0\neq 0 \] where the \(A_{\nu}\), \(s\) are constant matrices of dimension \(2 \times 2\) and the power series converges for \(|z| >R\). They give a complete list of canonical forms into which (1) can be reduced by a meromorphic or analytic transformation \(x=T(z)y\). They also give a procedure for calculating \(T(z)\) which reduces a given system to a canonical form. The importance of the results is that the complete information about the solution can be obtained from the computable transformation and the knowledge of solutions of some special equations. Basic results are stated as follows: (i) There exists a meromorphic transformation \(x=T(z)y\) at \(\infty\) such that \(y'=B(z)y\) with \(B(z)\) given by \[ (2) \qquad \qquad B(z)=z^{r-1}\sum_{\nu=0}^{r}B_{\nu}z^{-\nu}, \] (ii) There exists an analytic transformation \(x=T(z)y\) at \(\infty\) such that \(y'=B(z)y\) with \(B(z)\) given by (2) or \[ B(z)=z^{r-1}\sum_{\nu =0}^{r}B_{\nu}z^{-\nu}+z^{-1-k} \left[ \begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix} \right] , \] where \(k\) is a positive integer and \(B_{\nu}\) are lower triangular. Throughout the paper it is assumed that \(r=1\) and \(A_0\) has distinct eigenvalues, and a complete listing is made of canonical forms which need to be solved in order to represent the singularities of solutions of the general equations. A system of Birkhoff invariants are introduced and computed from a formal solution. Given a system of Birkhoff invariants, the authors show also how to explicitly construct a list of all special examples having this set of invariants.
Mathematics Subject Classification ID: 34M05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 3558226
reviewed by: Mithat Idemen
MaRDI profile type: Publication
cites work: Q5848416
cites work: Q5847356
cites work: Q3254327
cites work: On the Order of Solutions of Analytic Linear Differential Equations
cites work: On a Result of G. D. Birkhoff on Linear Differential Systems
cites work: Reduction of Ordinary Differential Equations to the Birkhoff Canonical Form
description: scientific article; zbMATH DE number 3558226