Asymptotics of Solutions of Linear Differential Equations with Holomorphic Coefficients in the Neighborhood of an Infinitely Distant Point

Maria Korovina

doi:10.3390/math8122249

Abstract

This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare problem on constructing the asymptotics of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhoods of irregular singular points. In this study we consider such equations for which the principal symbol of the differential operator has multiple roots. The asymptotics of a solution for the case of equations with simple roots of the principal symbol were constructed earlier.

Highlights

One of the fundamental problems of the analytic theory of ordinary differential equations with holomorphic coefficients is the problem of constructing the asymptotics of their solutions in the neighborhood of irregular singular points
Poincare has proved that the obtained divergent series are asymptotic and the idea was formulated that summation of the asymptotic series obtained can be performed by using an integral transform; in a particular case, it may be the Laplace transform
As a result of Theorem 1, we have constructed the uniform asymptotics of solutions of ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant point for the equations of the first type

Summary

Introduction

One of the fundamental problems of the analytic theory of ordinary differential equations with holomorphic coefficients is the problem of constructing the asymptotics of their solutions in the neighborhood of irregular singular points. This problem was formulated by Poincare in [1,2]. The problem of constructing the asymptotics of solutions for differential equations with holomorphic coefficients in the neighborhood of an infinitely distant point formulated by Poincare is still unsolved in the general case. Our study is devoted to solving this problem for a broad class of differential equations with holomorphic coefficients

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Conclusion