Asymptotic integration theory for $f'' + P(z)f = 0$

TL;DR

This paper clarifies and simplifies the asymptotic integration theory for second-order linear differential equations with polynomial coefficients, making the complex proofs and results more accessible and understandable.

Contribution

It provides complete explanations and clearer statements of the asymptotic integration theory, improving accessibility compared to previous sources like Hille's book.

Findings

Thorough description of solution growth and zero distribution

Simplified proofs and explanations of asymptotic integration results

Enhanced understanding of polynomial coefficient differential equations

Abstract

Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) $f''+P(z)f=~0$, where $P(z)$ is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E.~Hille's 1969 book \emph{Lectures on Ordinary Differential Equations} are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results. A considerable part of the presentation and explanation of the material is different from that in Hille's book.