Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) $f^{\prime \prime }+P\left(z\right)f=0$, where $P\left(z\right)$ 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 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.

渐近积分理论给出了一系列结果，这些结果提供了对 (*) 解的渐近增长和零分布的全面描述$F^{\text{'}\text{'}}+磷\left(z\right)F=0$， 在哪里$磷\left(z\right)$是多项式。这些结果已被几位作者用来寻找 (*) 解的有趣性质。话虽如此，但许多人表示，例如，在 E. Hille 1969 年的《常微分方程讲座》一书中，关于渐近积分理论的证明和讨论很难理解。本文的主要目的是通过对用于证明该理论的推理进行完整的解释并通过对结果进行完整而清晰的陈述来使该理论更易于理解和理解。材料的相当一部分呈现和解释与希勒的书中有所不同。