Mathematical Sciences: Error Bounds for Uniform Asymptotic Approximations

这项工作涉及针对渐近分析中许多重要且长期存在的问题构建误差界限理论。 将尝试导出具有合并鞍点的定积分的一致渐近展开的实际误差界，以及具有复平面中两个合并转折点的微分方程的一致渐近展开的实际误差界。 将研究渐近展开中指数小项的作用。 还将考虑简化一些现有展开式的可能性，以便它们可以更容易地在复平面中计算。 这项工作对于物理、生物和工程科学领域的研究人员来说应该具有很大的价值，因为这些领域的许多模型方程在适当的缩放下都涉及或大或小的参数。

This work involves the construction of error-bound theories for a number of important and long-standing problems in asymptotic analysis. Attempts will be made to derive realistic error bounds for uniform asymptotic expansions of definite integrals with coalescing saddle points, and for uniform asymptotic expansions of differerential equations with two coalescing turning points in the complex plane. The role of exponentially small terms in asymptotic expansions will be examined. The possibility of simplifying some existing expansions so they may be more readily computed in the complex plane will also be considered. This work should be of great value to researchers in the physical, biological, and engineering sciences where many of the model equations, with proper scaling, involve large or small parameters.