Mathematical Sciences: Error Bounds for Uniform Asymptotic Approximations

这项工作涉及对渐近分析中一些重要和长期存在的问题的误差界理论的构建。将尝试推导出具有合并鞍点的定积分的一致渐近展开式和具有复平面上两个合并拐点的微分方程的一致渐近展开式的实际误差界。指数小项在渐近展开式中的作用将被检验。还将考虑简化一些现有展开的可能性，以便在复平面上更容易计算。这项工作对物理、生物和工程科学领域的研究人员具有重要的价值，因为这些领域的许多模型方程，具有适当的缩放，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.