Mathematical Sciences: Error Bounds for Uniform Asymptotic Approximations

这项工作涉及为渐近分析中的一些重要和长期存在的问题构建误差界理论。我们将尝试得到复平面上具有凝聚鞍点的定积分的一致渐近展开式和具有两个凝聚转向点的微分方程解的一致渐近展开式的实际误差界。我们将考察指数小项在渐近展开中的作用。还将考虑简化一些现有展开式的可能性，以便在复杂的平面上更容易地计算它们。这项工作应该对物理、生物和工程科学的研究人员有很大的价值，这些科学中的许多模型方程，在适当的尺度下，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.