Mathematical Sciences: RUI: Uniform Asymptotic Approximations for Special Functions

数学科学：RUI：特殊函数的一致渐近逼近

• 批准号: 9404389
• 负责人: Timothy Dunster
• 金额: $ 6.05万
• 依托单位: San Diego State University Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-09-01 至 1997-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404389&HistoricalAwards=false
• 关键词: Mathematical; Sciences; RUI; Uniform; Asymptotic

9404389 Dunster Uniform asymptotic approximations are to be derived for a number of special functions, which will be more general than previous results and include explicit error bounds. The functions to be studied will include Mathieu, Legendre, Jacobi, and spheroidal functions, for the cases where the parameters or the independent variables are too large. The general theory of a coalescing turning point and simple pole as developed by the principal; investigator will be applied to these functions. In the case of Mathieu functions the principal investigator will extend the asymptotic results that were obtained earlier to larger parameter regimes, and to use these to investigate the distribution of the eigenvalues. In addition, it is proposed to derive new representations for solutions of second-order linear differential equations with a large parameter, and the Riemann zeta function of large argument, using convergent factorial series and exponentially-improved asymptotic expansions. The expansion of special functions and their properties plays an important role in explicit solutions of many of the equations of applied mathematics. This project deals with asymptotic approximation that includes error bounds. Error bounds are especially useful in numerical computations. Potential applications of these new results include tunneling in wave physics, elasticity, high-frequency scattering, and various problems in quantum mechanics. ***

9404389 Dunster均匀渐近近似值将用于许多特殊功能，这将比以前的结果更一般，并包括显式误差界。要研究的功能将包括Mathieu，Legendre，Jacobi和Spheroidal函数，对于参数或独立变量太大的情况。校长开发的合并转折点和简单极点的一般理论；研究人员将应用于这些功能。在Mathieu功能的情况下，主要研究者将将早期获得的渐近结果扩展到较大的参数状态，并使用这些结果来研究特征值的分布。此外，建议使用收敛的阶段序列和指数增强的渐近扩展来得出具有较大参数的二阶线性微分方程解决方案的新表示。 特殊功能及其特性的扩展在许多应用数学方程的显式解决方案中起着重要作用。 该项目处理包括误差界的渐近近似。误差界在数值计算中特别有用。 这些新结果的潜在应用包括波浪物理学，弹性，高频散射以及量子力学中的各种问题。 ***