Mathematical Sciences: RUI: Uniform Asymptotic Approximations for Special Functions

小行星9404389邓斯特 一致渐近近似是由一些特殊的功能，这将是更普遍的比以前的结果，包括明确的误差界。要研究的功能将包括马蒂厄，勒让德，雅可比和球函数，对于参数或自变量太大的情况下。一般理论的合并转折点和简单的极点作为开发的主要;调查将适用于这些功能。在马蒂厄函数的情况下，主要研究者将把先前获得的渐近结果扩展到更大的参数区域，并使用这些结果来研究特征值的分布。此外，利用收敛的阶乘级数和指数改进的渐近展开式，给出了二阶大参数线性微分方程解的新表示和大辐角Riemann zeta函数的新表示. 特殊函数的展开式及其性质在许多应用数学方程的显式求解中起着重要的作用。 这个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动物理中的隧穿、弹性、高频散射和量子力学中的各种问题。 ***

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. ***