Mathematical Sciences: RUI: Uniform Asymptotic Approximations for Special Functions

9404389将得到一些特殊函数的邓斯特一致渐近近似，它将比以前的结果更具一般性，并且包含显式的误差界。对于参数或自变量太大的情况，要研究的函数包括Mathieu、Legendre、Jacobi和球面函数。由校长发展的关于结合转折点和简单极点的一般理论；调查者将应用于这些函数。在Mathieu函数的情况下，主要研究者将把先前获得的渐近结果推广到更大的参数区间，并使用这些结果来研究本征值的分布。此外，利用收敛的阶乘级数和指数改进的渐近展开式，我们还提出了大参数二阶线性微分方程解的新表示以及大自变量的Riemann Zeta函数。特殊函数的展开式及其性质在许多应用数学方程的显式解中起着重要的作用。这个项目涉及包含误差界的渐近逼近。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物理中的隧道效应、弹性、高频散射以及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. ***