Mathematical Sciences: RUI: Uniform Asymptotic Approximations for Special Functions

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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. ***