Dunster, T. M. (2024). Simplified uniform asymptotic expansions for associated Legendre and conical functions. arXiv preprint arXiv:2410.03002.
This paper aims to derive simplified asymptotic expansions for associated Legendre and conical functions when either the degree (ν) or order (µ) is large. The goal is to provide expansions that are uniformly valid for a wide range of arguments, including at the singularity z=1, and have coefficients that are easy to compute.
The author utilizes Liouville-Green (LG) type expansions of exponential form to construct asymptotic expansions for the coefficient functions appearing in the solutions. This approach leads to simple expressions for the coefficients, involving recursively defined polynomials and the LG variable. The expansions are expressed in terms of modified Bessel functions and their analytic continuations.
The paper provides a new set of asymptotic expansions for associated Legendre and conical functions that are both accurate and computationally efficient. These expansions are uniformly valid for a wide range of arguments, overcoming limitations of previous approximations. The simplicity of the coefficient formulas makes these expansions particularly useful for numerical computations involving these functions.
This research contributes significantly to the field of special functions by providing improved asymptotic expansions for Legendre and conical functions. These functions have numerous applications in physics, engineering, and other scientific disciplines. The new expansions offer enhanced accuracy and computational efficiency, benefiting various applications involving these functions.
The paper focuses on the case where one parameter (ν or µ) is large while the other is bounded. Future research could explore expansions for cases where both parameters are large simultaneously, which presents a more complex scenario with coalescing turning points and poles. Additionally, extending the analysis to derive explicit error bounds for the new expansions would be beneficial.
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by T. M. Dunste... at arxiv.org 10-07-2024
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