关于弗雷斯涅尔 regime 中贝塞尔函数渐进行为的研究
On the asymptotics of Bessel functions in the Fresnel regime
AI总结：
本文提出了一种适用于深弗雷斯涅尔 regime 的贝塞尔函数渐进行为展开方法，通过非振荡相位函数的分析，为高效计算贝塞尔函数及相关量提供了理论和数值方案。
摘要：
我们介绍了一种适用于 |z| > ν（即深弗雷斯涅尔 regime）的贝塞尔函数 J_ν(z)、Y_ν(z) 的渐进行为展开方法，与仅适用于弗劳恩霍弗 regime（|z| > ν²）的传统展开不同。正如预期的那样，在弗劳恩霍弗 regime 中，我们的渐进行为退化为经典结果。该方法基于贝塞尔方程允许非振荡相位函数的观察，并利用经典公式获得该函数的渐进行为展开；这反过来又为高效计算 J_ν(z)、Y_ν(z) 以及各种相关量提供了分析工具和数值方案。通过多个数值示例展示了该技术的有效性。我们还观察到，该过程可以进一步推广到广泛的二阶微分方程类中，将在以后报告。
英文摘要：
We introduce a version of the asymptotic expansions for Bessel functions $J_ν(z)$, $Y_ν(z)$ that is valid whenever $|z| > ν$ (which is deep in the Fresnel regime), as opposed to the standard expansions that are applicable only in the Fraunhofer regime (i.e. when $|z| > ν^2$). As expected, in the Fraunhofer regime our asymptotics reduce to the classical ones. The approach is based on the observation that Bessel's equation admits a non-oscillatory phase function, and uses classical formulas to obtain an asymptotic expansion for this function; this in turn leads to both an analytical tool and a numerical scheme for the efficient evaluation of $J_ν(z)$, $Y_ν(z)$, as well as various related quantities. The effectiveness of the technique is demonstrated via several numerical examples. We also observe that the procedure admits far-reaching generalizations to wide classes of second order differential equations, to be reported at a later date.
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