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Numerical calculation of Bessel, Hankel and Airy functions

机译：贝塞尔，汉克尔和艾里函数的数值计算

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摘要

The numerical evaluation of an individual Bessel or Hankel function of large order and large argument is a notoriously problematic issue in physics. Recurrence relations are inefficient when an individual function of high order and argument is to be evaluated. The coefficients in the well-known uniform asymptotic expansions have a complex mathematical structure which involves Airy functions. For Bessel and Hankel functions, we present an adapted algorithm which relies on a combination of three methods: (i) numerical evaluation of Debye polynomials, (ii) calculation of Airy functions with special emphasis on their Stokes lines, and (iii) resummation of the entire uniform asymptotic expansion of the Bessel and Hankel functions by nonlinear sequence transformations. In general, for an evaluation of a special function, we advocate the use of nonlinear sequence transformations in order to bridge the gap between the asymptotic expansion for large argument and the Taylor expansion for small argument ("principle of asymptotic overlap"). This general principle needs to be strongly adapted to the current case, taking into account the complex phase of the argument. Combining the indicated techniques, we observe that it possible to extend the range of applicability of existing algorithms. Numerical examples and reference values are given.
机译：单个贝塞尔函数或汉克尔函数的高阶和大论点的数值评估是物理学中一个臭名昭著的问题。当要评估高阶和自变量的单个函数时，递归关系效率很低。众所周知的均匀渐近展开式中的系数具有复杂的数学结构，其中涉及艾里函数。对于Bessel和Hankel函数，我们提出一种经过改进的算法，该算法依赖于以下三种方法的组合：（i）Debye多项式的数值求值，（ii）特别关注其Stokes线的Airy函数的计算，以及（iii）求和贝塞尔和汉克尔函数通过非线性序列变换的整体均匀渐近展开。通常，为了评估特殊函数，我们提倡使用非线性序列变换来弥合大参数的渐近展开与小参数的泰勒展开之间的差距（“渐进重叠原理”）。考虑到论证的复杂阶段，需要对该通用原则进行严格调整以适应当前情况。结合指示的技术，我们观察到有可能扩展现有算法的适用范围。给出了数值示例和参考值。

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