复宗量扁球体波函数的计算
Calculation of oblate spheroidal wave functions with complex argument
摘要：
以前的一篇文章表明，用于计算这两种扁球径向函数的替代表达式可以使用双精度算法在非常大的参数范围内提供准确的值，即使传统表达式失败。假设尺寸参数c为实数。本文考虑的情况下，c = cr + ici是复杂的虚部ci经常用来代表损失的波的行为。只要ci非常小，c实数修改为复数算术的方法就可以很好地工作。本文介绍了必要的实质性变化，以获得更大的CI值的有用的结果。它表明，准确的特征值通常可以获得，即使矩阵方法用于获得他们的c实数提供越来越不准确的值，主要是那些相对较小的幅度，随着ci的增加。它还表明，一些特征值可以是prolate-like的值，以及近似使用渐近估计的prolate特征值，其中c被替换为-IC。提出了一种特征值排序的方法。必要的修改，以准确地计算复杂的c的径向和角函数进行了讨论。由此产生的Fortran程序coblfcn提供了一个合理的范围内的c，m阶，和径向坐标时，使用双精度运算有用的函数值。通过对Bouwkamp方法使用四倍精度来保证双精度特征值的准确性，可以改善结果。使用全四倍精度获得进一步的改进。Coblfcn可在www.mathieuandspheroidalwavefunctions.com上免费获得
英文摘要：
A previous article showed that alternative expressions for calculating oblate spheroidal radial functions of both kinds can provide accurate values over very large parameter ranges using double precision arithmetic, even where the traditional expressions fail. The size parameter c was assumed real. This paper considers the case where c = cr + ici is complex with an imaginary part ci often used to represent losses in wave behavior. The methods for c real modified to complex arithmetic work reasonably well as long as ci is very small. This paper describes the substantial changes necessary to obtain useful results for larger values of ci. It shows that accurate eigenvalues can usually be obtained even though the matrix methods used to obtain them for c real provide increasingly inaccurate values, primarily for those with relatively small magnitude, as ci increases. It also shows that some of the eigenvalues can be prolate-like with values that are well approximated using asymptotic estimates for prolate eigenvalues where c is replaced with -ic. A method to order the eigenvalues is presented. The modifications necessary to compute accurately the radial and angular functions for complex c are discussed. A resulting Fortran program coblfcn provides useful function values for a reasonably wide range of c, the order m, and the radial coordinate when using double precision arithmetic. The results can be improved by using quadruple precision for the Bouwkamp procedure to ensure accurate double precision eigenvalues. Further improvement is obtained using full quadruple precision. Coblfcn is freely available at www.mathieuandspheroidalwavefunctions.com
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