论文标题
用复杂的参数计算烧伤球波函数
Calculation of oblate spheroidal wave functions with complex argument
论文作者
论文摘要
上一篇文章表明，即使传统表达式失败，两种类型的分会球体径向函数的替代表达式可以在非常大的参数范围内提供准确的值。假定尺寸参数C真实。本文考虑了C = Cr + ICI复杂的情况，而假想部分CI通常用于表示波行行为中的损失。只要CI很小，C实际修改为复杂算术工作的方法就可以合理地进行。本文描述了获得更大的CI值的有用结果所需的实质性变化。它表明，即使用于获得c真实的矩阵方法的矩阵方法通常会越来越不准确的值，主要是随着CI的增加，即使用于c real的矩阵方法可提供越来越不准确的值，但通常可以获得准确的特征值。它还表明，某些特征值可以像以渐近的特征值替代-IC替换为-iC的渐近特征值一样近似地近似。提出了一种订购特征值的方法。讨论了精确计算复合物C的径向和角函数所需的修改。当使用双精度算术时，由此产生的FORTRAN程序COBLFCN为合理范围的C，顺序M和径向坐标提供了有用的功能值。可以通过对Bouwkamp程序进行四倍精度来确保准确的双精度特征值来改进结果。使用完整的四倍精度获得了进一步的改进。 COBLFCN可以在www.mathieuandspheroidalwavefunctions.com上免费获得。
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
