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Zbl 0996.65026
Gil, Amparo; Segura, Javier; Temme, Nico M.
Evaluation of the modified Bessel function of the third kind of imaginary orders. (English)
[J] J. Comput. Phys. 175, No.2, 398-411 (2002). ISSN 0021-9991

Summary: The evaluation of the modified Bessel function of the third kind of purely imaginary order $K_{ia}(x)$ is discussed; we also present analogous results for the derivative. The methods are based on the use of Maclaurin series, nonoscillatory integral representations, asymptotic expansions, an a continued fraction method, depending on the ranges of $x$ and $a$.\par We discuss the range of applicability of the different approaches considered and conclude that power series, the continued fraction method, and the nonoscillatory integral representation can be used to accurately compute the function $K_{ia}(x)$ in the range $0\le a\le 200$, $0\le x\le 100$; using a similar scheme the derivative $K_{ia}'(x)$ can also be computed within these ranges.

MSC 2000:: *65D20 Computation of special functions
33C10 Cylinder functions, etc.
33F05 Numerical approximation of special functions

Keywords: Bessel functions; series expansions; continued fraction; nonoscillatory integral representations; asymptotic expansions; Maclaurin series

Cited in: Zbl 1026.65015

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