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Evaluation of the modified Bessel function of the third kind of imaginary orders. (English) Zbl 0996.65026
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 $a$. 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:
 65D20 Computation of special functions, construction of tables 33C10 Bessel and Airy functions, cylinder functions, ${}_0F_1$ 33F05 Numerical approximation and evaluation of special functions
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##### References:
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