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, non-oscillatory integral representations, asymptotic expansions and 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 non-oscillatory integral representation can be used to accurately compute the function $K_{ia}(x)$ in the range $0le ale 200$, $0le xle 100$; using a similar scheme the derivative $K_{ia}^{prime}(x)$ can also be computed within these ranges.

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CWI
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Gil, A, Segura, J, & Temme, N.M. (2001). Evaluation of the modified Bessel function of the third kind of imaginary orders. Modelling, Analysis and Simulation [MAS]. CWI.