Evaluation of the Modified Bessel Function of the Third Kind of Imaginary Orders

The evaluation of the modified Bessel function of the third kind of purely imaginary order Kia(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, 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 nonoscillatory integral representation can be used to accurately compute the function Kia(x) in the range 0≤a≤200, 0≤x≤100; using a similar scheme the derivative K′ia(x) can also be computed within these ranges.