The use of a uniform Airy-type asymptotic expansion for the computation of the modified Bessel functions of the third kind of imaginary orders ($K_{ia}(x)$) near the transition point $x=a$, is discussed. In [2], an algorithm for the evaluation of $K_{ia}(x)$ was presented, which made use of series, a continued fraction method and non-oscillating integral representations. The range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point. We show how uniform Airy-type asymptotic expansions fill the gap left by the steepest descent method.

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CWI
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Gil, A, Segura, J, & Temme, N.M. (2002). Computation of the modified Bessel function of the third kind of imaginary orders: uniform Airy-type asymptotic expansion. Modelling, Analysis and Simulation [MAS]. CWI.