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.

Bessel and Airy functions, cylinder functions, ${}_0F_1$ (msc 33C10), Computation of special functions, construction of tables (msc 65D20), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60), Quadrature and cubature formulas (msc 65D32)
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.