Computation of the modified Bessel function of the third kind of imaginary orders: uniform Airy-type asymptotic expansion

The use of a uniform Airy-type asymptotic expansion for the computation of the modified Bessel functions of the third kind of imaginary orders (Kia(x)) near the transition point x=a, is discussed. In A. Gil et al., Evaluation of the modified Bessel functions of the third kind of imaginary orders, J. Comput. Phys. 17 (2002) 398–411, an algorithm for the evaluation of Kia(x) was presented, which made use of series, a continued fraction method and nonoscillating 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.