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Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion. (English) Zbl 1026.65015
Summary: The uses 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 The authors [J. Comput. Phys. 175, No. 2, 398-411 (2002; Zbl 0996.65026)] presented an algorithm for the evaluation of K ia (x), 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.
65D20Computation of special functions, construction of tables
33C10Bessel and Airy functions, cylinder functions, 0 F 1
33F05Numerical approximation and evaluation of special functions