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Zbl 1026.65015
Gil, Amparo; Segura, Javier; Temme, Nico M.
Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion. (English)
[J] J. Comput. Appl. Math. 153, No.1-2, 225-234 (2003). ISSN 0377-0427

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.

MSC 2000:: *65D20 Computation of special functions
33C10 Cylinder functions, etc.
33F05 Numerical approximation of special functions

Keywords: Bessel functions; nonoscillatory integral representations; uniform Airy-type asymptotic expansions; continued fraction method; steepest descent method

Citations: Zbl 0996.65026

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