zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Evaluation of the modified Bessel function of the third kind of imaginary orders. (English) Zbl 0996.65026
Summary: The evaluation of the modified Bessel function of the third kind of purely imaginary order $K_{ia}(x)$ is discussed; we also present analogous results for the derivative. The methods are based on the use of Maclaurin series, nonoscillatory integral representations, asymptotic expansions, an a continued fraction method, depending on the ranges of $x$ and $a$. We discuss the range of applicability of the different approaches considered and conclude that power series, the continued fraction method, and the nonoscillatory integral representation can be used to accurately compute the function $K_{ia}(x)$ in the range $0\le a\le 200$, $0\le x\le 100$; using a similar scheme the derivative $K_{ia}'(x)$ can also be computed within these ranges.

MSC:
65D20Computation of special functions, construction of tables
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
33F05Numerical approximation and evaluation of special functions
WorldCat.org
Full Text: DOI
References:
[1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions. (1972) · Zbl 0543.33001
[2] Boris, J. P.; Oran, E. S.: Numerical evaluation of oscillatory integrals such as the modified Bessel function $Ki{\zeta}$(x). J. comput. Phys. 17, 425 (1975) · Zbl 0302.65016
[3] Closas, Ll.; Rubio, J. Fernández: Calculo rapido de las funciones de Bessel modificadas $Iis(X)$ y sus derivadas. Stochastica 11, 53 (1987)
[4] Cody, W. J.; Hillstrom, K. E.: Chebyshev approximations for the Coulomb phase shift. Math. comput. 24, 671 (1970) · Zbl 0225.65018
[5] Dunster, T. M.: Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. anal. 21, 1594 (1990) · Zbl 0722.34051
[6] Dunster, T. M.: Conical functions with one or both parameters large. Proc. rov. Soc. Edinburgh 119A, 311 (1991) · Zbl 0736.33002
[7] Ehrenmark, U. T.: The numerical inversion of two classes of kontorovich--lebedev transform by direct quadrature. J. comput. Appl. math. 61, 43 (1995) · Zbl 0841.65124
[8] Gautschi, W.: Computational aspects of three-term recurrence relations. SIAM rev. 9, 24 (1967) · Zbl 0168.15004
[9] Kerimov, M. K.; Skorokhodov, S. L.: Calculation of modified Bessel functions in the complex domain. Comput. math. Math. phys. 25, 15 (1984) · Zbl 0568.65010
[10] Kerimov, M. K.; Skorokhodov, S. L.: Zh. vychisl. Mat. mat. Fiz.. 24, 650 (1984)
[11] Kiyono, T.; Murashima, S.: A method of evaluation of the function $Kis(x)$. Mem. fac. Eng. Kyoto univ. 35, 102 (1973)
[12] Lear, J. D.; Sturm, J. E.: An integral representation for the modified Bessel function of the third kind, computable for large, imaginary order. Math. comput. 21, 496 (1967) · Zbl 0155.22202
[13] Lebedev, N. N.: Special functions and their applications. (1972) · Zbl 0271.33001
[14] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974; reprinted by, A. K. Peters, 1997. · Zbl 0303.41035
[15] R. Piessens, and, E. De Doncker, subroutine DQAGIE (obtained from SLATEC Public Domain Library).
[16] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P.: Numerical recipes in Fortran 77. (1992) · Zbl 0778.65002
[17] Rappoport, J. M.: Canonical vector polynomials for the computation of the complex order Bessel functions with the tau method. Comput. math. Appl. 41, 399 (2001) · Zbl 0986.65015
[18] Temme, N. M.: Special functions: an introduction to the classical functions of mathematical physics. (1996) · Zbl 0856.33001
[19] Temme, N. M.: Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Meth. appl. Anal. 1, 14 (1994) · Zbl 0848.33003
[20] Temme, N. M.: On the numerical evaluation of the modified Bessel function of the third kind. J. comput. Phys. 19, 324 (1975) · Zbl 0334.65013
[21] Thompson, I. J.; Barnett, A. R.: Modified Bessel functions $I{\nu}(z)$ and $K{\nu}(z)$ of real order and complex argument, to selected accuracy. Comput. phys. Commun. 47, 245 (1987)
[22] Weniger, E. J.; Čı\acute{}žek, J.: Rational approximations for the modified Bessel functions of the second kind. Comput. phys. Commun. 59, 471 (1990) · Zbl 0875.65036