The modified bessel function of the second kind is the function which is one of the solutions to the modified
Bessel differential equation . The modified Bessel functions of the second kind
are sometimes called the Basset functions, modified Bessel functions of the third
kind (Spanier and Oldham 1987, p. 499), or Macdonald functions (Spanier and
Oldham 1987, p. 499; Samko et al. 1993, p. 20). The modified Bessel
function of the second kind is implemented in the Wolfram
Language as BesselK [nu ,
z ].
is closely related to the modified
Bessel function of the first kind and Hankel function ,
(Watson 1966, p. 185). A sum formula for is
(4)
where is the digamma function
(Abramowitz and Stegun 1972). An integral formula is
(5)
which, for , simplifies to
(6)
Other identities are
(7)
for and
The special case of gives as the integrals
(Abramowitz and Stegun 1972, p. 376).
See also Bessel Function of the Second Kind ,
Continued Fraction
Constants ,
Modified Bessel
Function of the First Kind
Related Wolfram sites http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/
Explore with Wolfram|Alpha
References Abramowitz, M. and Stegun, I. A. (Eds.). "Modified Bessel Functions and ." §9.6 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 374-377, 1972. Arfken, G. "Modified Bessel
Functions, and ." §11.5 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 610-616,
1985. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.;
and Vetterling, W. T. "Modified Bessel Functions of Integral Order"
and "Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel
Functions." §6.6 and 6.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 229-245, 1992. Samko, S. G.;
Kilbas, A. A.; and Marichev, O. I. Fractional
Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 20,
1993. Spanier, J. and Oldham, K. B. "The Basset ." Ch. 51 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 499-507, 1987. Watson,
G. N. A
Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge
University Press, 1966. Referenced on Wolfram|Alpha Modified Bessel
Function of the Second Kind
Cite this as:
Weisstein, Eric W. "Modified Bessel Function of the Second Kind." From MathWorld --A Wolfram Web Resource.
https://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html
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