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Modified Bessel Function of the Second Kind

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BesselK

The modified Bessel function of the second kind is the function K_n(x) 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].

K_n(x) is closely related to the modified Bessel function of the first kind I_n(x) and Hankel function H_n(x),

K_n(x)=1/2pii^(n+1)H_n^((1))(ix)
(1)
=1/2pii^(n+1)[J_n(ix)+iN_n(ix)]
(2)
=pi/2(I_(-n)(x)-I_n(x))/(sin(npi))
(3)

(Watson 1966, p. 185). A sum formula for K_n(x) is

K_n(z)=1/2(1/2z)^(-n)sum_(k=0)^(n-1)((n-k-1)!)/(k!)(-1/4z^2)^k+(-1)^(n+1)ln(1/2z)I_n(z)+(-1)^n1/2(1/2z)^nsum_(k=0)^infty[psi(k+1)+psi(n+k+1)]((1/4z^2)^k)/(k!(n+k)!),
(4)

where psi is the digamma function (Abramowitz and Stegun 1972). An integral formula is

K_nu(z)=(Gamma(nu+1/2)(2z)^nu)/(sqrt(pi))int_0^infty(costdt)/((t^2+z^2)^(nu+1/2))
(5)

which, for nu=0, simplifies to

K_0(x)=int_0^inftycos(xsinht)dt=int_0^infty(cos(xt)dt)/(sqrt(t^2+1)).
(6)

Other identities are

K_n(z)=(sqrt(pi))/((n-1/2)!)(1/2z)^nint_1^inftye^(-zx)(x^2-1)^(n-1/2)dx
(7)

for n>-1/2 and

K_n(z)=sqrt(pi/(2z))(e^(-z))/((n-1/2)!)int_0^inftye^(-t)t^(n-1/2)(1-t/(2z))^(n-1/2)dt
(8)
=sqrt(pi/(2z))(e^(-z))/((n-1/2)!)sum_(r=0)^(infty)((n-1/2)!)/(r!(n-r-1/2)!)(2z)^(-r)int_0^inftye^(-t)t^(n+r-1/2)dt.
(9)
BesselK0ReIm
BesselK0Contours

The special case of n=0 gives K_0(z) as the integrals

K_0(z)=int_0^inftycos(zsinht)dt
(10)
=int_0^infty(cos(zt))/(sqrt(t^2+1))dt
(11)

(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

https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Modified Bessel Functions I and K." §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, I_nu(x) and K_nu(x)." §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 K_nu(x)." 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 Resource. https://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html

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