Modified Bessel Function of the Second Kind

The function $K_n(x)$ which is one of the solutions to the Modified Bessel Differential Equation. The above plot shows $K_n(x)$ for $n=1$, 2, ..., 5. $K_n(x)$ is closely related to the Modified Bessel Function of the First Kind $I_n(x)$ and Hankel Function $H_n(x)$,

$$K_n(x) \equiv {\textstyle{1\over 2}}\pi i^{n+1}H_n^{(1)}(ix)$$ (1)

$$= {\textstyle{1\over 2}}\pi i^{n+1}[J_n(ix)+iN_n(ix)]$$ (2)

$$= {\pi\over 2} {I_{-n}(x)-I_n(x)\over\sin(n\pi)}$$ (3)

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

$K_n(z)={\textstyle{1\over 2}}({\textstyle{1\over 2}}z)^{-n}\sum_{k=0}^{n-1} {(n... ...k!} (-{\textstyle{1\over 4}}z^2)^k+(-1)^{n+1}\ln({\textstyle{1\over 2}}z)I_n(z)$
$+(-1)^n {\textstyle{1\over 2}}({\textstyle{1\over 2}}z)^n \sum_{k=0}^\infty [\psi(k+1)+\psi(n+k+1)]{({\textstyle{1\over 4}}z^2)^k\over k!(n+k)!},\quad$	(4)

where $\psi$ is the Digamma Function (Abramowitz and Stegun 1972). An integral formula is

$$K_\nu(z) = {\Gamma(\nu+{\textstyle{1\over 2}})(2z)^\nu\over\sqrt{\pi}} \int_0^\infty {\cos t\,dt\over(t^2+z^2)^{\nu+1/2}}$$ (5)

which, for $\nu=0$, simplifies to

$$K_0(x) = {\int_0^\infty \cos(x\sinh t)\,dt} = {\int_0^\infty {\cos(xt)\,dt\over \sqrt{t^2+1}}}.$$ (6)

Other identities are

$$K_n(z) = {\sqrt{\pi}\over(n-{\textstyle{1\over 2}})!}({\textstyle{1\over 2}}z)^n \int^\infty_1 e^{-zx}(x^2-1)^{n-1/2}\,dx$$ (7)

for $n > -1/2$ and

$$K_n(z) = \sqrt{\pi\over 2z} {e^{-z}\over (n-{\textstyle{1\over 2}})!} \int_0^\infty e^{-t}t^{n-1/2}\left({1-{t\over 2z}}\right)^{n-1/2} \,dt$$ (8)

$$= \sqrt{\pi\over 2z} {e^{-z}\over (n-{\textstyle{1\over 2}})!} \sum... ...r r!(n-r-{\textstyle{1\over 2}})!}(2z)^{-r}\int^\infty_0 e^{-t}t^{n+r-1/2}\,dt.$$ (9)

The modified Bessel function of the second kind is sometimes called the Basset Function.

References

Abramowitz, M. and Stegun, C. 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.

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.