The Art of Interface

Article 11 — Appendix A.37

K — modified Bessel function of the second kind

Category. Mathematics.

Abstract. Modified Bessel function of the second kind of the real (fractional) order: definition, plot, properties and identities.

Reference. This article is a part of Librow professional formula calculator project.

See also. Iν — modified Bessel function of the first kind, Yν — Bessel function of the second kind.

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1. Definition

By definition modified Bessel function is solution of the modified Bessel equation

z2 w′′ + z w′ − (z2 + ν2) w = 0

As second order equation it has two solutions, second of which has singularity at 0 and is called modified Bessel function of the second kind — Kν. Parameter ν is called order of the function.

First solution has no singularity at 0 and is called modified Bessel function of the first kind — Iν.

2. Plot

Modified Bessel functions of the second kind defined everywhere on the real axis, at 0 functions have singularity, so, their domain is (−∞, 0)∪(0, +∞). Plots of the first three representatives of the second kind modified Bessel function family depicted below — fig. 1.

Fig. 1. Plots of the modified Bessel functions of the second kind y = K0(x), y = K1(x), y = K2(x). Fig. 1. Plots of the modified Bessel functions of the second kind y = K0(x), y = K1(x) and y = K2(x).

3. Identities

Connection to the ordinary Bessel functions:

Kν(z) = π /2 iν+1[Jν(iz) + iYν(iz)]

Connection to the modified Bessel function of the first kind:

Kν(z) = π / [2 sin(πν)] [I−ν(z) − Iν(z)]

Next order recurrence:

Kν+1(z) = 2ν /z Kν(z) + Kν−1(z)

Negative argument:

Kν(−z) = e−iπν Kν(z) − i π Iν(z) = cos(πν) Kν(z) − i [sin(πν) Kν(z) + π Iν(z)]

For the case of integer order ν=n the negative argument identity can be simplified down to:

Kn(−z) = (−1)n Kn(z) − i π In(z)

and for the case of half-integer order ν=n+1/2 the identity can be simplified down to:

Kn+1/2(−z) = −i [(−1)n Kn+1/2(z) + π In+1/2(z)]

Reflection — negative order:

K−ν(z) = Kν(z)

4. Support

Modified Bessel function of the second kind Kν of the real (fractional) order and complex argument is supported by professional version of the Librow calculator.

5. Interface

Modified Bessel function call looks like

K(order, argument);

or

BesselK(order, argument);

where order is the function real order, and argument — function argument.

6. How to use

To calculate modified Bessel function of the second kind of the 0 order of the number:

K(0, 1.5−i);

or:

BesselK(0, 1.5−i);

To calculate modified Bessel function of the second kind of the 1.2 order of the current result:

K(1.2, rslt);

or:

BesselK(1.2, rslt);

To calculate modified Bessel function of the second kind of the 2.5 order of the number z in memory:

K(2.5, mem[z]);

or:

BesselK(2.5, mem[z]);