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K — modified Bessel function of the second kind

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. Graph

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

Fig. 1. Graphs y = K0(x), y = K1(x), y = K2(x). Fig. 1. Graphs 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. Derivative

Modified Bessel function of the second kind derivative:

K′ν(z) = −Kν−1(z) − ν Kν(z) /z K′ν(z) = ν Kν(z) /z − Kν+1(z) K′ν(z) = −[Kν−1(z) + Kν+1(z)] /2

In particular:

K′0(z) = −K1(z)

5. Interface

Modified Bessel function call looks like

K(ν, z);

where ν is the function real order, and z — function complex 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);

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

K(1.2, rslt);

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

K(2.5, mem[z]);

7. Support

Modified Bessel function of the second kind of real order and complex argument is supported in professional version of the Librow calculator.