Bessel-K Function Calculator

The calculator on this page calculates the modified Bessel function Kn(z) for real numbers. Bessel functions for complex numbers can be found in the complex numbers section.

To calculate, enter the ordinal number and the argument z. Then click the 'Calculate' button. The ordinal number must be an integer and the argument z must be positive.

Bessel function Kn(z) of the second kind

The Bessel-K function calculates the modified Bessel function Kn(z) of the second kind. The modified Bessel functions do not exhibit oscillating but rather exponential behavior. Bessel-K (n, z) is a solution of the modified Bessel differential equation.

The result can be a complex number if z is negative and n is a decimal number.

Plot of the Bessel-K function with ordinal numbers 0, 1 and 2

Definition of the Bessel functions

Bessel functions of the first kind (Jν)

The Bessel function of the first genus of the nth order is defined as:

\(\displaystyle J_{\nu}(z) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m! \Gamma(m + \nu + 1)} \left(\frac{z}{2}\right)^{2m + \nu} \)

Here \( \Gamma \) is the gamma function.

At the origin (\( z = 0 \)) these functions are finite for integer values of \( \nu \).
For non-integer values of \( \nu \) there are two linearly independent solutions.
For integer values of \( \nu \) the relationship holds:

\(\displaystyle J_{-\nu}(z) = (-1)^{\nu} J_{\nu}(z) \)

Bessel functions of the second genus (Yν)

The Bessel function of second genus nth order is defined as:

\(\displaystyle Y_{\nu}(z) = \frac{J_{\nu}(z) \cos(\nu \pi) - J_{-\nu}(z)}{\sin(\nu \ pi)} \)

Modified Bessel functions (Iν, Kν)

The modified Bessel function of the first kind of nth order is defined as:

\(\displaystyle I_{\nu}(z) = i^{-\nu} J_{\nu}(iz) \)

The modified Bessel function of the second type nth order is defined as:

\(\displaystyle K_{\nu}(z) = \frac{\pi}{2} \frac{I_{-\nu}(z) - I_{\nu}(z)}{\sin(\nu \pi)} \)