Modified Bessel Functions of the Second Kind: K

The functions \( K_0(x) \) and \( K_1(x) \) are the second solutions of the modified Bessel equation

\[ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - (x^2 + \nu^2)\, y = 0, \]

decaying exponentially with \( x \). They can be written as

\[ K_\nu(x) = \frac{\pi}{2}\,\frac{I_{-\nu}(x) - I_\nu(x)}{\sin(\nu\pi)}, \]

with the large- \( x \) asymptotic \( K_\nu(x) \sim \sqrt{\pi/(2x)}\, e^{-x} \).

Function shape

K0 and K1

Domain and special values

\( x \)	\( K_0(x) \) / \( K_1(x) \)
\( x \le 0 \)	NaN (undefined for \( x<0 \), diverges as \( x\to 0^+ \))
\( x \to 0^+ \)	\( +\infty \)
\( +\infty \)	\( 0 \)

Usage

use bessels, only: besselk0, besselk1, bk
 
real(BK) :: x(2), y(2)
x = [0.5_bk, 2.0_bk]
 
y = besselk0(x)            ! elemental over the array
print *, besselk1(2.0_bk)

Accuracy and performance

besselk0/besselk1 use two branches with rational (boost-style, Holoborodko) approximations: \( x \le 1 \) uses the \( \log x \cdot I_\nu(x) \) correction form, and \( x > 1 \) a scaled \( \sqrt{x}\,e^{x} K_\nu(x) \) rational form. Against the netlib RKBESL reference they are several times faster (~6 ns/eval vs ~27–44 ns/eval) at matching accuracy:

K0 / K1 relative error vs reference

See also

Modified Bessel Functions of the First Kind: I

Bessel Functions of the Second Kind: Y

fortran-bessels