modified_bessel_k0

→Tensor

modified_bessel_k0(x: Tensor)

source

Modified Bessel function of the second kind, order 0.

Computes $K_0(x)$ , the exponentially decaying solution of the modified Bessel equation at order 0. Appears as the Green's function of the 2D Helmholtz operator and as the log-density kernel of certain heavy-tailed distributions.

Parameters

xTensor

Input tensor; only

x > 0

is in the domain. Any floating-point dtype.

Returns

Tensor

$K_0(x)$ element-wise, same shape and dtype as x.

Notes

Asymptotic forms:

K_0(x) \sim -\log(x/2) - \gamma \;\text{as } x \to 0^+, \qquad K_0(x) \sim \sqrt{\frac{\pi}{2x}}\, e^{-x} \;\text{as } x \to \infty.

Implementation: Abramowitz & Stegun §9.8 polynomial branches at the cutoff $x = 2$ , accurate to $\approx 10^{-7}$ . For numerically stable evaluation at large arguments use scaled_modified_bessel_k0.

Examples

>>> import lucid
>>> from lucid.special import modified_bessel_k0
>>> modified_bessel_k0(lucid.tensor([0.5, 1.0, 5.0]))
Tensor([0.9244, 0.4210, 0.0037])