modified_bessel_k1

→Tensor

modified_bessel_k1(x: Tensor)

source

Modified Bessel function of the second kind, order 1.

Computes $K_1(x)$ , the order-one analogue of $K_0$ . Diverges as $1/x$ near the origin and decays exponentially for large argument. Appears in Generalised-Inverse-Gaussian and Normal-Inverse-Gaussian log-densities.

Parameters

xTensor

Input tensor; only

x > 0

is in the domain.

Returns

Tensor

$K_1(x)$ element-wise, same shape and dtype as x.

Notes

Asymptotic forms:

K_1(x) \sim \frac{1}{x} \;\text{as } x \to 0^+, \qquad K_1(x) \sim \sqrt{\frac{\pi}{2x}}\, e^{-x} \;\text{as } x \to \infty.

Implementation: Abramowitz & Stegun §9.8 two-branch polynomial, accurate to $\approx 10^{-7}$ . Use scaled_modified_bessel_k1 for numerically stable evaluation at large x.

Examples

>>> import lucid
>>> from lucid.special import modified_bessel_k1
>>> modified_bessel_k1(lucid.tensor([0.5, 1.0, 5.0]))
Tensor([1.6564, 0.6019, 0.0040])