bessel_y1

→Tensor

bessel_y1(x: Tensor)

source

Bessel function of the second kind, order 1.

Computes $Y_1(x)$ , the singular-at-origin order-one solution of Bessel's equation. Diverges as $-2/(\pi x)$ at the origin and decays as $\sqrt{2/(\pi x)}$ for large x.

Parameters

xTensor

Input tensor; only

x > 0

is in the domain. Any floating-point dtype.

Returns

Tensor

$Y_1(x)$ element-wise, same shape and dtype as x.

Notes

Asymptotic forms:

Y_1(x) \sim -\frac{2}{\pi x} \;\text{as } x \to 0^+, \qquad Y_1(x) \sim \sqrt{\frac{2}{\pi x}} \sin\!\left(x - \frac{3\pi}{4}\right) \;\text{as } x \to \infty.

Implementation: Numerical Recipes §6.5 rational fit on $(0, 8]$ (Wronskian-corrected with the $J_1 \log$ term), trigonometric asymptotic on $(8, \infty)$ . Accuracy is $\approx 10^{-7}$ .

Examples

>>> import lucid
>>> from lucid.special import bessel_y1
>>> bessel_y1(lucid.tensor([1.0, 5.0, 10.0]))
Tensor([-0.7812, 0.1479, 0.2490])