bessel_y0

→Tensor

bessel_y0(x: Tensor)

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Bessel function of the second kind, order 0.

Computes $Y_0(x)$ , the singular-at-origin partner of $J_0$ . Together $J_0$ and $Y_0$ span the solution space of Bessel's equation at order 0; $Y_0$ is the one that diverges (logarithmically) at the origin.

Parameters

xTensor

Input tensor; only

x > 0

is in the domain. Any floating-point dtype.

Returns

Tensor

$Y_0(x)$ element-wise, same shape and dtype as x.

Notes

Asymptotic behaviour:

Y_0(x) \sim \frac{2}{\pi}\left(\log\frac{x}{2} + \gamma\right) \; \text{as } x \to 0^+, \qquad Y_0(x) \sim \sqrt{\frac{2}{\pi x}} \sin\!\left(x - \frac{\pi}{4}\right) \;\text{as } x \to \infty.

Implementation: Numerical Recipes §6.5 rational fit on $(0, 8]$ with the $J_0 \log$ Wronskian term, and the standard A&S 9.4.5 trigonometric asymptotic on $(8, \infty)$ . Accuracy is $\approx 10^{-7}$ . Y_0 diverges to $-\infty$ as $x \to 0^+$ .

Examples

>>> import lucid
>>> from lucid.special import bessel_y0
>>> bessel_y0(lucid.tensor([1.0, 5.0, 10.0]))
Tensor([0.0883, -0.3085, 0.0557])