bessel_j0

→Tensor

bessel_j0(x: Tensor)

source

Bessel function of the first kind, order 0.

Computes $J_0(x)$ , the regular-at-origin solution of Bessel's equation $x^2 y'' + x y' + x^2 y = 0$ for order 0. Appears throughout wave propagation, optical diffraction (the Airy disk profile), and vibration of circular membranes.

Parameters

xTensor

Input tensor; any floating-point dtype.

Returns

Tensor

$J_0(x)$ element-wise, same shape and dtype as x.

Notes

Series representation:

J_0(x) = \sum_{k=0}^\infty \frac{(-1)^k}{(k!)^2} \left(\frac{x}{2}\right)^{2k}.

The implementation uses the Abramowitz & Stegun §9.4 two-branch polynomial fit: a power series in $(x/3)^2$ for $|x| \le 3$ and an asymptotic $\sqrt{2/(\pi x)} \cos(x - \pi/4) \cdot f(3/|x|)$ form for $|x| > 3$ . Accuracy is $\approx 10^{-7}$ on the real line. J_0 is even, $J_0(0) = 1$ , and the function decays as $\sqrt{2/(\pi x)}$ for large x.

Examples

>>> import lucid
>>> from lucid.special import bessel_j0
>>> bessel_j0(lucid.tensor([0.0, 1.0, 2.4048, 5.0]))
Tensor([1.0000, 0.7652, 0.0000, -0.1776])