bessel_j1

→Tensor

bessel_j1(x: Tensor)

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Bessel function of the first kind, order 1.

Computes $J_1(x)$ , the regular order-one solution of Bessel's equation. Closely related to $J_0$ by $J_0'(x) = -J_1(x)$ ; appears as the radial derivative of cylindrical waves and in the antenna-pattern of a uniformly illuminated circular aperture.

Parameters

xTensor

Input tensor; any floating-point dtype.

Returns

Tensor

$J_1(x)$ element-wise, same shape and dtype as x.

Notes

Series representation:

J_1(x) = \sum_{k=0}^\infty \frac{(-1)^k}{k!\, (k+1)!} \left(\frac{x}{2}\right)^{2k+1}.

Implementation: Abramowitz & Stegun §9.4 two-branch polynomial, accurate to $\approx 10^{-7}$ . J_1 is odd, $J_1(0) = 0$ , with maximum $\approx 0.5819$ at $x \approx 1.8412$ .

Examples

>>> import lucid
>>> from lucid.special import bessel_j1
>>> bessel_j1(lucid.tensor([0.0, 1.0, 5.0]))
tensor([0., 0.4401, -0.3276])
>>> bool(bessel_j1(lucid.tensor(3.8317)).abs() < 1e-4)  # the first zero
True

>>> import lucid >>> from lucid.special import bessel_j1 >>> bessel_j1(lucid.tensor([0.0, 1.0, 5.0])) tensor([0., 0.4401, -0.3276]) >>> bool(bessel_j1(lucid.tensor(3.8317)).abs() < 1e-4) # the first zero True