modified_bessel_i0

→Tensor

modified_bessel_i0(x: Tensor)

source

Modified Bessel function of the first kind, order 0.

Convenience alias for lucid.i0. Computes $I_0(x)$ , the order-zero regular solution of the modified Bessel equation, used in the von Mises density and rotational kernel families.

Parameters

xTensor

Input tensor; any floating-point dtype.

Returns

Tensor

$I_0(x)$ element-wise, same shape and dtype as x.

Notes

Series form:

I_0(x) = \sum_{k=0}^\infty \frac{1}{(k!)^2} \left(\frac{x}{2}\right)^{2k}.

Even function; $I_0(0) = 1$ ; grows as $e^{|x|}/\sqrt{2\pi |x|}$ for large argument — use i0e for numerical stability at large |x|.

Examples

>>> import lucid
>>> from lucid.special import modified_bessel_i0
>>> modified_bessel_i0(lucid.tensor([0.0, 1.0, 3.0]))
Tensor([1.0000, 1.2661, 4.8808])