i1 — special

i1

→Tensor

i1(x: Tensor)

source

Modified Bessel function of the first kind, order 1.

Computes $I_1(x)$ , the order-one solution of the modified Bessel equation $x^2 y'' + x y' - (x^2 + 1) y = 0$ that is regular at the origin. Used in Rician statistics, Bessel / von Mises – Fisher distributions, and rotational-symmetric kernels.

Parameters

xTensor

Input tensor; any floating-point dtype.

Returns

Tensor

$I_1(x)$ element-wise, same shape and dtype as x.

Notes

Series definition:

I_1(x) = \sum_{k=0}^\infty \frac{1}{k!\,(k+1)!}\left(\frac{x}{2}\right)^{2k+1}.

The implementation uses the two-branch Abramowitz & Stegun §9.8 polynomial approximation: a power series in $(x/3.75)^2$ for $|x| \le 3.75$ , and an asymptotic expansion in $3.75/|x|$ scaled by $e^{|x|}/\sqrt{|x|}$ for $|x| > 3.75$ . Accuracy is roughly $10^{-7}$ over the real line. I_1 is odd, I_1(0) = 0, and $I_1(x) \sim e^{x}/\sqrt{2\pi x}$ as $x \to \infty$ .

Examples

>>> import lucid
>>> from lucid.special import i1
>>> i1(lucid.tensor([0.0, 1.0, 5.0]))
Tensor([0.0000, 0.5652, 24.3356])