modified_bessel_i1

→Tensor

modified_bessel_i1(x: Tensor)

source

Modified Bessel function of the first kind, order 1.

Convenience wrapper around i1. Computes $I_1(x)$ , the order-one regular solution of the modified Bessel equation, forming a natural pair with $I_0$ for log-density evaluation of Rician / Bessel-family distributions.

Parameters

xTensor

Input tensor; any floating-point dtype.

Returns

Tensor

$I_1(x)$ element-wise.

Notes

Series form:

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

Odd function; $I_1(0) = 0$ ; grows like $I_0$ at infinity. Use i1e for stable evaluation at large |x|.

Examples

>>> import lucid
>>> from lucid.special import modified_bessel_i1
>>> modified_bessel_i1(lucid.tensor([0.0, 1.0, 3.0]))
Tensor([0.0000, 0.5652, 3.9534])