Module bessel 

The Bessel family and what is built on it.

One directory, seven kernels, all sharing the two recurrence directions (J/I are the minimal solutions and walk down on ratios, while Y/K are dominant and walk up) and the envelope-relative accuracy contract the oscillating members need. Read jy first for that contract. It is the decision everything else is graded by.

module	what	order
ik	$I_n$, $K_n$: fitted rationals at 0 and 1, recurrences and an asymptotic arm above	whole, const or per-lane
jy	$J_n$, $Y_n$: the same shape for the oscillating pair	whole
half	all four at half-integer order, where they are elementary	k/2
spherical	$j_n$, $y_n$, $i_n$, $k_n$: half’s walks seeded in the spherical normalization	whole
jy_real	$J_\nu$, $Y_\nu$ at arbitrary real order: series, Steed, Temme, Hankel	real
ik_real	$I_\nu$, $K_\nu$ at arbitrary real order, generic over a real or complex argument	real
airy	$\mathrm{Ai}$, $\mathrm{Bi}$ and derivatives, as Bessel functions at thirds	-

The order dispatch (which of these a runtime BesselOrder reaches) is in the per-element impls (specialized/pd.rs, ps.rs). The entry-point stamping and the BesselDetails trait are in specialized/bessel.rs.

Modules

airy

The Airy functions $\mathrm{Ai}$, $\mathrm{Bi}$ and their derivatives.

half

Bessel functions at half-integer order, where all four families are elementary.

ik

Modified Bessel functions of the first kind, orders 0 and 1, scaled and unscaled.

ik_real

Modified Bessel $I_\nu$ and $K_\nu$ at arbitrary real order.

jy

The oscillatory Bessel functions $J_0$, $J_1$, $Y_0$, $Y_1$.

jy_real

Bessel functions at arbitrary real order, starting with the large-x Hankel arm.

ratio

A_nu(x) = I_nu(x) / I_{nu-1}(x), the modified Bessel ratio, and its inverse.

spherical

The spherical Bessel functions $j_n$, $y_n$, $i_n$, $k_n$.