Certified Bessel functions

Keep letter-and-order names in context

The short conventional names are recognizable in a formula about Bessel functions but ambiguous in a general calculator. The plugin therefore keeps them under the .bessel namespace instead of installing bare J0, J1, Y0, and Y1 variables.

Runnable RiXRun cell

.Plugin.Load("bessel"); values := [ .bessel.J0(0), .bessel.J1(1), .bessel.Y0(1), .bessel.Y1(1) ]; values |>> ((value) -> .numerics.Refine(value, {= absoluteWidth=1/1000, maxWork=1200 }));

Loading bessel automatically loads its Numerics and Oracle dependencies. All functions return certified refinable real values rather than Float samples. The Y family currently accepts positive real arguments only; an unresolved or invalid real domain produces structured :unknown evidence.

Use the universal implementation directly when needed

The algorithms remain part of Numerics so every certified real provider can use them. Their explicit universal names also make the relationship clear:

Runnable RiXRun cell

.Plugin.Load("numerics"); result := .numerics.Refine(.numerics.BesselJ0(1), {= absoluteWidth=1/10000, maxWork=400 }); [result[:status], result[:interval], result[:certified]];

For ordinary calculator code, prefer .bessel.J0(1). The longer Numerics names are mainly useful to code that deliberately works with the universal algorithm layer.

Select any integer order

J(n,x) and Y(n,x) keep the family letter in the Bessel namespace while making the order explicit:

Runnable RiXRun cell

.Plugin.Load("bessel"); results := [.bessel.J(2, 1), .bessel.J(-3, 1), .bessel.Y(2, 1)]; results.Map((value) -> .numerics.Refine(value, {= absoluteWidth=1/1000, maxWork=7000 }));

Negative orders use J_-n=(-1)^n J_n and Y_-n=(-1)^n Y_n. The Y family requires a certifiably positive argument.

Modified families

The same namespace provides integer-order I and K:

Runnable RiXRun cell

.Plugin.Load("bessel"); modified := [.bessel.I(2, 1), .bessel.K(2, 1)]; modified.Map((value) -> .numerics.Refine(value, {= absoluteWidth=1/1000, maxWork=12000 }));

I accepts real arguments and follows integer parity. K(-n,x)=K(n,x) and requires x>0 on the real branch.