BesselJZero¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

BesselJZero[n, k]

gives the k-th positive zero of BesselJ[n, x]. Stays symbolic for symbolic arguments.

Examples (3)¶

Every input below was run against the current Mathilda build and its output recorded.

Applications (3)¶

No numeric path yet -- stays symbolic

In[1]:= BesselJZero[0, 1]
Out[1]= BesselJZero[0, 1]

A bare, unevaluated head

In[2]:= Head[BesselJZero[n, k]]
Out[2]= BesselJZero

The Hadamard product it exists to name

In[3]:= Product[1 - x^2/BesselJZero[n, k]^2, {k, 1, Infinity}]
Out[3]= Gamma[1 + n] BesselJ[n, x] (2/x)^n

Algorithm¶

besseljzero.c -- BesselJZero[n, k], the k-th positive zero of BesselJ[n, x].

Currently a symbolic placeholder: it stays unevaluated for all arguments so that the Hadamard-product recogniser (Product`BesselZero) can match the canonical infinite product

  Product[1 - x^2/BesselJZero[n,k]^2, {k,1,Inf}] = Gamma[n+1] (2/x)^n BesselJ[n,x].

A future numeric path can evaluate BesselJZero[n, k] for exact numeric n and positive-integer k via McMahon asymptotics + Newton refinement on BesselJ.

Memory contract: takes ownership of res but must not free it; returns NULL (leave unevaluated) or an owned closed form.

Implementation notes¶

Algorithm. builtin_besseljzero is a symbolic placeholder: for every argument it returns NULL, so BesselJZero[n, k] stays unevaluated. It carries no numeric path yet. Its purpose is to be the canonical name for the k-th positive zero of BesselJ[n, x] so that the infinite-product recogniser in the Product subsystem (Product\BesselZero`) can match the Hadamard product

Product[1 - x^2/BesselJZero[n, k]^2, {k, 1, Infinity}]
    = Gamma[n + 1] (2/x)^n BesselJ[n, x].

A future numeric path could evaluate BesselJZero[n, k] for exact numeric n and positive-integer k via McMahon asymptotics plus Newton refinement on BesselJ, as the source header notes.

Data structures. None of its own — it is a bare EXPR_FUNCTION head that the evaluator leaves intact. The product identity above is applied by the Product family, not by BesselJZero.

Complexity / limits. O(1) (immediate decline). Attributes: Listable, NumericFunction, Protected. No numeric evaluation is implemented, so even BesselJZero[0, 1] remains symbolic; the symbol is useful today only as the product index that produces the closed Bessel form.

Attributes: Listable, NumericFunction, Protected.

References¶

Source: src/special_functions/besseljzero.c
Specification index: Mathilda_spec.md
Tests: tests/test_compile_coverage.c
Tests: tests/test_sum_product_families.c

Notes & additional examples¶

Notes¶

BesselJZero[n, k] names the k-th positive zero of BesselJ[n, x]. It is currently a symbolic placeholder: it carries no numeric evaluator, so every call — even BesselJZero[0, 1] — stays unevaluated.

Its reason for existing today is the infinite-product recogniser in the Product subsystem, which matches the canonical Hadamard product

Product[1 - x^2/BesselJZero[n, k]^2, {k, 1, Infinity}] = Gamma[1 + n] (2/x)^n BesselJ[n, x]

and so reproduces the closed Bessel form directly. BesselJZero is Listable, NumericFunction and Protected.