BesselJZero

The $k$-th positive zero of the Bessel function $J_\nu$, for real $\nu > -1$ and positive integer $k$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimBesselJZero↗Fungrim entry1e6344↗da7fb1↗Wolfram LanguageBesselJZero✓↗

BesselJZero(nu, k)the $k$-th positive zero of $J_\nu$.

Typed signatures

Type	Package	Applies
(number, integer) -> number	analytic	always

Domain: Special functions

Details

• compute-engine's native BesselJ only evaluates numerically at integer order, so the zero-finder here carries its own real $J_\nu$ series (term-ratio, stable for the double-precision range zero-finding needs) rather than depending on it — which matters for exactly the half-integer orders Fungrim's identities use.
• A zero is located in doubles (McMahon's expansion, then a bracketed series search) and refined by Newton's method in arbitrary precision: the alternating series cancels about $x/2.3$ digits, so a double alone is good to roughly $16 - x/2.3$. Past $j = 400$ it stays symbolic.
• McMahon's asymptotic expansion seeds a bracket around the $k$-th zero, then bisection (with a few closing Newton steps) converges it.
• Matches mpmath's besseljzero(nu, k) and Wolfram's BesselJZero[nu, k].

Examples

3 more kept as data, checked against the oracles

Basic #

$j_{3/2,1} = 4.4934\ldots$ — also the first positive root of $\tan x = x$

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In$N(\mathrm{BesselJZero}(\frac{3}{2}, 1))$Out$4.493\,409\,457\,909\,064\,175\,31$

$j_{0,1} = 2.4048\ldots$

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In$N(\mathrm{BesselJZero}(0, 1))$Out$2.404\,825\,557\,695\,772\,768\,62$

the third zero of $J_0$

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In$N(\mathrm{BesselJZero}(0, 3))$Out$8.653\,727\,912\,911\,012\,216\,95$

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In$\operatorname{Series}(\mathrm{BesselJZero}(n, x), x, 0, 1)$Out$\mathrm{BesselJZero}(n, 0)+x\mathrm{BesselJZero}^{(0, 1)}(n, 0)+O\left(x^2\right)$

Scope #

The first three zeros of $J_0$ at engine precision

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In$N(\bigl\lbrack\mathrm{BesselJZero}(0, 1), \mathrm{BesselJZero}(0, 2), \mathrm{BesselJZero}(0, 3)\bigr\rbrack)$Out$\bigl\lbrack2.404\,825\,557\,695\,772\,768\,62, 5.520\,078\,110\,286\,310\,649\,6, 8.653\,727\,912\,911\,012\,216\,95\bigr\rbrack$

$j_{0,1}$ to 20 significant digits, refined in arbitrary precision from the double zero

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In$N(\mathrm{BesselJZero}(0, 1), 20)$Out$2.404\,825\,557\,695\,772\,768\,6$

$j_{0,3}$ to 30 significant digits

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In$N(\mathrm{BesselJZero}(0, 3), 30)$Out$8.653\,727\,912\,911\,012\,216\,954\,198\,712\,66$

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In$\mathrm{Simplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{BesselJZero}(n, x), x, 0, 2)))$Out$\frac{1}{2}(\mathrm{BesselJZero}^{(0, 2)}(n, 0)x^2)+x\mathrm{BesselJZero}^{(0, 1)}(n, 0)+\mathrm{BesselJZero}(n, 0)$

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In$\operatorname{Series}(\mathrm{BesselJZero}(n, x), x, \mathrm{x0}, 1)$Out$\mathrm{BesselJZero}(n, \mathrm{x0})+(x-\mathrm{x0})\frac{∂}{∂\mathrm{x0}} \mathrm{BesselJZero}(n, \mathrm{x0})+O\left((x-\mathrm{x0})^2\right)$

Applications #

Zero of $J_1$ divided by $\pi$ at high precision

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In$N(\frac{\mathrm{BesselJZero}(1, 1)}{π})$Out$1.219\,669\,891\,266\,504\,513\,619\,177\,543\,587\,529\,225$

Properties #

$\tan(j_{3/2,1}) = j_{3/2,1}$ — the sinc function's first turning point

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In$\mathrm{chop}(\tan(N(\mathrm{BesselJZero}(3/2, 1)))-N(\mathrm{BesselJZero}(3/2, 1)))$Out$0.0$

About

Examples	14
Checked against	Wolfram Language 14 agree
Crosswalk	FungrimFungrim entry ×2Wolfram Language

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/bessel-zeros.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

BesselJZero[nu, k]; mpmath.besseljzero(nu, k).

See also: Sinc