Positive zeros of the Kelvin function $\operatorname{bei}_0$

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Numbers

$n$

$x_n$

5.026223951953151329939079653458968130556133531388572232891515916979516961045398603507418620778932592

9.455406303277153573107566071814137375271881305275345102861410667488110461152041295611697930265522231

13.89348785265946720438597418646755972553818310988912234940618342554844523741339658054839722275532495

18.33398345577453030795968129281413207948931288412915943984103249793735665902846080583640394210113636

22.77543929381786931149608299195695516022708690046445718932874938424359419557835499500378171709105303

27.21737318108423715510401832291713008146825260322100099412639640342607995959572003435891681170646493

31.65957941363257596823656264158484023183399177114099775482171586196193567961608677472613987242088218

36.10195548491730125371613300620248849243268541463294932296174942122735753059556989520704359658592359

40.54444459090275659282298371785053903333776334863073134350564584081249795964345155134747972785318138

10:

44.98701271615374515051706750961564110406591997046305417246623971633651424989576546237006699078233153

11:

49.42963824844000089675124850782513696487739625548999927211577990618471306634346659050211608741289050

12:

53.87230679805301491907256815644899055046650442809187296349392122963731263232909816406569988861045948

13:

58.31500841348885069965992702728206376794719082504023623649306921602700697334194239887634971174129256

14:

62.75773599291734135063738423555738286698057342084918453915964647294916094594403407314862315542640835

15:

67.20048433233293258984169345643985132511745804317951568689585833478454912019672920632553417026573420

16:

71.64324953132709851956502112179564965972768050395475416925092813144600782609903872904644479117541233

17:

76.08602860896750026786569646344948482186972714253022724191072962347647196630957107760396088212663934

18:

80.52881924793824051943816175029230736342950336808073397852949042123598439022539281678296412361841218

19:

84.97161961961098861381992683794702459301219779383376109102723227467810732696464126475582839192063928

20:

89.41442826167742013744492086378836498387471599380651201231642925903882646097557529232713479596338421

Definition

The number $x_n$ is the $n$th strictly positive zero, in increasing order, of the Kelvin function $\operatorname{bei}_0(x)=\Im J_0(xe^{3\pi i/4})$ [1].

Parameters

$n$

— positive zero index ($n\geq1$)

Formulas

(1)

$\operatorname{bei}_0(x)=\Im I_0(xe^{\pi i/4})$, where $I_0$ is the order-zero modified Bessel function of the first kind [1].

(2)

$\displaystyle \operatorname{bei}_0(x)=\sum_{m=0}^{\infty}\frac{(-1)^m x^{4m+2}}{4\cdot16^m((2m+1)!)^2}$ [2].

Comments

(3)

$J_0$ is the order-zero Bessel function of the first kind. It is entire, so no branch choice is needed. The argument is $x$ itself, and $x_n$ is a dimensionless zero location. The order is fixed at zero, often omitted in the notation $\operatorname{bei}(x)$ [1].

(4)

The power series gives a double zero at the origin, $\operatorname{bei}_0(x)=x^2/4+O(x^6)$. This zero is excluded from the indexing [2].

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 130
for n in range(1, 21):
    seed = mp.sqrt(2) * (n + mp.mpf(1)/8) * mp.pi
    root = mp.findroot(lambda x: mp.bei(0, x),
                       (seed-mp.mpf('0.25'), seed+mp.mpf('0.25')))
    print(n, mp.nstr(root, 100))

Links

[1]

DLMF 10.61: Definitions and Basic Properties

[2]

DLMF 10.65: Power Series

[3]

DLMF 10.71.8: Modulus and Wronskian integral identity

[4]

DLMF 10.75(xii): Tables of Kelvin zeros

[5]

Abramowitz and Stegun, Handbook of Mathematical Functions, p. 381: zeros of functions of order zero

Similar tables

Values of $\operatorname{bei}_\nu$ — Values of $\operatorname{bei}_\nu(x)$; the present table gives positive zero locations at order zero.

Values of $\operatorname{ber}_\nu$ — The companion $\operatorname{ber}_0$ is the real part of the same rotated Bessel value.

Positive zeros of $\operatorname{kei}_0$ — Positive zeros of $\operatorname{kei}_0$, the imaginary component of the decaying Kelvin pair.

Positive zeros of $\operatorname{ker}_0$ — Positive zeros of $\operatorname{ker}_0$, the real component of the decaying Kelvin pair.

Zeros of Bessel $J_\alpha$ — Positive real-axis zeros of $J_\alpha$; taking a rotated imaginary component defines a different zero family.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds all first twenty positive order-zero zeros, $1\leq n\leq20$. This finite reference depth covers several oscillations beyond the first few nodes while permitting exhaustive root counting and independent hundred-digit checks. The cutoff is a noncanonical reference choice, with a twenty-root precedent recorded in [4]; higher orders and derivative zeros are outside this family)

How they were obtained:

The attached generator uses 800-bit Arb evaluations of $I_0(xe^{\pi i/4})$ to prove opposite signs at rational endpoints enclosing each root. Every returned interval has positive width and lies inside a counted root interval. Its decimal representation contains the root within one unit of the last written place.

The index certificate covers the positive axis through the twentieth root. For $F=I_0(xe^{\pi i/4})=u+iv$ and $W=uv'-vu'$, the differential equation gives $(xW)'=x|F|^2$. The origin expansion implies $W>0$ for $x>0$, so the continuous phase increases strictly [1] [3]. The power series proves $\operatorname{bei}_0(x)>0$ for $0<x\leq1$. A Gronwall bound then proves that the phase changes by less than $\pi/2$ in each of 708 consecutive intervals of width $1/8$. Endpoint signs count every zero without omissions. The derivation and the numerical bounds are attached in certificate.md and checks-evidence.json.

An independent 900-bit real power-series computation, with an explicit geometric tail bound, proves opposite signs at the endpoints of every written hundred-digit interval. It also rejects intervals displaced by ten units of the last place. Origin and nonzero small-argument controls precede the checks. All roots agree with hypergeometric computations at 160 and 210 decimal digits, and the first five agree with the printed values [5] within half a unit in their fifth decimal place. The rotated Bessel representations, their derivatives and the differential equation agree at every root midpoint. The Programs snippet gives numerical approximations; the attached generator supplies the certificates.