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

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Numbers

$n$

$x_n$

2.848917820795139852749704738728343991410270980728809734081213885628856836225978697828301247917273597

7.238829447632408344098499643929549162903700720920175232475103074964294637678074782049938746153028565

11.67396354964249215827580228814886302628689883528643972897835836403711932603437457862801480276753422

16.11356382738790184873250694149594101949548938961680139833202780842244050663724379046007592779900413

20.55463157881566164124652728480744298893359098202309360581951323366460278755271033680584190395026999

24.99636286082847420994711309822796781804396009221021188670572834698519267069187458085194877352449483

29.43845014569280549219240424760608980544398839386106653983343592030379743084589292687536746775462076

33.88075048249828116728764182473718563969941472096699750817850130433448190594786010371996292490923070

38.32318841103040187820693477943042963871655361688736067622912790051239828861106433915830836173722915

10:

42.76572034084897766859297855676685698923835550276015690172203946539070285820179480769835872630392440

11:

47.20831933463302770306509723023665548192098392136690657634411593959814660528364180266275071026057181

12:

51.65096784939712926676283534640553596507633992163211882616261174509020985135037914007116242369343725

13:

56.09365396980591515800199311766551460739828927232684226147544012868548079503714710016005864408181473

14:

60.53636931917853842220510662125693603921262549319000193841590481537761974164707021306849330336975330

15:

64.97910783667615896140179926139065894219367230310591637025839035512484540827047133470873713121542145

16:

69.42186502875825215149673523563757176548702089617443827416046477469247492191163458536280721299670256

17:

73.86463749334113636629390370293702217169084959738330900669859790885509390782732046740124744103419162

18:

78.30742260739024221459842196723045559372524993143145053429791819158429818921852581610411509163020888

19:

82.75021831600452445018263629447811886443850181550114654968280767661491100442481009866912569934325010

20:

87.19302298650247235243779639149963319832843925215158096471200429765856293983789298339324630415645246

Definition

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

Parameters

$n$

— positive zero index ($n\geq1$)

Formulas

(1)

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

(2)

$\displaystyle \operatorname{ber}_0(x)=\sum_{m=0}^{\infty}\frac{(-1)^m x^{4m}}{16^m((2m)!)^2}$ [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 $\operatorname{ber}(x)$ [1].

(4)

The origin is excluded from the indexing. The power series gives $\operatorname{ber}_0(0)=1$ [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(3)/8) * mp.pi
    root = mp.findroot(lambda x: mp.ber(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 $\operatorname{ber}_\nu$ — Values of $\operatorname{ber}_\nu(x)$; the present table gives positive zero locations at order zero.

Values of $\operatorname{bei}_\nu$ — The companion $\operatorname{bei}_0$ is the imaginary 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.

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

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

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 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 whole positive axis segment 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 origin series rules out a zero in $[0,1]$. A Gronwall bound proves that the phase changes by less than $\pi/2$ in each consecutive interval of width $1/8$ thereafter. Endpoint signs therefore count every zero without omissions. The full 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 both endpoints of every written hundred-digit interval. It also rejects intervals displaced by ten units of the last place. Nonzero origin and small-argument controls precede the checks. All roots agree with independent 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 and their derivatives agree at every root midpoint. The Programs snippet gives numerical approximations; the attached generator supplies the certificates.