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

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Numbers

$n$

$x_n$

3.914667606843246082248522786841159362118245140033896876481409624718963835262054932638562117525876313

8.344225062937332035330306995680109871104448891136815893175819047092013252661186877211103702704340229

12.78255714860019849323674616307008350157749908713178247759248107566022491834210454618259218821337034

17.22314372343476945553233733582037505929493832333115158170731820298792882608575312173488920306326503

21.66464214063516449913316958499225512146845366272185563598952929066578604229752630639047276027519252

26.10659928055842432482534866691990083750505445880136552118581501356547049663846942635504033127414684

30.54881957534176276623116858512292019252028483693620716701376281421984822185777673229783560095678302

34.99120479023713671166738599458398054906369570316370304603480589476735014664362740553919586851520903

39.43370017261625990952801121478285546060557424678615714767574096393268574055096568650914468741778385

10:

43.87627279110208307389416096754234474464038793981541630158416632365915265166045935390772139174965917

11:

48.31890164989348503351879028460588372842478117645921891710904499015501958976257645501349420937247469

12:

52.76157273073524710735013864653861983972744937699333482982544643013324793386556965423508275327482553

13:

57.20427631674848858496381430361508026326128867217941589690617718411196358273560894086117274017457561

14:

61.64700546019694678704954194815370276436090652467559867662835450221503556503971101218881803464484884

15:

66.08975506165933815220329865337526933758717426896201893636364660807851164323711726307384250831352504

16:

70.53252129372920528008427257072071235994960772087805897950871690958796923933694733756114001780402466

17:

74.97530122768217821092244160484786053428706847194925257437723195357853480497994320023581587269825798

18:

79.41809258433820737907118234860465621870214385178784929383453495995453347375099545678891193189561653

19:

83.86089356344997016860349727627448518248301508703980885098610085456021054542019708173955325378656914

20:

88.30370272418294129413487914267720979440661205409591801421930924671381962629154266132153386723621304

Definition

The number $x_n$ is the $n$th strictly positive zero, in increasing order, of the Kelvin function $\operatorname{kei}_0(x)=\Im K_0(xe^{\pi i/4})$, where $K_0$ is taken on its principal branch [1].

Parameters

$n$

— zero index ($n\geq1$)

Formulas

(1)

Put $G(x)=K_0(xe^{\pi i/4})=u(x)+iv(x)$ and $W(x)=u(x)v'(x)-v(x)u'(x)$, where $K_0$ is the modified Bessel function of the second kind. Then $(xW)'=x|G|^2$ and $xW(x)=-\int_x^\infty t|G(t)|^2\,dt<0$. Consequently the continuous phase of $G$ strictly decreases and every positive zero of $v$ is simple [1] [3].

Comments

(2)

The order is fixed at zero. Each entry is the dimensionless location $x_n$, using $x$ itself as the argument. The origin is excluded; $\operatorname{kei}_0(x)$ tends to $-\pi/4$ as $x\to0^+$ [2].

Programs

(P1)

Python

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

Links

[1]

DLMF 10.61: Kelvin functions, definitions and basic properties

[2]

DLMF 10.65: Kelvin function power series

[3]

DLMF 10.67: Kelvin function asymptotic expansions

[4]

DLMF 10.32.9: Integral representation of K

[5]

DLMF 10.75(xii): Tables of Kelvin zeros

[6]

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

Similar tables

Values_of_the_Kelvin_function_4 — Values of $\operatorname{kei}_\nu(x)$; the present table gives positive zero locations at order zero.

Values_of_the_Kelvin_function_3 — The companion function $\operatorname{ker}_0$ is the real part of the same rotated $K_0$ value.

Zeros_of_Bessel_J_functions — Positive zeros of Bessel functions on the real axis.

Zeros_of_Bessel_Y_functions — Positive zeros of the other real Bessel solution.

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it contains every positive order-zero zero with $1\leq n\leq20$. The root index is intrinsic; the cutoff twenty is a practical reference depth covering the first nodes and further oscillations while allowing a complete certified count. DLMF lists a twenty-root reference table [5]. Higher orders and derivative zeros are outside this table)

How they were obtained:

The attached generator uses Arb complex-ball evaluations of $K_0$ at 753 bits. Each root lies in a rational bracket of radius $10^{-108}$ with certified opposite endpoint signs; the returned ball contains that bracket, and the written decimal denotes uncertainty of one unit in its last place.

The ordinal indices are certified as well. An analytic series bound excludes roots in $(0,1/4]$. On each of 353 consecutive cells of width $1/4$ ending at $88.5$, a 384-bit calculation proves that the image of $G(x)=K_0(xe^{\pi i/4})$ lies in a disk excluding zero. The derivative bound $|G'(x)|\leq K_1(x/\sqrt2)$ follows from [4]. The disk condition and strict phase monotonicity give at most one zero per cell, so endpoint signs count exactly twenty. The largest certified disk-radius-to-center-modulus ratio is less than $0.513$. The attached root-counting.md supplies the proof.

An independent implementation of the real logarithmic and harmonic-number series [2], with explicit geometric tail bounds at 1024 bits, certifies opposite signs at both endpoints of every written 100-significant-digit interval. All endpoint balls are finite with radii below $10^{-200}$. Controls include the nonzero identity $K_{1/2}(1)=\sqrt{\pi/2}/e$, the origin series, and rejection of intervals displaced by ten last-place units. The first five roots agree with the printed values [6] within $10^{-5}$; this is a low-precision control, not the basis of the hundred-digit bounds. The Programs snippet illustrates numerical approximation; the attached generator supplies the certificates.