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

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Numbers

$n$

$x_n$

1.718542959623231276081960694445212805718652553585179201271097383628989318254312507468372694771022913

6.127279134969613893617615453355747276618342460053124195427847908739148288135673732592313271525657426

10.56294270835859841254180678952230556247715482631475835815244341180372537617477813915521765324854611

15.00268812153473997996920400372798868508363520398060016078136688528260110522775237653772265723544733

19.44381663384066931711867533443106772765147234027635021754002024726366536385080649444248853861400332

23.88557890442561985232493439135809301033783428696676242229987685158871418169274294607513497504928535

28.32768408211700717711961294504563971032649809024193633260602835837178633808988583506481585515144422

32.76999567089426280721128606705089902468502993710856580460539157173843045941482997097875213083534231

37.21244113036517712043712749671080814924247387866195594288268295980906267173685907342248842326746773

10:

41.65497834599069048238375813185785038049066050579249608527821767779151011186826620356168707236798093

11:

46.09758119133442549954346465358771336161976892661158357750533964877641861381380396245285692997518895

12:

50.54023259882956197259723943429938902194110093241064638977419094528586984602301742636998765464971868

13:

54.98292094678785842500808459349760576093004343187259716651361501452064743953245938216116850384870369

14:

59.42563804782621159574690287010106772720020843894161729486345083143636856692274420112571997337033358

15:

63.86837796757472667270070508411864274844955430738302449962765461001012101250168581881272265096022896

16:

68.31113629958708131936645745814535999207050068688922959064694914018103537111875884212647030855389744

17:

72.75390970334112357829774942145746881760365027924326705960243456927886587086095225176378048975608568

18:

77.19669560031245041921863029032539785614071842837132423270188392743170297075400234290805562427157404

19:

81.63949196842686315668582621880556055133006722700095549881002214790802225804189484635775254306722171

20:

86.08229719964172581000592967096755042050689073992589292867761380705597249541745418373429767847428370

Definition

The number $x_n$ is the $n$th strictly positive zero, in increasing order, of the Kelvin function $\operatorname{ker}_0(x)=\Re K_0(xe^{\pi i/4})$, 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 $W(x)=u(x)v^{\prime}(x)-v(x)u^{\prime}(x)$, 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 $u$ is simple [1] [3].

Comments

(2)

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

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 130
n = 1
seed = mp.sqrt(2) * (n - mp.mpf(5)/8) * mp.pi
root = mp.findroot(lambda x: mp.ker(0, x), (seed-mp.mpf(1)/4, seed+mp.mpf(1)/4))
print(mp.nstr(root, 100))

Links

[1]

DLMF 10.61: Kelvin functions, definitions and differential equations

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

Values of the Kelvin function $\operatorname{kei}_\nu(x)$ — The companion function $\operatorname{kei}_0$ is the imaginary part of the same rotated $K_0$ value.

Zeros of Bessel functions of the first kind $J_\alpha$ — Positive zeros of Bessel functions on the real axis.

Zeros of Bessel functions of the second kind $Y_\alpha$ — 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 zero lies in a rational bracket of radius $10^{-108}$ whose endpoints have certified opposite signs. The returned interval encloses that bracket. Each written 100-significant-digit decimal denotes uncertainty of one unit in its last place.

The ordinal indices are certified as well. An analytic series bound excludes zeros in $(0,1/4]$. On each of 344 consecutive cells of width $1/4$ ending at $86.25$, 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 argument.

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 decimal 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 expansion, and rejection of intervals displaced by ten last-place units. The differential equation and the sign of $W$ are checked at every listed root. The first five zeros agree with the printed values [6] within $10^{-5}$; that comparison is a low-precision control. The Programs snippet illustrates numerical approximation; the attached generator supplies the certificates.