—   order ($\nu\geq0$)

—   argument ($x\geq0$)

For real $\nu\geq0$ and $x>0$, put $a=x/\sqrt2$. Then $\displaystyle\operatorname{kei}_\nu(x)=-\int_0^\infty e^{-a\cosh t}\sin\!\left(a\cosh t+\frac{\nu\pi}{2}\right)\cosh(\nu t)\,dt$, obtained from the integral representation of $K_\nu$ [2].

$\displaystyle\operatorname{kei}_\nu(x)=\Im\!\left[\frac{\pi i}{2}H_\nu^{(1)}(xe^{3\pi i/4})\right]$, where $H_\nu^{(1)}=J_\nu+iY_\nu$ is the Hankel function of the first kind, and $J_\nu$ and $Y_\nu$ are the principal Bessel functions [1].

For $x>0$, $\operatorname{kei}_{\nu-1}(x)+\operatorname{kei}_{\nu+1}(x)=-\dfrac{\nu\sqrt2}{x}\bigl(\operatorname{kei}_\nu(x)+\operatorname{ker}_\nu(x)\bigr)$, where $\operatorname{ker}_\nu$ is the companion Kelvin function [3].

For an integer $n\geq0$ and $x>0$, $\operatorname{kei}_{-n}(x)=(-1)^n\operatorname{kei}_n(x)$ [1].

$K_\nu$ denotes the principal branch of the modified Bessel function of the second kind. The rotated argument has modulus $x$ and angle $\pi/4$, so it lies in the right half-plane, away from the branch cut of $K_\nu$ on the negative real axis. The phase factor $e^{-\nu\pi i/2}$ is part of the Kelvin convention [1].

The Kelvin function $\operatorname{ker}_\nu(x)$ is the real part of the same phase multiple of the rotated Bessel value. The notation $\operatorname{kei}(x)$ without an order means $\operatorname{kei}_0(x)$ [1].

The continuous order-zero value at the origin is $\operatorname{kei}_0(0)=-\pi/4$, as follows from the power series [4].

The Kelvin function $\operatorname{ber}_\nu$ and $\operatorname{bei}_\nu$ are the real and imaginary parts of $J_\nu(xe^{3\pi i/4})$, where $J_\nu$ is the Bessel function of the first kind. They provide the other standard Kelvin pair [1].

It preserves the existing order-zero extremal heights and finite initial value where present. Order zero gains the sparse references $x=1/2,1,2,3,5,8$ and $x=\sqrt2,2\sqrt2,4\sqrt2$. The arguments $x=\sqrt2,2\sqrt2,4\sqrt2$ give equal integer-sized real and imaginary components in the rotated Bessel argument; this is a normalization-based reference, not a universal physical scale. Orders $1,2,3,1/2$ are sampled only at $x=1$, with order $1$ also at $x=2$, to cover adjacent integer-order relations and the elementary half-order reduction. These finite exploratory slices avoid a Cartesian grid and make no demand claim for each radius. [7] [8] [9]

The reference inputs and finite cutoffs are explicit choices rather than a completeness or application-demand claim. The attached balanced-selection.json specifies every argument exactly. Existing entries retain their identifiers, decimal strings and annotations.

For $\nu=0$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(2525/512,5051/1024)$, $(9629/1024,4815/512)$, $(7095/512,14191/1024)$, $(9373/512,18747/1024)$, $(23299/1024,5825/256)$, $(6963/256,27853/1024)$, respectively. These labels identify arguments, not the values at them.

The retained pre-extension subfamily has the following argument definitions: These are intrinsic extremal heights, with the finite origin limit where it exists. The exact argument is the unique zero of the derivative in the explicitly listed rational interval. The intervals identify roots; their endpoints are not evaluation arguments. Six isolated critical arguments are retained per selected order. Numbering orders this finite selection, without claiming a global count of all preceding zeros. The order-zero Kelvin functions are the standard ber, bei, ker, kei functions. [5]. [6]

The attached generate.py forwards to balanced_generate.py, the standalone generator for the entire current selection. Its default offline check recomputes both new and retained values. The previous entry point is preserved byte-for-byte as pre-balanced-generator.py and in revision history. Earlier supporting sources and checks are preserved as provenance for their original subfamilies; balanced-provenance.json identifies their source hashes and the retained website revision.

Python

import mpmath as mp
mp.mp.dps = 150
left = mp.mpf('2525') / 512
right = mp.mpf('5051') / 1024
argument = mp.findroot(lambda point: mp.diff(
    lambda value: mp.kei(0, value), point), (left, right))
print(mp.nstr(mp.kei(0, argument), 100))

DLMF 10.61: Kelvin Functions, Definitions and Basic Properties

DLMF 10.32.9: Integral representation of the modified Bessel function K

DLMF 10.63: Kelvin Functions, Recurrence Relations and Derivatives

DLMF 10.65: Kelvin Functions, Power Series

Argument selection: 10.61

Argument selection: 10.63

DLMF 10.61

DLMF 10.63

DLMF 10.73

Values of the Kelvin function $\operatorname{ker}_\nu(x)$ —   $\operatorname{ker}_\nu$ and $\operatorname{kei}_\nu$ are the real and imaginary parts of the same phase multiple of a rotated $K_\nu$ value

Values of modified Bessel functions of the second kind $K_\nu(x)$ —   $\operatorname{kei}_\nu$ is the imaginary part of a phase multiple of $K_\nu$ at a rotated argument

Values of Bessel functions of the first kind $J_\nu(x)$ —   $J_\nu$ occurs in the Hankel representation of $\operatorname{kei}_\nu$ and defines the first Kelvin pair after rotation

Values of Bessel functions of the second kind $Y_\nu(x)$ —   $Y_\nu$ is the second Bessel solution in the Hankel representation of $\operatorname{kei}_\nu$

Values of modified Bessel functions of the first kind $I_\nu(x)$ —   $I_\nu$ is the other standard solution of the modified Bessel equation and gives the first Kelvin pair after rotation

Values of the Kelvin function $\operatorname{ber}_\nu(x)$ —   $\operatorname{ber}_\nu$ is the real component of the first Kelvin pair

Values of the Kelvin function $\operatorname{bei}_\nu(x)$ —   $\operatorname{bei}_\nu$ is the imaginary component of the first Kelvin pair

Values of the Struve function $\mathbf{H}_\nu(x)$ —   $\mathbf{H}_\nu$ solves the inhomogeneous Bessel equation, while Kelvin functions arise from the homogeneous equation at rotated arguments

Values of the modified Struve function $\mathbf{L}_\nu(x)$ —   $\mathbf{L}_\nu$ solves the inhomogeneous modified Bessel equation, while $\operatorname{kei}_\nu$ comes from a rotated solution of the homogeneous equation

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it preserves the existing order-zero extremal heights and finite initial value where present. Order zero gains the sparse references $x=1/2,1,2,3,5,8$ and $x=\sqrt2,2\sqrt2,4\sqrt2$. The arguments $x=\sqrt2,2\sqrt2,4\sqrt2$ give equal integer-sized real and imaginary components in the rotated Bessel argument; this is a normalization-based reference, not a universal physical scale. Orders $1,2,3,1/2$ are sampled only at $x=1$, with order $1$ also at $x=2$, to cover adjacent integer-order relations and the elementary half-order reduction. These finite exploratory slices avoid a Cartesian grid and make no demand claim for each radius)

How they were obtained:

Retained-entry provenance (unchanged numerical data): Arb ball arithmetic encloses the function and its argument uncertainty at 773 bits for 100 requested significant digits. Critical arguments are recomputed from their defining rational intervals, with opposite endpoint signs, a Gronwall bound proving the second derivative has a fixed sign throughout the interval, and a separately certified narrow bracket. Turning points use exact integers and enclosed square roots. Exact origin values remain exact integers. Every stored interval contains an independently recomputed 900-bit enclosure and agrees with mpmath at 160 and 210 decimal digits. Kelvin and Struve values additionally agree with numerical quadrature of elementary integral representations; these agreement checks do not establish the proven bounds. Coulomb checks include the unit Wronskian, the vanishing turning-point coefficient, and the free-case Bessel identities. Nonzero controls and a deliberately displaced value test the comparisons. See checks-evidence.json for per-entry measurements. Earlier grid-specific checks and sources remain historical evidence and do not certify this selection.

more

Balanced extension: balanced_generate.py recomputes the entire attached selection, including retained values. New values use Arb ball arithmetic with exact rational inputs or enclosed algebraic and hyperbolic inputs at 773 working bits for 100 significant digits. Exact origin anchors are stored as integers. The attached critical-argument source is preserved and hash-identified; its certified argument enclosures are propagated through the evaluation. All old numerical strings, higher precision and manual entry annotations are preserved. balanced-checks.json records full independent higher-precision Arb recomputation and independent mpmath evaluations at 160 and 210 digits for every new value, with finite-width controls and family-specific identities or integral checks. The generator does not claim authorship of retained data; their prior supporting attachments remain unchanged, and the previous entry point is preserved as pre-balanced-generator.py.

digits: 100 (Unknown key)