Values of the derivative of the Airy function of the second kind $\operatorname{Bi}'(x)$ - NumberDB

Values of the derivative of the Airy function of the second kind $\operatorname{Bi}'(x)$

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Numbers

$x$

$\operatorname{Bi}'$

0:

0.4482883573538263579148237103988283908662267992122620610828087783723307550097806471850465744007363629

$b_{1}$:

0.6019578879762395637411828125926287325714065132437963731094999265060531527822461706343960845455394008

comment: The argument is specified by (6).

$b_{2}$:

-0.7603101414928010888448240018336352472733686882657445665794019384235437034195168153869302496888239099

comment: The argument is specified by (6).

$b_{3}$:

0.8369910126192610944085858035913127990983566046572619289088689247689304282249664566512139675229115538

comment: The argument is specified by (6).

$b_{4}$:

-0.8894799014265396245553725258959372817971258762203783647585276411597443310694993972661815638485200576

comment: The argument is specified by (6).

$b_{5}$:

0.9299836385680266344781216825745925328550348725118470506227480152830676165660114438211295403402929065

comment: The argument is specified by (6).

$b_{6}$:

-0.9632344301904237743737576103990675242326662558780245157368198774356707030726383913608795804669718006

comment: The argument is specified by (6).

$b_{7}$:

0.9915863705176604410407293309768266765739285316123746198889782422578675092760090875753486837160974089

comment: The argument is specified by (6).

$b_{8}$:

-1.016389659221248942483138674598403490096555301632789191653111744924193781020121829236003613057917967

comment: The argument is specified by (6).

$b_{9}$:

1.038494286048009376019010462988873410657691171933519635551781204326918688771543541285714509112003545

comment: The argument is specified by (6).

$b_{10}$:

-1.058471844394023368251427454746849945243071003643359914089930518141203961553065396864981038591611580

comment: The argument is specified by (6).

$b_{11}$:

1.076726148316514007016529434839265911675617476912654401507699177533461743001649173250836957445473986

comment: The argument is specified by (6).

$b_{12}$:

-1.093553623307425288447343617924129057474646353621544040578923525058321150828220006076448802452490401

comment: The argument is specified by (6).

$b_{13}$:

1.109178636525735616303193911995737745303388683100527774448169928742800935362626683307911968992850690

comment: The argument is specified by (6).

$b_{14}$:

-1.123775327286271309247630404164691938134338132623135933636844232524351661001732832239702526292346571

comment: The argument is specified by (6).

$b_{15}$:

1.137481708119596394809975480606455854516761096297472721141873149426097971161849208823940002323219151

comment: The argument is specified by (6).

$b_{16}$:

-1.150409114329055039856909606507592166731672750875442309114253532095376873286535550971273406229730038

comment: The argument is specified by (6).

$b_{17}$:

1.162648735788382025013017605705863355069036895048450990449944614786624048769326486245503451587573770

comment: The argument is specified by (6).

$b_{18}$:

-1.174276253118058449137954751302455336549354375198781591110990088405269384964040768714792097887371784

comment: The argument is specified by (6).

$b_{19}$:

1.185355204648904120061078488262812333825903632424619776345215654957228061702059331149243788180277122

comment: The argument is specified by (6).

$b_{20}$:

-1.195939481079752461470863037028782834059416330535361022747830731288387069398379722908642501373235790

comment: The argument is specified by (6).

-1/2:

0.5059337136238471665702604378969036163730513656761924189028459319434157648207993009265312445241845856

comment: A bounded simple-real reference; see (7).

1/2:

0.5445725641405923018271640182178233256545288982318871440949459393297924156275125945017892101930906919

comment: A bounded simple-real reference; see (7).

-1:

0.5923756264227923508167792291816009732767958833673629057820719939482987979074233408436948982835696949

comment: A bounded simple-real reference; see (7).

1:

0.9324359333927756329594514536744353442695653752386283954928699120385701053212359914165137166283123733

comment: A bounded simple-real reference; see (7).

-2:

0.2787951669211695226850975694109832414030005934516310002397323516499229052306788963796746603879555016

comment: A bounded simple-real reference; see (7).

2:

4.100682049932889889382034079177935294390244613775137119837709189590545889936745602999116219404486847

comment: A bounded simple-real reference; see (7).

-4:

-0.1166705674383408936795672429766645983332920738953310796038548883567684328318830998264056353237428433

comment: A bounded simple-real reference; see (7).

4:

161.9266835046134018430949242854763484403832327919302102012625776636770361296310813831604357524395098

comment: A bounded simple-real reference; see (7).

-8:

-0.1594504978129813893499357336501562246136560242355853962398859270219076513452378388030061711693442834

comment: A bounded simple-real reference; see (7).

8:

3354342.312744538876507746496527586779617045446747452048778442171670485851294882972644823892978650641

comment: A bounded simple-real reference; see (7).

$-2/\sqrt[3]{3}$:

0.5840772524753836222628637884631495699346395613945924868590711200281231775603842050871774485646589121

comment: A unit-cubic normalization reference; see (8).

$-1/\sqrt[3]{3}$:

0.5437176386401158396793380758973071534914182708417593337410030306241740365147968534288531764261609038

comment: A unit-cubic normalization reference; see (8).

$1/\sqrt[3]{3}$:

0.6499142120959702312447419875131555461106164591196588900934162004271310433136645469487467341231810728

comment: A unit-cubic normalization reference; see (8).

$2/\sqrt[3]{3}$:

1.595321145874211197124379897430175312821343117540407351253416774914697146888324343527757823356299240

comment: A unit-cubic normalization reference; see (8).

Definition

The derivative $\operatorname{Bi}'$ of the Airy function of the second kind [2]. This table gives its values at intrinsic and bounded reference arguments.

Parameters

$x$

— argument ($x\in\mathbb{R}$)

Formulas

(1)

$\operatorname{Bi}'(0)=3^{1/6}/\Gamma(1/3)$ [3].

(2)

For real $x$, $\operatorname{Ai}(x)\operatorname{Bi}'(x)-\operatorname{Ai}'(x)\operatorname{Bi}(x)=1/\pi$ [4].

(3)

$\int_0^\infty u[e^{-u^3+ku}+\cos(u^3+ku)]\,du=\pi3^{-2/3}\operatorname{Bi}^{\prime}(k/3^{1/3})$.

Comments

(4)

The normalisation is the standard pair of solutions of Airy\'s differential equation [2]. Values on both sides of the origin are listed: $\operatorname{Ai}$ decays and $\operatorname{Bi}$ grows for positive $x$, while both oscillate for negative $x$.

(5)

At $x=0$ the value is a Gamma-function value, $\operatorname{Bi}'(0)=3^{1/6}/\Gamma(1/3)$ [3], and $\Gamma(1/3)$ and $\Gamma(2/3)$ are in the table of Gamma values.

(6)

Let $0>b_1>b_2>\cdots$ be the negative real zeros of $\operatorname{Bi}$, ordered from the origin [6]. The selected values are the slopes at the successive zeros, not the zero locations. The origin is the initial derivative anchor.

(7)

The exploratory reference arguments $x=\pm1/2,\pm1,\pm2,\pm4,\pm8$ give a sparse set of simple real inputs on both sides of the Airy turning point. They complement root-indexed values with near-origin, moderate and larger arguments. This is a bounded identification reference, not a claim that each point has a documented application [7].

(8)

The change of variable $t=3^{1/3}u$ in the defining cubic integrals gives $x=k/3^{1/3}$ when the cubic coefficient is one and the linear coefficient is the integer $k$. The finite range $1\leq|k|\leq2$ is an exploratory simple-coefficient reference choice, not a source-prescribed cutoff [8].

(9)

The current generate.py reproduces the complete attached selection, including the retained entries. Their original written values and entry comments are preserved rather than reassigned to a new source. The selection manifest records the predecessor revision and source hashes; earlier generators and checks remain historical provenance, not the enumeration of this extension.

Programs

(P1)

Python

from generate import generator
from numberdb._write import to_text

for params, result in generator.all_entries():
    print(params["x"], to_text(result["number"], generator.digits_for(params)))

Links

[1]

Wikipedia: Airy function

[2]

DLMF 9.2: Airy's equation

[3]

DLMF 9.2: Initial values

[4]

DLMF 9.2: Wronskians

[5]

OEIS A358561: decimal expansion of $\operatorname{Bi}'(0)$

[6]

DLMF: definitions and natural argument families

[7]

DLMF 9.2: Airy differential equation

[8]

DLMF: defining cubic integral representations

Similar tables

Values of the Airy function of the first kind $\operatorname{Ai}(x)$ — the other solution of the same equation

Values of the derivative of the Airy function of the first kind $\operatorname{Ai}'(x)$ — the other solution of the same equation; their Wronskian is (2)

Values of the Airy function of the second kind $\operatorname{Bi}(x)$ — the function whose derivative this is

Zeros of the Airy function of the first kind Ai — stores the zeros of $\operatorname{Ai}$

Zeros of the Airy function of the second kind Bi — stores the zeros of $\operatorname{Bi}$

Local extrema of the Airy function of the first kind Ai — stores the zeros of $\operatorname{Ai}'$, where $\operatorname{Ai}$ has local extrema

Local extrema of the Airy function of the second kind Bi — stores the zeros of $\operatorname{Bi}'$, where $\operatorname{Bi}$ has local extrema

Values of the Gamma function at rational numbers — contains the Gamma values in the initial values at $x=0$

Values of Bessel functions of the first kind $J_\nu(x)$ — Airy functions at negative arguments are related to Bessel functions of orders $1/3$ and $2/3$

Values of Bessel functions of the second kind $Y_\nu(x)$ — gives the companion Bessel solution appearing in the connection formulas for negative Airy arguments

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the origin and the first twenty slopes at zeros specified in (6); the selection is not exhaustive over the real domain. In addition, it includes the simple real arguments $\pm1/2,\pm1,\pm2,\pm4,\pm8$ and unit-cubic arguments $k/3^{1/3}$ for $1\leq|k|\leq2$. The exploratory families are identification references with practical finite bounds, not claims of documented applications for every input. All prior entries are retained.)

How they were obtained:

All retained values and metadata are preserved, including their precision. New nonexact values have 100 significant digits. balanced_values.py reproduces the complete selection in balanced-selection.json with Arb balls, including uncertainty in exact irrational arguments and certified roots. A 640-bit guard is supplemented by measured accuracy checks and precision escalation.

more

Every written interval contains a freshly recomputed certified enclosure; every value is also checked with independent mpmath computations at 50 and 100 decimal guard digits. Exact origin anchors stay exact. Structural identities, root endpoint signs where applicable, and selected transformed definite integrals supplement these checks. See balanced-checks-evidence.json. Earlier checks-evidence.json and source attachments describe earlier selections and remain for provenance.