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

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

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Numbers

$x$

$\operatorname{Bi}$

0:

0.6149266274460007351509223690936135535947281886485965050408787530142965193055206405293873433452675692

$b^{\prime}_{1}$:

-0.4549443836396573543890502762305885108631795350893794086920014639075743646053644369031429966388152089

comment: The argument is specified by (5).

$b^{\prime}_{2}$:

0.3965228360944646560805181373672310327746939001299627514195174497304071288880777379069703166425789375

comment: The argument is specified by (5).

$b^{\prime}_{3}$:

-0.3679691614869592564534311059279052560356436886925648129969930536639885455681963748549198938685062717

comment: The argument is specified by (5).

$b^{\prime}_{4}$:

0.3494991168318044408050529562458262144929210281354964493764510942483468555345150001619891067970731755

comment: The argument is specified by (5).

$b^{\prime}_{5}$:

-0.3360262401336600411674890394790926214423543461453298213114250813400835769334954261561306851650570239

comment: The argument is specified by (5).

$b^{\prime}_{6}$:

0.3255097364326236035057391248763071516508589832241548356406392302459731484635488249006808120815563850

comment: The argument is specified by (5).

$b^{\prime}_{7}$:

-0.3169346537076649432834327333461092431312712447031146109970379913570043612968308585254377527759511125

comment: The argument is specified by (5).

$b^{\prime}_{8}$:

0.3097259407690768433320522424041910151958615564599530440272155538451960275027664937215889990148614100

comment: The argument is specified by (5).

$b^{\prime}_{9}$:

-0.3035276648433455736680831048014089916195889992224384441448082970959109591641902705678886850638114685

comment: The argument is specified by (5).

$b^{\prime}_{10}$:

0.2981049111448003441478709902142557483957008453979001978099080154835784117322305848277518279530615364

comment: The argument is specified by (5).

$b^{\prime}_{11}$:

-0.2932948758166831405311846858437926952881275514503405389011873678712455676001416512003290004993146006

comment: The argument is specified by (5).

$b^{\prime}_{12}$:

0.2889803065671025872317235261662245813500075669143629302175990409132037766314120605545061039895423581

comment: The argument is specified by (5).

$b^{\prime}_{13}$:

-0.2850740906528823299407738540343340850698082189078828092152853528172785683701671365897996175223954017

comment: The argument is specified by (5).

$b^{\prime}_{14}$:

0.2815098244689997270918704645251010904668651211727090361001401251935890194928815577294876789125734645

comment: The argument is specified by (5).

$b^{\prime}_{15}$:

-0.2782357870262160775923472449641283925675793453802832342373763493292556844332556474757481905716746944

comment: The argument is specified by (5).

$b^{\prime}_{16}$:

0.2752109464392148542108169617317684140223484164954783006278406672145620440861288953302551002672192906

comment: The argument is specified by (5).

$b^{\prime}_{17}$:

-0.2724022309288061808381288110571275504805776059778796108921024769027513704668428775703445648791984828

comment: The argument is specified by (5).

$b^{\prime}_{18}$:

0.2697826139865425952871187039266224627784072648488892663114048002763432158947214136684532042486704669

comment: The argument is specified by (5).

$b^{\prime}_{19}$:

-0.2673297395916449815569537038852564966592908154827444919484873619919641624946293170587510191126289849

comment: The argument is specified by (5).

$b^{\prime}_{20}$:

0.2650249150795161691862106970188163098773955990357725038951446367030217605209046933809208900946738204

comment: The argument is specified by (5).

-1/2:

0.3803526597510538501697124937262645026709261850134749759835389296543251425165916912418965260948392892

comment: A bounded simple-real reference; see (6).

1/2:

0.8542770431031554933000487987952431808567874005047396269340087415514085646293252655590229480643045376

comment: A bounded simple-real reference; see (6).

-1:

0.1039973894969446118886899909785991446370181113925239724352569732346381583781607572237058000535328422

comment: A bounded simple-real reference; see (6).

1:

1.207423594952871259436378817028286995385348944644442537538620971681617349026550483112472823650305294

comment: A bounded simple-real reference; see (6).

-2:

-0.4123025879563984880832340546114610420345348344724047288238769642753796616753812761702252538581759744

comment: A bounded simple-real reference; see (6).

2:

3.298094999978214710280604425223452422003975963403620787682923544837159147310985629250352203056941645

comment: A bounded simple-real reference; see (6).

-4:

0.3922347057069992895544918276468434832416121723328910995953531732714224517412370234650234624352322324

comment: A bounded simple-real reference; see (6).

4:

83.84707140846813992258049046104627624061181948502415561777460288274695876310729444037238191884939443

comment: A bounded simple-real reference; see (6).

-8:

-0.3312515807511378599698762392757070198124344449306921774254187891706234472477186845694486915508108118

comment: A bounded simple-real reference; see (6).

8:

1199586.004124459930881654499598420899578783127710177823328778141231718255640574547410381549350976396

comment: A bounded simple-real reference; see (6).

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

-0.1269848921460825838653928757366695333540743479619508317719156389590630850106172816882492569409736431

comment: A unit-cubic normalization reference; see (7).

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

0.2788818604144509205240460438919242623251747233180586328866051048759002035963441712278273688877871632

comment: A unit-cubic normalization reference; see (7).

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

0.9689991788799079886814475343047742347865523536746387626700561728552147829105441778266351624886021251

comment: A unit-cubic normalization reference; see (7).

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

1.682270755776719591374313416081952782737661326127086718893840079862315887645093727631822514955786192

comment: A unit-cubic normalization reference; see (7).

Definition

The Airy function $\operatorname{Bi}$ is the real solution of $w^{\prime\prime}(x)=xw(x)$ with initial values $\operatorname{Bi}(0)=1/(3^{1/6}\Gamma(2/3))$ and $\operatorname{Bi}^{\prime}(0)=3^{1/6}/\Gamma(1/3)$ [3]. This table gives its values at intrinsic and bounded reference arguments.

Parameters

$x$

— argument ($x\in\mathbb{R}$)

Formulas

(1)

$\operatorname{Bi}(0)=1/(3^{1/6}\Gamma(2/3))$ [3].

(2)

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

(3)

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

Comments

(4)

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

(5)

Let $0>b_1^{\prime}>b_2^{\prime}>\cdots$ be the negative real zeros of $\operatorname{Bi}^{\prime}$, ordered from the origin [6]. The selected values are the successive extremal heights, not the locations of the extrema. The origin is the initial-value anchor.

(6)

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].

(7)

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].

(8)

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 A358559: 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; their Wronskian is (2)

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

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

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 extremal heights specified in (5); 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.