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

Values of the Airy function of the first kind $\operatorname{Ai}(x)$

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Numbers

$x$

$\operatorname{Ai}$

0:

0.3550280538878172392600631860041831763979791741991772405833265103008100424501267129571742460540402717

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

0.5356566560156998611448602006953228548697317372329832593648404519597763592711322031778764666765192421

comment: The argument is specified by (5).

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

-0.4190154780325639543033713334465359795642227640011038207965882992682353078067950766071768435134950748

comment: The argument is specified by (5).

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

0.3804064686281532824017091751921365362825282891157382234977085784928367616767781078620507846696215453

comment: The argument is specified by (5).

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

-0.3579079437122922467589985390756460938629038223510948572853878252826341612923191009900771485375316955

comment: The argument is specified by (5).

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

0.3423012444116238257878067578110959467572126385339893159289263263950456241732446472040642533015322489

comment: The argument is specified by (5).

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

-0.3304762291479668663097346406633665475262948955684823179533424763064856648970137353636776290991839919

comment: The argument is specified by (5).

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

0.3210222881947164018629144990452626935563924464783966148591887689427731282196219002711715071359650853

comment: The argument is specified by (5).

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

-0.3131853909786817909914771015870301544706035686781488844343320287686951750415415192208584640425504663

comment: The argument is specified by (5).

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

0.3065172938827772002658739169567118363171258559165032641927982512528084459106621330064199481958598085

comment: The argument is specified by (5).

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

-0.3007308293226446446933688366197039557398206214229288910557863540089894023294915189849284901756573191

comment: The argument is specified by (5).

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

0.2956314810019132007739400895661577874460465757539564293214503993509917183464382321640529739919189340

comment: The argument is specified by (5).

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

-0.2910816772035390565976348867246890492559344383248175027024356438821073467835915235443519011712271296

comment: The argument is specified by (5).

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

0.2869807069992019534562405282726014646832829372052459591434542369853940823623223655575384311012653641

comment: The argument is specified by (5).

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

-0.2832527361250212752431824018654940738472184073100799540631198358423164788399773610495185279545907951

comment: The argument is specified by (5).

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

0.2798393053604110449760006916469452456916043754043436527177740680926210925303321551640602617216897708

comment: The argument is specified by (5).

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

-0.2766944450689298393882768648480481012007360925228792229328356459105714749354561574554413315949566988

comment: The argument is specified by (5).

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

0.2737813856466851413700862033214047269640006496239790454134310179688535716542221057445832577669040427

comment: The argument is specified by (5).

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

-0.2710702785769711720135487151133934262360852821430727189998367988595209382468706984328873038082495413

comment: The argument is specified by (5).

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

0.2685365782821764580308536164562198363403771232084311445344724144263230995135481214435756201005278686

comment: The argument is specified by (5).

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

-0.2661598682157092155171377712940553427060682983846510760907578331306514363150981742007530710661780820

comment: The argument is specified by (5).

-1/2:

0.4757280916105395887986437782813071504876098160591710878821805899071093720740111649747301964264977787

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

1/2:

0.2316936064808334897691252545099217396183864753577499817922638265232339844931838503861302580597311124

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

-1:

0.5355608832923521187995165656388747074669308976836170027706315172054360568644817353292767459848259814

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

1:

0.1352924163128814155241474235154663061749441429883307060091020547576335348022657236634871099087486832

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

-2:

0.2274074282016855759919244360378737994607722254170967164957900340170790486510887973777793656962577680

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

2:

0.03492413042327437913532208079180760976106021389758320718866991321215972168770820861293170885792461996

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

-4:

-0.07026553294928951509908431163180311641823742785379146759843349896453084072583925266113103705117915165

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

4:

0.0009515638512048018736214999689001287600276878022737792596183295509525202656533653029817761460523261306

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

-8:

-0.05270505035638620262208267579388862081638303072854352688367921774637953016518782153362642364861936103

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

8:

4.692207616099231625649081703488224455252887658634213853311893662240473605151848180125905358668352460e-8

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

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

0.4948237529249096716093335424108646111164150807642533436503903585225386184905968597738656329612225649

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

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

0.5100323112100781916039615785098188120465749278142143834069557428568190046206050206396905358765083257

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

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

0.1904920731151781416385496045423164778244485256091190562750614536994786972488113881388196923681763534

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

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

0.08348893157885108183182906215498748413709999389358170489408793008920537194573278259558940952133930154

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

Definition

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

Parameters

$x$

— argument ($x\in\mathbb{R}$)

Formulas

(1)

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

(2)

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

(3)

$\int_0^\infty\cos(u^3+ku)\,du=\pi3^{-1/3}\operatorname{Ai}(k/3^{1/3})$.

Comments

(4)

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

(5)

Let $0>a_1^{\prime}>a_2^{\prime}>\cdots$ be the negative real zeros of $\operatorname{Ai}^{\prime}$, 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 A284867: decimal expansion of $\operatorname{Ai}(0)$

[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 derivative of the Airy function of the first kind $\operatorname{Ai}'(x)$ — its derivative

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

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

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.