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

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

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Numbers

$x$

$\operatorname{Ai}'$

0:

-0.2588194037928067984051835601892039634790911383549345822100018138561027726767902806541964058272753843

$a_{1}$:

0.7012108227206913624906916560315387022534715359933855379643126691976127710398608518377493521240067154

comment: The argument is specified by (6).

$a_{2}$:

-0.8031113696548639636343099280505852678839311283227355584017762597352663868547439347507237134461849081

comment: The argument is specified by (6).

$a_{3}$:

0.8652040258941519308429566587860278285878835833007741580190905679710363210221321497767088632933462131

comment: The argument is specified by (6).

$a_{4}$:

-0.9108507370496018030736526440918938574480162051681514152126527774545693381907498436074364581303213466

comment: The argument is specified by (6).

$a_{5}$:

0.9473357094415677655949310009899116771469101854235837094796361539913623475245200614723355871919806707

comment: The argument is specified by (6).

$a_{6}$:

-0.9779228085694986109515960484273489045871857055836224940588623848151457282686769687906287000352291029

comment: The argument is specified by (6).

$a_{7}$:

1.004370122660311968497799456809582888009760853728417341239641948190211788569373950594798524442465002

comment: The argument is specified by (6).

$a_{8}$:

-1.027738688820786176723305014920047279658011420333904688882674694821150912313589227865633973726904012

comment: The argument is specified by (6).

$a_{9}$:

1.048720648588189547955819536844809054793280954002668393407896906046648173714474841191934250453956889

comment: The argument is specified by (6).

$a_{10}$:

-1.067793859157427834651424574182770215274273104930874676094718882623807053817160336694649061321077512

comment: The argument is specified by (6).

$a_{11}$:

1.085302831350700032067143808829872492434114953510140037491982586323319774680517726243587662559408577

comment: The argument is specified by (6).

$a_{12}$:

-1.101504570277496811797553250185723087569340969913028813926439643597354845817547197010769382553894207

comment: The argument is specified by (6).

$a_{13}$:

1.116596177932656084545192817245263947935834404638422160236201217694790791511757313462773525772056014

comment: The argument is specified by (6).

$a_{14}$:

-1.130732310493187890175616553030514535232047384197904379235275290745145649761443009007942448575988128

comment: The argument is specified by (6).

$a_{15}$:

1.144036673273552677214724919036279507528304733717002695851493728058287206542114972611703630962112472

comment: The argument is specified by (6).

$a_{16}$:

-1.156609849116565463890191886252969395663981466217886598992189539711490446650989728566449121557689301

comment: The argument is specified by (6).

$a_{17}$:

1.168534784487524818317469117771759762291100445492598744280898281211220370244964566668619008480198988

comment: The argument is specified by (6).

$a_{18}$:

-1.179880729870145572917852685071615652006370204153307984892709019789786974647599023943543357001343901

comment: The argument is specified by (6).

$a_{19}$:

1.190706131158776652844314552263921905870989107149852334474045197203966709591164628065038497207446949

comment: The argument is specified by (6).

$a_{20}$:

-1.201060791519823280015985286945228407123189536861441178186395105774881162185156741330678412556688692

comment: The argument is specified by (6).

-1/2:

-0.2040816703395473861448172017949446083748692923967239852851209555900383808270922948466519047669094656

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

1/2:

-0.2249105326646838931359969903285832148250296356108928371362859557207680278598497074788531425605908780

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

-1:

-0.01016056711664520939504546984535756184189039546667066410539992972289181906672324397320081546018098660

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

1:

-0.1591474412967932127875002524972296865738892015116109694151995214891733834680021191886361558205320733

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

-2:

0.6182590207416910414062642913324752829157779451241469421598977665455728146715584141244688071497345446

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

2:

-0.05309038443365363170399918587870349124856099004587799263040296617761990531959672949111407830185987372

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

-4:

-0.7906285753685813802964544458279401013974397253692758957316929042559671440035492525285201227943342005

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

4:

-0.001958640950204178900138140918409032580845283313417362449886327399908245103441459839497729585503343834

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

-8:

0.9355609381983065510255224621326357323636987997800806725840767514878940450418883195124665559908901919

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

8:

-1.341439297906786574291153707932024241570012221325286972531616150994375149088833090014541829429010225e-7

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

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

0.2306934913340010467845118236281661375953596008440531371528265925688020318077615236919139853303498582

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

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

-0.1470024699403574160887620692734863315924503723104327670342255702782551810585021780814929816318988304

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

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

-0.2007291490168720600161301387649203384506142936405397335382713218180439544087160545816914656358763913

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

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

-0.1100406861106811491308990228353131028275237427920820814675322901966827996635607248894561832678518799

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

Definition

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

Parameters

$x$

— argument ($x\in\mathbb{R}$)

Formulas

(1)

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

(2)

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

(3)

$-\int_0^\infty u\sin(u^3+ku)\,du=\pi3^{-2/3}\operatorname{Ai}^{\prime}(k/3^{1/3})$.

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

(6)

Let $0>a_1>a_2>\cdots$ be the negative real zeros of $\operatorname{Ai}$, 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 A284868: 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 Airy function of the first kind $\operatorname{Ai}(x)$ — the function whose derivative this is

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

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

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.