A284867 -id:A284867 - cp's OEIS frontend

A284868 Decimal expansion of the derivative Ai'(0) (negated), where Ai is the Airy function of the first kind.

2, 5, 8, 8, 1, 9, 4, 0, 3, 7, 9, 2, 8, 0, 6, 7, 9, 8, 4, 0, 5, 1, 8, 3, 5, 6, 0, 1, 8, 9, 2, 0, 3, 9, 6, 3, 4, 7, 9, 0, 9, 1, 1, 3, 8, 3, 5, 4, 9, 3, 4, 5, 8, 2, 2, 1, 0, 0, 0, 1, 8, 1, 3, 8, 5, 6, 1, 0, 2, 7, 7, 2, 6, 7, 6, 7, 9, 0, 2, 8, 0, 6, 5, 4, 1, 9, 6, 4, 0, 5, 8, 2, 7, 2, 7, 5, 3, 8, 4, 3, 1, 3, 3, 7, 1

Offset: 0

Jean-François Alcover, Apr 04 2017

			-0.2588194037928067984051835601892039634790911383549345822100018138561...

Simon Plouffe, Airy Functions Constant C2
Eric Weisstein's MathWorld, Airy Functions
Wikipedia, Airy Function

Cf. A096714, A096715, A269892, A269893, A073005 (Gamma(1/3)), A284867 (Ai(0)).

Mathematica
```
RealDigits[AiryAi'[0], 10, 105][[1]]
```

-derivnum(x=0,airy(x)[1]) \\ Charles R Greathouse IV, Apr 26 2019

Ai'(0) = -1/(3^(1/3)*Gamma(1/3)).

A358559 Decimal expansion of Bi(0), where Bi is the Airy function of the second kind.

Original entry on oeis.org

6, 1, 4, 9, 2, 6, 6, 2, 7, 4, 4, 6, 0, 0, 0, 7, 3, 5, 1, 5, 0, 9, 2, 2, 3, 6, 9, 0, 9, 3, 6, 1, 3, 5, 5, 3, 5, 9, 4, 7, 2, 8, 1, 8, 8, 6, 4, 8, 5, 9, 6, 5, 0, 5, 0, 4, 0, 8, 7, 8, 7, 5, 3, 0, 1, 4, 2, 9, 6, 5, 1, 9, 3, 0, 5, 5, 2, 0, 6, 4, 0, 5, 2, 9, 3

Offset: 0

Dumitru Damian, Nov 22 2022

			0.61492662744600073515092236909361355359472818864859650504087875301429651...

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ISBN 978-0-12-525856-2, 1974.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 56, page 559.

[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.2.5.
Eric Weisstein's World of Mathematics, Airy Functions.
Wikipedia, Airy Function.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), this sequence (Bi(0)), A358561 (Bi'(0)), A358564(Gi(0)).

RealDigits[AiryBi[0], 10, 120][[1]] (* Amiram Eldar, Nov 28 2022 *)

PARI
```
airy(0)[2]
```
PARI
```
airy(0)[1]*sqrt(3)
```
PARI
```
3^(1/3)*gamma(1/3)/(2*Pi)
```

airy_bi(0).n(algorithm='scipy', prec=250)

Bi(0) = A284867*A002194.
Bi(0) = A358564*3.
Bi(0) = 1/(3^(1/6)*A073006).
Bi(0) = A073005/(3^(1/6)*A186706).
Bi(0) = A073005/(3^(1/6)*2*A093602).
Bi(0) = 3^(1/3)*A073005/(2*A000796).
Bi(0) = A252799/(3^(1/6)*BarnesG[5/3]).
Bi(0) = 3^(1/4)/(2^(2/9) * Pi^(1/3) * AGM(2,(sqrt(2+sqrt(3))))^(1/3)), where AGM is the arithmetic-geometric mean.

A358561 Decimal expansion of the derivative Bi'(0), where Bi is the Airy function of the second kind.

Original entry on oeis.org

4, 4, 8, 2, 8, 8, 3, 5, 7, 3, 5, 3, 8, 2, 6, 3, 5, 7, 9, 1, 4, 8, 2, 3, 7, 1, 0, 3, 9, 8, 8, 2, 8, 3, 9, 0, 8, 6, 6, 2, 2, 6, 7, 9, 9, 2, 1, 2, 2, 6, 2, 0, 6, 1, 0, 8, 2, 8, 0, 8, 7, 7, 8, 3, 7, 2, 3, 3, 0, 7, 5, 5, 0, 0, 9, 7, 8, 0, 6, 4, 7, 1, 8, 5, 0, 4

Offset: 0

Dumitru Damian, Nov 22 2022

			0.44828835735382635791482371039882839086622679921226206108280877837233075...

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ISBN 978-0-12-525856-2, 1974.

[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.2.6.
Eric Weisstein's World of Mathematics, Airy Functions.
Wikipedia, Airy Function.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), A358559 (Bi(0)), this sequence (Bi'(0)), A358564 (Gi(0)).

RealDigits[AiryBi'[0], 10, 120][[1]] (* Amiram Eldar, Nov 28 2022 *)

PARI
```
derivnum(x=0, airy(x)[2])
```

airy_bi_prime(0).n(algorithm='scipy', prec=250)

Bi'(0) = A284868*A002194.
Bi'(0) = 3*Gi'(0), where Gi' is the derivative of the inhomogeneous Airy function of the first kind.
Bi'(0) = 3^(1/6)/A073005.
Bi'(0) = A073006*3^(1/6)/A186706.
Bi'(0) = A073006*3^(1/6)/2*A093602.
Bi'(0) = 3^(2/3)*A073006/(2*A000796).
Bi'(0) = 3^(1/4)*AGM(2,(sqrt(2+sqrt(3))))^(1/3)/(2^(7/9) * Pi^(2/3)), where AGM is the arithmetic-geometric mean.

A358564 Decimal expansion of Gi(0), where Gi is the inhomogeneous Airy function of the first kind (also called Scorer function).

Original entry on oeis.org

2, 0, 4, 9, 7, 5, 5, 4, 2, 4, 8, 2, 0, 0, 0, 2, 4, 5, 0, 5, 0, 3, 0, 7, 4, 5, 6, 3, 6, 4, 5, 3, 7, 8, 5, 1, 1, 9, 8, 2, 4, 2, 7, 2, 9, 5, 4, 9, 5, 3, 2, 1, 6, 8, 3, 4, 6, 9, 5, 9, 5, 8, 4, 3, 3, 8, 0, 9, 8, 8, 3, 9, 7, 6, 8, 5, 0, 6, 8, 8, 0, 1, 7, 6, 4, 6, 2

Offset: 0

Dumitru Damian, Nov 22 2022

			0.204975542482000245050307456364537851198242729549532168346959584338098839...

Scorer, R. S., Numerical evaluation of integrals of the form Integral_{x=x1..x2} f(x)*e^(i*phi(x))dx and the tabulation of the function Gi(z)=(1/Pi)*Integral_{u=0..oo} sin(u*z+u^3/3) du, Quart. J. Mech. Appl. Math. 3 (1950), 107-112.

Amparo Gil, Javier Segura, and Nico Temme, On nonoscillating integrals for computing inhomogeneous Airy functions, Mathematics of Computation 70.235 (2001): 1183-1194.
[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.12.6.
Allan J. MacLeod, Computation of inhomogeneous Airy functions, Journal of Computational and Applied Mathematics, Volume 53, Issue 1, 1994, Pages 109-116, ISSN 0377-0427.
Wikipedia, Scorer's function.
Index entries for transcendental numbers.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), A358559 (Bi(0)), A358561 (Bi'(0)), this sequence (Gi(0)).

First[RealDigits[N[ScorerGi[0],90]]] (* Stefano Spezia, Nov 28 2022 *)

PARI
```
airy(0)[2]/3
```
PARI
```
1/(3^(7/6)*gamma(2/3))
```
PARI
```
sqrt(3)*gamma(1/3)/(3^(7/6)*2*Pi)
```

1/(3^(3/4)*2^(2/9)*Pi^(1/3)*sqrtn(agm(2,(sqrt(2+sqrt(3)))),3))

1/(3^(7/6)*gamma(2/3)).n(algorithm='scipy', prec=250)

Gi(0) = A358559/3.
Gi(0) = A284867/A002194.
Gi(0) = Hi(0)/2, where Hi is the inhomogeneous Airy function of the second kind.
Gi(0) = 1/(3^(7/6)*A073006).
Gi(0) = A073005/(3^(7/6)*A186706).
Gi(0) = A073005/(3^(7/6)*2*A093602).
Gi(0) = A073005/(3^(4/6)*2*A000796).
Gi(0) = A252799/(3^(7/6)*BarnesG(5/3)).
Gi(0) = 1/(3^(3/4) * 2^(2/9) * Pi^(1/3) * AGM(2,(sqrt(2+sqrt(3))))^(1/3)), where AGM is the arithmetic-geometric mean.

A385253 Decimal expansion of the function AiryAi(x) at x=1.

Original entry on oeis.org

1, 3, 5, 2, 9, 2, 4, 1, 6, 3, 1, 2, 8, 8, 1, 4, 1, 5, 5, 2, 4, 1, 4, 7, 4, 2, 3, 5, 1, 5, 4, 6, 6, 3, 0, 6, 1, 7, 4, 9, 4, 4, 1, 4, 2, 9, 8, 8, 3, 3, 0, 7, 0, 6, 0, 0, 9, 1, 0, 2, 0, 5, 4, 7, 5, 7, 6, 3, 3, 5, 3, 4, 8, 0, 2, 2, 6, 5, 7, 2, 3, 6, 6, 3, 4, 8, 7, 1, 0, 9, 9, 0, 8, 7, 4, 8, 6, 8, 3, 2, 1, 3, 0, 5, 3

Offset: 0

Artur Jasinski, Jul 29 2025

Second derivative at x=1 has that same value.

			0.1352924163128814155241474235154663...

Cf. A014402, A176730, A284867, A284868, A269893.

AiryAi(1) ; evalf(%) ; # R. J. Mathar, Jul 31 2025

Mathematica
```
RealDigits[AiryAi[1], 10, 105][[1]]
```

airy(1)[1] \\ Michel Marcus, Jul 29 2025

Equals AiryAi''(1).

cp's OEIS Frontend

A284868 Decimal expansion of the derivative Ai'(0) (negated), where Ai is the Airy function of the first kind.

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A358559 Decimal expansion of Bi(0), where Bi is the Airy function of the second kind.

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A358561 Decimal expansion of the derivative Bi'(0), where Bi is the Airy function of the second kind.

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A358564 Decimal expansion of Gi(0), where Gi is the inhomogeneous Airy function of the first kind (also called Scorer function).

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A385253 Decimal expansion of the function AiryAi(x) at x=1.

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