A358561 -id:A358561 - cp's OEIS frontend

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

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.

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.

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A358559 Decimal expansion of 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).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

SageMath

Formula