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

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

Author

Dumitru Damian, Nov 22 2022

Keywords

• cons
• nonn

Examples

			0.204975542482000245050307456364537851198242729549532168346959584338098839...
		

References

• 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.

Links

• 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.

Crossrefs

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

Programs

Mathematica

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)

PARI

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

SageMath

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

Formula

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.