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

0,1

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

Table of n, a(n) for n=0..86.

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 Function.

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.

Example

0.204975542482000245050307456364537851198242729549532168346959584338098839...

Mathematica

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

Prog

(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)

Crossrefs

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

Sequence in context: A021492 A342201 A287314 * A077119 A002938 A111938

Adjacent sequences: A358561 A358562 A358563 * A358565 A358566 A358567

Keyword

cons,nonn

Author

Dumitru Damian, Nov 22 2022

Status

approved