A358564 - OEIS

A358564

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

%I #19 Nov 26 2024 14:53:08

%S 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,

%T 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,

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

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

%D 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.

%H Amparo Gil, Javier Segura, and Nico Temme, <a href="https://doi.org/10.1090/S0025-5718-00-01268-0">On nonoscillating integrals for computing inhomogeneous Airy functions</a>, Mathematics of Computation 70.235 (2001): 1183-1194.

%H [DLMF] NIST Digital Library of Mathematical Functions, <a href="https://dlmf.nist.gov/9.12.E6">Eq. 9.12.6</a>.

%H Allan J. MacLeod, <a href="https://doi.org/10.1016/0377-0427(94)90196-1">Computation of inhomogeneous Airy functions</a>, Journal of Computational and Applied Mathematics, Volume 53, Issue 1, 1994, Pages 109-116, ISSN 0377-0427.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Scorer's_function">Scorer's function</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Gi(0) = A358559/3.

%F Gi(0) = A284867/A002194.

%F Gi(0) = Hi(0)/2, where Hi is the inhomogeneous Airy function of the second kind.

%F Gi(0) = 1/(3^(7/6)*A073006).

%F Gi(0) = A073005/(3^(7/6)*A186706).

%F Gi(0) = A073005/(3^(7/6)*2*A093602).

%F Gi(0) = A073005/(3^(4/6)*2*A000796).

%F Gi(0) = A252799/(3^(7/6)*BarnesG(5/3)).

%F 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.

%e 0.204975542482000245050307456364537851198242729549532168346959584338098839...

%t First[RealDigits[N[ScorerGi[0],90]]] (* _Stefano Spezia_, Nov 28 2022 *)

%o (PARI) airy(0)[2]/3

%o (PARI) 1/(3^(7/6)*gamma(2/3))

%o (PARI) sqrt(3)*gamma(1/3)/(3^(7/6)*2*Pi)

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

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

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

%K cons,nonn

%O 0,1

%A _Dumitru Damian_, Nov 22 2022