A231786 Negative initial slope of the Thomas-Fermi equation, y"(x) = Sqrt( y(x)^3 / x), with boundary conditions y(0) = 1 and y(Infinity) = 0.

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

Offset

1,2

References

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999, p. 167.

Max Born, Atomic Physics, Blackie & Son Ltd., 8th. ed., 1969, p. 200.

Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th edition, World Scientific (Singapore, 2009), p. 422.

J. Schwinger, Quantum Mechanics: Symbolism of Atomic Measurements, Springer (2001), p. 419.

Links

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

P. Amore, J. Boyd, and F. Fernández, Accurate calculation of the solutions to the Thomas-Fermi equations, arXiv:1205.1704 [quant-ph], 2012-2014.

John P. Boyd, Rational Chebyshev series for the Thomas-Fermi function: Endpoint singularities and spectral methods, Journal of Computational and Applied Mathematics, 244 (2013), p. 90-101.

S. Esposito, Majorana solution of the Thomas-Fermi equation, Am. J. Phys. 70 (8), p 852-856. (2002); arXiv:physics/0111167 [physics.atom-ph], 2001.

R. P. Feynman, N. Metropolis, and E. Teller, Equations of State of Elements Based on the Generalized Fermi-Thomas Theory, Phys. Rev. 75, p 1561-1573 (1949); doi:10.1103/PhysRev.75.1561.

J. Schwinger, Thomas-Fermi model: The Leading correction, Phys. Rev A, vol 22 (1980), p. 1827-1832.

Example

y'(0) = -1.588...

Mathematica

nn = 150; Clear[a]; a[0] = 1; a[1] = 9 - Sqrt[73]; (* a[m]:=N[_, x] will give about x/3 digits and will calculate up to about a[4 x]. Example: to find 1000 digits, x needs to be 3000 and will calculate a[m] upto about a[12500] *) a[m_] := a[m] = N[(Sum[((n + 7) a[n - 1] + (n + 1) a[n + 1] - 2 (n + 4) a[n]) a[m - n], {n, m - 2}] + (a[1] (m + 6)) a[m - 2] + ((m + 7) - 2 a[1] (m + 3)) a[m - 1])/(2 (m + 8) - a[1] (m + 1)), nn]; RealDigits[N[(3/16)^(1/3) Sum[a[n], {n, 0, #[[2]]}], #[[1]]] &[{-MantissaExponent[a[#]][[2]] - 1, #} &[NestWhile[# + 1 &, 0, Precision[a[#]] > 5 &] - 1]]][[1]]

Crossrefs

Sequence in context: A039678 A259234 A131040 * A007450 A303816 A342647

Adjacent sequences: A231783 A231784 A231785 * A231787 A231788 A231789

Keyword

nonn,cons

Author

Peter J. C. Moses, Nov 13 2013

Status

approved