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
KEYWORD
nonn,cons
AUTHOR
Peter J. C. Moses, Nov 13 2013
STATUS
approved