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A231786
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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.
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1
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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
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OFFSET
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1,2
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REFERENCES
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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.
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LINKS
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EXAMPLE
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y'(0) = -1.588...
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MATHEMATICA
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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]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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