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A261745
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Decimal expansion of -sm(-1), where sm(t) is the Dixonian elliptic function sm(t).
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0
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1, 2, 0, 5, 4, 1, 5, 1, 5, 1, 4, 0, 2, 9, 8, 3, 1, 5, 4, 8, 3, 1, 4, 1, 1, 3, 7, 5, 7, 8, 4, 4, 8, 8, 0, 1, 2, 0, 7, 2, 7, 0, 4, 1, 9, 1, 8, 8, 2, 2, 4, 9, 5, 8, 1, 0, 9, 3, 2, 7, 1, 8, 2, 3, 5, 4, 4, 7, 6, 4, 8, 8, 1, 0, 6, 5, 5, 1, 1, 2, 5, 5, 6, 3, 2, 1, 7, 0, 3, 6, 5, 2, 2, 9, 3, 7, 9, 8, 9, 0, 8, 0, 6, 7, 9
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OFFSET
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1,2
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COMMENTS
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In the context of particle physics and in the case of a Yule branching process with two types of particles, this constant appears in the asymptotic expression of the probability that all particles be of the second type at time t as exp(-t)*smh(1).
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LINKS
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EXAMPLE
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1.205415151402983154831411375784488012072704191882249581...
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MATHEMATICA
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sm[z_] := 6*WeierstrassP[z, {0, 1/27}]/(1 - 3*WeierstrassPPrime[z, {0, 1/27}]); N[-sm[-1], 105] // RealDigits // First
(* or, without using the Weierstrass P function: *) nint[y_?NumericQ] := NIntegrate[1/(1 + w^3)^(2/3), {w, 0, y}, WorkingPrecision -> 105]; smh[t_] := y /. FindRoot[nint[y] == t, {y, t}, WorkingPrecision -> 105]; N[smh[1], 105] // RealDigits // First
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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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