OFFSET
1,2
COMMENTS
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).
LINKS
Eric van Fossen Conrad and Philippe Flajolet, The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion, Séminaire Lotharingien de Combinatoire 54 (2006), Article B54g, page 20.
Eric Weisstein's MathWorld, Weierstrass Elliptic Function
EXAMPLE
1.205415151402983154831411375784488012072704191882249581...
MATHEMATICA
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
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Aug 30 2015
STATUS
approved