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A366174
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Decimal expansion of a constant related to the asymptotics of A106336.
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1
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4, 9, 8, 3, 3, 4, 7, 9, 7, 9, 3, 3, 6, 0, 3, 4, 2, 2, 6, 0, 6, 3, 5, 9, 2, 6, 4, 0, 2, 8, 5, 0, 0, 1, 6, 4, 4, 3, 0, 6, 9, 4, 2, 8, 2, 3, 3, 0, 5, 1, 2, 9, 0, 2, 0, 1, 9, 9, 6, 8, 5, 3, 4, 9, 8, 3, 4, 0, 8, 7, 7, 4, 8, 5, 5, 2, 2, 7, 8, 4, 0, 2, 8, 9, 1, 1, 4, 7, 7, 3, 5, 2, 9, 2, 4, 3, 0, 3, 5, 0, 6, 2, 6, 0, 3, 1
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OFFSET
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0,1
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LINKS
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FORMULA
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EXAMPLE
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0.4983347979336034226063592640285001644306942823305129020199685349834...
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MATHEMATICA
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val = Sqrt[(s^2*(-1 + r^2*s^2)*Log[r^2*s^2]^2 * QPochhammer[-1, r*s]^2)/ (2*Pi*(-2*r^2*s^2*(-1 + r^2*s^2)*Log[r^2*s^2]^2* Derivative[0, 1][QPochhammer][-1, r*s]^2 + r*s*(-1 + r^2*s^2)*Log[r^2*s^2]*QPochhammer[-1, r*s]* ((Log[r^2*s^2] + 4*Log[1 - r^2*s^2] + 4*QPolyGamma[0, 1, r^2*s^2])* Derivative[0, 1][QPochhammer][-1, r*s] + r*s*Log[r^2*s^2] * Derivative[0, 2][QPochhammer][-1, r*s]) + 2*r^4*s^3*(-1 + r^2*s^2) * Log[r^2*s^2]^2*QPochhammer[-1, r*s]^3 * Derivative[0, 2][QPochhammer][r^2*s^2, r^2*s^2] + QPochhammer[-1, r*s]^2*(16*r^2*s^2*ArcTanh[1 - 2*r^2*s^2] + (-1 + r^2*s^2)*Log[r^2*s^2]^2 - 8*Log[1 - r^2*s^2] - 4*(-1 + r^2*s^2)* Log[1 - r^2*s^2]^2 - 8*(-1 + r^2*s^2)*(-1 + Log[1 - r^2*s^2])* QPolyGamma[0, 1, r^2*s^2] + (4 - 4*r^2*s^2)* QPolyGamma[0, 1, r^2*s^2]^2 + 4*(-1 + r^2*s^2)*(QPolyGamma[1, 1, r^2*s^2] - 2*r^2*s^2*Log[r^2*s^2]* Derivative[0, 0, 1][QPolyGamma][0, 1, r^2*s^2]))))] /. FindRoot[{2*s == QPochhammer[-1, r*s]*QPochhammer[r^2*s^2], (-2*Log[1 - r^2*s^2] - 2*QPolyGamma[0, 1, r^2*s^2])/ Log[r^2*s^2] + (r* s*(Derivative[0, 1][QPochhammer][-1, r*s] + r*QPochhammer[-1, r*s]^2* Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2]))/ QPochhammer[-1, r*s] == 1}, {r, 1/3}, {s, 2}, WorkingPrecision -> 600]; RealDigits[Chop[val], 10, -Floor[Log[10, Abs[Im[val]]]] - 3][[1]] (* r = A106335, s = A106334 *)
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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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