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A244381 Decimal expansion of 'lambda', a constant such that exp(lambda*Pi) is the best-known upper bound (as given by Julian Gevirtz) of the John constant for the unit disk. 1

%I

%S 6,2,7,8,3,4,2,6,7,6,8,7,2,1,3,1,6,8,2,8,3,8,3,0,5,6,6,2,9,2,4,8,8,6,

%T 8,7,6,4,5,1,8,7,3,4,2,4,3,4,9,3,9,4,3,4,3,4,3,8,4,3,5,1,5,1,9,7,3,6,

%U 0,9,1,2,2,1,9,4,9,0,6,3,6,6,6,5,7,2,2,9,8,4,2,7,8,6,8,1,5,0,2,2,7,9

%N Decimal expansion of 'lambda', a constant such that exp(lambda*Pi) is the best-known upper bound (as given by Julian Gevirtz) of the John constant for the unit disk.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 7.4 John Constant, p. 467.

%H Julian Gevirtz, <a href="http://dx.doi.org/10.1090/S0002-9939-1981-0627673-0">An upper bound for the John constant.</a>

%F Positive solution of the equation Pi/(exp(2*Pi*lambda)-1) = sum_(k=1..infinity) k/(k^2+lambda^2)*exp(-k*(Pi/(2*lambda))).

%e 0.62783426768721316828383...

%t eq = Pi/(Exp[2*Pi*x] - 1) == Sum[(k/(k^2 + x^2))*Exp[-k*(Pi/(2*x))], {k, 1, Infinity}]; lambda = x /. FindRoot[eq, {x, 1/2}, WorkingPrecision -> 102] // Re; RealDigits[lambda] // First

%Y Cf. A244382.

%K nonn,cons,easy

%O 0,1

%A _Jean-Fran├žois Alcover_, Jun 27 2014

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Last modified January 31 18:30 EST 2023. Contains 359980 sequences. (Running on oeis4.)