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%I #11 Jun 27 2014 16:17:50
%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-François Alcover_, Jun 27 2014
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