A244382 - OEIS

%I #16 Dec 09 2017 07:43:50

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

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

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

%N Decimal expansion of 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. 466.

%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 exp(lambda*Pi), where lambda is the positive solution of the equation Pi/(exp(2*Pi*lambda)-1) = Sum_{k > 0} k/(k^2+lambda^2)*exp(-k*(Pi/(2*lambda))).

%e 7.187903351641062294405117549244421...

%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[Exp[lambda*Pi]] // First

%t RealDigits[N[E^(Pi Root[{(E^(2 Pi #) - 1) Beta[E^(-Pi/(2 #)), 1 - I #, -1] + (E^(2 Pi #) - 1) Beta[ E^(-Pi/(2 #)), 1 + I #, -1] + 2 Pi # &, 0.6278342676872}]), 100] // Chop][[1]] // Most (* _Eric W. Weisstein_, Dec 08 2017 *)

%Y Cf. A244381 (lambda).

%K nonn,cons,easy

%O 1,1

%A _Jean-François Alcover_, Jun 27 2014