OFFSET
1,1
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 7.4 John Constant, p. 466.
LINKS
Julian Gevirtz, An upper bound for the John constant.
FORMULA
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))).
EXAMPLE
7.187903351641062294405117549244421...
MATHEMATICA
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
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 *)
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jun 27 2014
STATUS
approved