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%I #19 Jun 02 2017 12:27:57
%S 0,3,9,4,9,2,9,2,2,7,7,1,6,6,3,5,8,9,5,1,6,4,0,3,7,4,6,9,9,0,8,1,4,6,
%T 1,1,2,0,1,0,6,6,0,4,5,8,2,4,3,0,7,0,6,6,6,9,5,0,2,7,8,7,4,2,6,6,4,5,
%U 3,8,1,5,4,5,7,7,0,9,1,7,3,4,5,7,9,7,3,4,7,5,4,8,6,5,9,1,0,7,1,2,0
%N Decimal expansion of r_0, a universal radius associated with mapping properties of analytic functions on the unit disk and with Dirichlet's integral.
%H Steven R. Finch, <a href="/A249414/a249414.pdf">Dirichlet Integral</a>, May 15, 2008. [Cached copy, with permission of the author]
%F r_0 = (2^(1/4)*K(-sqrt(2)) - K(-1/sqrt(2)))/(2^(1/4)*K(-sqrt(2)) + K(-1/sqrt(2))), where K is the complete elliptic integral of the first kind.
%e 0.039492922771663589516403746990814611201066045824307066695...
%p evalf((EllipticK(sqrt(2-sqrt(2))) - EllipticK(sqrt(sqrt(2)-1))) / (EllipticK(sqrt(2-sqrt(2))) + EllipticK(sqrt(sqrt(2)-1))), 120); # _Vaclav Kotesovec_, Oct 28 2014
%t r0 = (2^(1/4)*EllipticK[-Sqrt[2]] - EllipticK[-1/Sqrt[2]])/(2^(1/4)*EllipticK[-Sqrt[2]] + EllipticK[-1/Sqrt[2]]); Join[{0}, RealDigits[r0, 10, 100] // First]
%t Prepend[RealDigits[2/(1 + EllipticK[Sqrt[2] - 1]/EllipticK[2 - Sqrt[2]]) - 1, 10, 100][[1]], 0] (* _Jan Mangaldan_, Jan 04 2017 *)
%K nonn,cons,easy
%O 0,2
%A _Jean-François Alcover_, Oct 28 2014
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