A249414 Decimal expansion of r_0, a universal radius associated with mapping properties of analytic functions on the unit disk and with Dirichlet's integral.

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, 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, 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

Offset

0,2

Links

Table of n, a(n) for n=0..100.

Steven R. Finch, Dirichlet Integral, May 15, 2008. [Cached copy, with permission of the author]

Formula

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.

Example

0.039492922771663589516403746990814611201066045824307066695...

Maple

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

Mathematica

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]
Prepend[RealDigits[2/(1 + EllipticK[Sqrt[2] - 1]/EllipticK[2 - Sqrt[2]]) - 1, 10, 100][[1]], 0] (* Jan Mangaldan, Jan 04 2017 *)

Crossrefs

Sequence in context: A021721 A212002 A199452 * A011428 A070356 A143237

Adjacent sequences: A249411 A249412 A249413 * A249415 A249416 A249417

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Oct 28 2014

Status

approved