A262427 Decimal expansion of the complete elliptic integral of the first kind at sqrt(2*sqrt(2) - 2).
2, 3, 2, 7, 1, 8, 5, 1, 4, 2, 4, 3, 6, 5, 3, 8, 7, 5, 0, 6, 0, 5, 0, 3, 6, 2, 8, 5, 6, 1, 8, 3, 5, 7, 0, 7, 7, 5, 1, 5, 1, 8, 1, 7, 5, 8, 2, 3, 2, 5, 4, 1, 1, 7, 4, 7, 9, 3, 2, 0, 8, 1, 9, 9, 4, 4, 6, 1, 1, 8, 8, 2, 5, 7, 3, 1, 3, 6, 0, 4, 9, 5, 7, 8, 2, 2, 5, 9, 0, 0, 7, 0, 1, 1, 0, 6, 6, 1, 0, 5, 6, 2, 3, 7, 1
Offset: 1
Examples
2.3271851424365387506050362856183570775151817582325411747932...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..10000
- M. L. Glasser and V. E. Wood, A closed form evaluation of the elliptic integral, Math. Comp. 25 (1971), 535-536.
Crossrefs
Cf. A130786.
Programs
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Magma
SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)^(3/2)*Sqrt(4 + 2*Sqrt(2))/(4*Gamma(5/8)*Gamma(7/8)); // G. C. Greubel, Oct 07 2018
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Maple
evalf(sqrt(2)*EllipticK(sqrt(2)-1), 120); # Vaclav Kotesovec, Sep 22 2015 evalf(int(2/sqrt(1-x^8), x=0..1), 120); # Christian N. Hofmann, Jun 28 2023
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Mathematica
K[x_] := EllipticK[x^2/(x^2 - 1)]/Sqrt[1 - x^2]; RealDigits[ K[Sqrt[2 Sqrt[2] - 2]], 10, 105][[1]]
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PARI
ellk(k)=intnum(t=0,1,1/sqrt((1-t^2)*(1-(k*t)^2))) sqrt(2)*ellk(sqrt(2)-1) \\ Charles R Greathouse IV, Apr 18 2016
Formula
Equals Pi^(3/2)*sqrt(4 + 2*sqrt(2))/(4*Gamma(5/8)*Gamma(7/8)).
Also equals sqrt(2)*K(sqrt(2) - 1).
Also equals 2*Integral_{x=0..1} 1/sqrt(1-x^8) dx. - Christian N. Hofmann, Jun 24 2023
Also equals Pi^(3/2)*cos(Pi/4)*cos(Pi/8)/(Gamma(5/8)*Gamma(7/8)). - Christian N. Hofmann, Aug 20 2023
Equals Gamma(1/8)^2 / (2^(11/4) * Gamma(1/4)). - Vaclav Kotesovec, Apr 15 2024