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A256667 Decimal expansion of Integral_{x=0..Pi/2} sqrt(2-sin(x)^2) dx, an elliptic integral once studied by John Landen.

%I A256667 #16 Feb 16 2025 08:33:25
%S A256667 1,9,1,0,0,9,8,8,9,4,5,1,3,8,5,6,0,0,8,9,5,2,3,8,1,0,4,1,0,8,5,7,2,1,
%T A256667 6,4,5,9,5,4,9,8,3,8,0,7,3,2,3,6,3,7,3,6,0,5,4,0,2,4,8,3,2,8,3,7,3,5,
%U A256667 9,7,9,0,0,6,0,7,1,6,4,9,6,0,5,3,3,0,9,0,5,4,4,7,2,5,6,1,1,2,4,1,4,1,1,0,2
%N A256667 Decimal expansion of Integral_{x=0..Pi/2} sqrt(2-sin(x)^2) dx, an elliptic integral once studied by John Landen.
%C A256667 Arclength on sine from origin to first maximum point. - _Clark Kimberling_, Jul 01 2020
%D A256667 Mark Pinsky, Björn Birnir, Probability, Geometry and Integrable Systems (Cambridge University Press 2007), p. 289.
%H A256667 G. C. Greubel, <a href="/A256667/b256667.txt">Table of n, a(n) for n = 1..10000</a>
%H A256667 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>
%H A256667 Wikipedia, <a href="http://en.wikipedia.org/wiki/John_Landen">John Landen</a>
%F A256667 Equals (1/sqrt(2*Pi))*(Gamma(3/4)^2 + 4*Gamma(5/4)^2).
%F A256667 Equals sqrt(2)*E(Pi/2 | 1/2), where E(phi|m) is the elliptic integral of the second kind.
%F A256667 Equals (L^2 + Pi)/(2*L), where L is the lemniscate constant 2.622...
%e A256667 1.91009889451385600895238104108572164595498380732363736...
%t A256667 RealDigits[(1/Sqrt[2*Pi])*(Gamma[3/4]^2 + 4*Gamma[5/4]^2), 10, 105] // First
%o A256667 (PARI) default(realprecision, 100); (1/sqrt(2*Pi))*(gamma(3/4)^2 + 4*gamma(5/4)^2) \\ _G. C. Greubel_, Oct 07 2018
%o A256667 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (1/Sqrt(2*Pi(R)))*(Gamma(3/4)^2 + 4*Gamma(5/4)^2); // _G. C. Greubel_, Oct 07 2018
%Y A256667 Cf. A062539 (Lemniscate constant), A068465 (Gamma(3/4)), A068467 (Gamma(5/4)).
%K A256667 nonn,cons,easy
%O A256667 1,2
%A A256667 _Jean-François Alcover_, Apr 07 2015