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

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Apr 07 2015

Keywords

• nonn
• cons
• easy

Comments

Arclength on sine from origin to first maximum point. - Clark Kimberling, Jul 01 2020

Examples

			1.91009889451385600895238104108572164595498380732363736...
		

References

• Mark Pinsky, Björn Birnir, Probability, Geometry and Integrable Systems (Cambridge University Press 2007), p. 289.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's MathWorld, Lemniscate Constant
• Wikipedia, John Landen

Crossrefs

Cf. A062539 (Lemniscate constant), A068465 (Gamma(3/4)), A068467 (Gamma(5/4)).

Programs

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

Mathematica

RealDigits[(1/Sqrt[2*Pi])*(Gamma[3/4]^2 + 4*Gamma[5/4]^2), 10, 105] // First

PARI

default(realprecision, 100); (1/sqrt(2*Pi))*(gamma(3/4)^2 + 4*gamma(5/4)^2) \\ G. C. Greubel, Oct 07 2018

Formula

Equals (1/sqrt(2*Pi))*(Gamma(3/4)^2 + 4*Gamma(5/4)^2).

Equals sqrt(2)*E(Pi/2 | 1/2), where E(phi|m) is the elliptic integral of the second kind.

Equals (L^2 + Pi)/(2*L), where L is the lemniscate constant 2.622...