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

1,2

Comments

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

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

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...

From Artur Jasinski, Apr 29 2026: (Start)

Equals EllipticE(-1).

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

Equals 2*Integral_{x=0..Pi/2} 1/((1 + sin(x)^2)*sqrt(1 + sin(x)^2)) dx.

Equals A335930/2.

Equals A105419/4. (End)

Example

1.91009889451385600895238104108572164595498380732363736...

Mathematica

RealDigits[(1/Sqrt[2*Pi])*(Gamma[3/4]^2 + 4*Gamma[5/4]^2), 10, 105] // First
(* Alternative: *)
RealDigits[EllipticE[-1], 10, 105][[1]] (* Artur Jasinski, Apr 29 2026 *)

Prog

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

Crossrefs

Cf. A062539 (Lemniscate constant), A068465 (Gamma(3/4)), A068467 (Gamma(5/4)), A105419, A335930.

Sequence in context: A155783 A382006 A257097 * A175764 A388595 A388388

Adjacent sequences: A256664 A256665 A256666 * A256668 A256669 A256670

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Apr 07 2015

Status

approved