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...
EXAMPLE
1.91009889451385600895238104108572164595498380732363736...
MATHEMATICA
RealDigits[(1/Sqrt[2*Pi])*(Gamma[3/4]^2 + 4*Gamma[5/4]^2), 10, 105] // First
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
KEYWORD
AUTHOR
Jean-François Alcover, Apr 07 2015
STATUS
approved