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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.
2
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
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
Cf. A062539 (Lemniscate constant), A068465 (Gamma(3/4)), A068467 (Gamma(5/4)).
Sequence in context: A166734 A155783 A257097 * A175764 A269948 A121935
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved