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A256667
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Decimal expansion of Integral_{x=0..Pi/2} sqrt(2-sin(x)^2) dx, an elliptic integral once studied by John Landen.
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1
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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
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OFFSET
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1,2
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COMMENTS
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Arclength on sine from origin to first maximum point. - Clark Kimberling, Jul 01 2020
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REFERENCES
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Mark Pinsky, Björn Birnir, Probability, Geometry and Integrable Systems (Cambridge University Press 2007), p. 289.
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LINKS
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FORMULA
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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...
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EXAMPLE
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1.91009889451385600895238104108572164595498380732363736...
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MATHEMATICA
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RealDigits[(1/Sqrt[2*Pi])*(Gamma[3/4]^2 + 4*Gamma[5/4]^2), 10, 105] // First
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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