A358450 Decimal expansion of 2*EllipticK(i) - EllipticE(i), reciprocal of A088375.
7, 1, 1, 9, 5, 8, 6, 5, 9, 7, 7, 8, 2, 6, 3, 8, 0, 1, 5, 1, 2, 4, 5, 8, 5, 4, 8, 8, 0, 5, 3, 9, 7, 7, 6, 7, 7, 2, 7, 7, 7, 1, 1, 4, 4, 1, 0, 7, 9, 8, 5, 8, 0, 2, 2, 7, 6, 5, 7, 3, 3, 7, 5, 4, 2, 7, 1, 9, 2, 6, 8, 6, 4, 6, 3, 2, 4, 9, 2, 8, 9, 6, 9, 7, 2, 0
Offset: 0
Examples
0.7119586597782638015124585488053977677277711441...
Crossrefs
Cf. A088375.
Programs
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Maple
Digits := 100: a := sqrt(2)*GAMMA(5/4)^2: b := Pi/4: evalf((a/b - b/a)*b^(1/2), Digits)*10^90: ListTools:-Reverse(convert(floor(%), base, 10));
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Mathematica
With[{a = Sqrt[2]*Gamma[5/4]^2, b = Pi/4}, RealDigits[(a/b - b/a)*b^(1/2), 10, 120][[1]]] (* Amiram Eldar, Nov 19 2022 *) -
PARI
2*ellK(I) - ellE(I) \\ Charles R Greathouse IV, Feb 04 2025
Formula
Equals (a/b - b/a)*b^(1/2), where a = sqrt(2)*Gamma(5/4)^2 and b = Pi/4.
Equals sqrt(2) * (EllipticK(sqrt(2)/2) - EllipticE(sqrt(2)/2)).
Equals Integral_{x=0..Pi/2} cos(x)^2 / sqrt(1 + sin(x)^2).