A249282 Decimal expansion of K(1/4), where K is the complete elliptic integral of the first kind.
1, 6, 8, 5, 7, 5, 0, 3, 5, 4, 8, 1, 2, 5, 9, 6, 0, 4, 2, 8, 7, 1, 2, 0, 3, 6, 5, 7, 7, 9, 9, 0, 7, 6, 9, 8, 9, 5, 0, 0, 8, 0, 0, 8, 9, 4, 1, 4, 1, 0, 8, 9, 0, 4, 4, 1, 1, 9, 9, 4, 8, 2, 9, 7, 8, 9, 3, 4, 3, 3, 7, 0, 2, 8, 8, 2, 3, 4, 6, 7, 6, 0, 4, 0, 6, 4, 5, 0, 9, 7, 3, 9, 3, 6, 6, 1, 2, 5, 7, 0, 3, 3
Offset: 1
Examples
1.685750354812596042871203657799076989500800894141089...
Links
- Steven R. Finch, Gergonne-Schwarz Surface, April 12, 2013. [Cached copy, with permission of the author]
- Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.
Programs
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Maple
evalf(EllipticK(1/2), 120); # Vaclav Kotesovec, Apr 22 2015
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Mathematica
RealDigits[EllipticK[1/4], 10, 102] // First
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PARI
ellK(1/2) \\ Charles R Greathouse IV, Feb 04 2025
Formula
From Paul D. Hanna, Mar 25 2024: (Start)
K(1/4) = Pi/2 * Sum_{n>=0} binomial(2*n,n)^2/16^n * (1/4)^n.
K(1/4) = Pi/2 * sqrt( Sum_{n>=0} binomial(2*n,n)^3/16^n * (m*(1-m))^n ), where m = 1/4. (End)