A064853 Decimal expansion of the Lemniscate constant.
5, 2, 4, 4, 1, 1, 5, 1, 0, 8, 5, 8, 4, 2, 3, 9, 6, 2, 0, 9, 2, 9, 6, 7, 9, 1, 7, 9, 7, 8, 2, 2, 3, 8, 8, 2, 7, 3, 6, 5, 5, 0, 9, 9, 0, 2, 8, 6, 3, 2, 4, 6, 3, 2, 5, 6, 3, 3, 6, 4, 3, 4, 0, 7, 6, 0, 1, 5, 8, 1, 1, 7, 4, 1, 4, 0, 8, 2, 8, 5, 0, 0, 4, 6, 0, 5, 9, 1, 0, 6, 5, 9, 2, 2, 8, 5, 8, 1, 8, 6, 8, 9
Offset: 1
Examples
5.244115108584239620929679...
Links
- Harry J. Smith, Table of n, a(n) for n = 1..5000
- Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 15.
- Lorenz Milla, World record computation of Lemniscate Constant (2,000,000,000,000 digits)
- Eric Weisstein's World of Mathematics, Lemniscate Constant.
- Eric Weisstein's World of Mathematics, Lemniscate.
- Index entries for transcendental numbers.
Programs
-
Magma
SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(1/4)^2/Sqrt(2*Pi(R)); // G. C. Greubel, Oct 07 2018
-
Mathematica
First@RealDigits[ N[ Gamma[ 1/4 ]^2/Sqrt[ 2 Pi ], 102 ] ]
-
PARI
{ allocatemem(932245000); default(realprecision, 5080); x=gamma(1/4)^2/sqrt(2*Pi); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b064853.txt", n, " ", d)); } \\ Harry J. Smith, Jun 20 2009 -
PARI
gamma(1/2)*gamma(1/4)/gamma(3/4) \\ Charles R Greathouse IV, Oct 29 2021
Formula
Equals Gamma(1/4)^2/sqrt(2*Pi). - G. C. Greubel, Oct 07 2018
From Stefano Spezia, Sep 23 2022: (Start)
Equals 4*Integral_{x=0..Pi/2} 1/sqrt(2*(1 - (1/2)*sin(x)^2)) dx [Gauss, 1799] (see Faulhuber et al.).
Equals 2*sqrt(2)*A093341. (End)