A242670 - OEIS

%I #25 Apr 06 2017 02:28:22

%S 6,7,4,3,0,0,1,4,1,9,2,5,0,3,8,4,1,7,1,4,8,4,8,1,4,6,3,1,1,9,6,3,0,7,

%T 9,5,8,0,0,3,2,0,3,5,7,6,5,6,4,3,5,6,1,7,6,4,7,9,7,9,3,1,9,1,5,7,3,7,

%U 3,4,8,1,1,5,2,9,3,8,7,0,4,1,6,2,5,8,0,3,8,9,5,7,4,6,4,5,0,2,8,1,3,3,9

%N Decimal expansion of the (real) period of the elliptic function sn(x,1/2).

%C One can notice that, for real values of x, sin((gamma(3/4)^2/sqrt(Pi))*x) is extremely close to sn(x,1/2) (though different).

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.4.6 Elliptic Functions, p. 24.

%H G. C. Greubel, <a href="/A242670/b242670.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html">Elliptic Integral of the First Kind</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/JacobiEllipticFunctions.html">Jacobi Elliptic Functions</a>

%H Jean-François Alcover, <a href="/A242670/a242670.gif">Compare sin(x) with sn(x,1/2)</a>

%F 8*EllipticK(-1/3)/sqrt(3), where EllipticK is the complete elliptic integral of the first kind.

%F Also equals (4*Pi/sqrt(3))*2F1(1/2, 1/2; 1; -1/3), where 2F1 is the hypergeometric function.

%F Also equals 4*EllipticK(1/4). - _Jan Mangaldan_, Jan 06 2017

%e 6.7430014192503841714848146311963079580032035765643561764797931915737348...

%p Re(evalf(8*EllipticK(I/sqrt(3))/sqrt(3), 120)); # _Vaclav Kotesovec_, Apr 22 2015

%t RealDigits[8*EllipticK[-1/3]/Sqrt[3], 10, 103] // First

%t RealDigits[4 EllipticK[1/4], 10, 103] // First (* _Jan Mangaldan_, Jan 06 2017 *)

%K nonn,cons,easy

%O 1,1

%A _Jean-François Alcover_, Jul 10 2014