A242670 Decimal expansion of the (real) period of the elliptic function sn(x,1/2).

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, 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, 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

Offset

1,1

Comments

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).

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Eric Weisstein's MathWorld, Elliptic Integral of the First Kind

Eric Weisstein's MathWorld, Jacobi Elliptic Functions

Jean-François Alcover, Compare sin(x) with sn(x,1/2)

Formula

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

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

Also equals 4*EllipticK(1/4). - Jan Mangaldan, Jan 06 2017

Example

6.7430014192503841714848146311963079580032035765643561764797931915737348...

Maple

Re(evalf(8*EllipticK(I/sqrt(3))/sqrt(3), 120)); # Vaclav Kotesovec, Apr 22 2015

Mathematica

RealDigits[8*EllipticK[-1/3]/Sqrt[3], 10, 103] // First
RealDigits[4 EllipticK[1/4], 10, 103] // First (* Jan Mangaldan, Jan 06 2017 *)

Crossrefs

Sequence in context: A097410 A367312 A195792 * A196761 A362769 A195776

Adjacent sequences: A242667 A242668 A242669 * A242671 A242672 A242673

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jul 10 2014

Status

approved