A212002 Decimal expansion of (2*Pi)^2.

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

Offset: 2

Author

Omar E. Pol, Aug 11 2012

Keywords

• nonn
• cons

Comments

This constant appears in Kepler's 3rd Law, T^2 = (2*Pi)^2/GM*a^3 where a is the semi-major axis of a planet orbiting the Sun, T is its period, and GM is the standard gravitational parameter. - Raphie Frank, Dec 13 2012

García & Marco give a generalized zeta regularization by which this is the value of the product of the primes. - Charles R Greathouse IV, Jun 17 2013

Examples

			39.4784176043574344753379639995046045412547976289631...
		

Links

• J.-P. Allouche, The zeta-regularized product of odious numbers, arXiv:1906.10532 [math.NT], 2019.
• Hyperphysics, Kepler's Laws
• E. Muñoz García and R. Pérez Marco, The product over all primes is 4Pi^2, Communications in Mathematical Physics, Vol. 277, No. 1 (2008), pp. 69-81.
• Hengguang Li and Jeffrey S. Ovall, A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential, Discrete and Continuous Dynamical Systems Series B, Volume 20, Number 5, July 2015, pp. 1377-1391. Also doi:10.3934/dcdsb.2015.20.1377
• Wikipedia, Pendulum.
• Index entries for transcendental numbers

Crossrefs

Cf. A000796, A002388, A019692, A212003.

Programs

Mathematica

RealDigits[(2*Pi)^2,10,120][[1]] (* Harvey P. Dale, Mar 27 2019 *)

PARI

4*Pi^2 \\ Charles R Greathouse IV, Jun 17 2013

Formula

Equals Product_{k=1..10, gcd(k,10)==1} Gamma(k/10) = Gamma(1/10)*Gamma(3/10)*Gamma(7/10)*Gamma(9/10). - Amiram Eldar, Jun 12 2021

Equals lim_{n->oo} |B(2*n)/B(2*n+2)|*(2*n+1)*(2*n+2), where B(n) denotes the n-th Bernoulli number. - Peter Luschny, Dec 09 2021