A019727 Decimal expansion of sqrt(2*Pi).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

Pickover says that the expression: lim_{n->oo} e^n(n!) / (n^n * sqrt(n)) = sqrt(2*Pi) is beautiful because it connects Pi, e, radicals, factorials and infinite limits. - Jason Earls, Mar 16 2001

Appears in the formula of the normal distribution. - Johannes W. Meijer, Feb 23 2013

sqrt(2*Pi)*sqrt(n) is the expected height of a labeled random tree of order n (see Rényi, Szekeres, 1967, formula (4.6)). - Hugo Pfoertner, May 18 2023

The constant in the formula known as "Stirling's approximation" (or "Stirling's formula"). It is sometimes called Stirling constant. The formula without the exact value of the constant was discovered by the French mathematician Abraham de Moivre (1667-1754), and was published in his book (1730). The exact value of the constant was found by the Scottish mathematician James Stirling (1692-1770) and was published in his book "Methodus differentialis" (1730). - Amiram Eldar, Jul 08 2023

Examples

			2.506628274631000502415765284811045253006986740609938316629923576342293....
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
• Clifford A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 307.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• Mohammad K. Azarian, An Expression for Pi, Problem #870, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. Solution appeared in Vol. 40, No. 1, January 2009, pp. 62-64.
• Abraham de Moivre, Miscellanea Analytica de Seriebus et Quadraturis, London, England: J. Tonson & J. Watts, 1730, pp. 96-106.
• K. Kimoto, N. Kurokawa, C. Sonoki, and M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.
• Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
• A. Rényi and G. Szekeres, On the height of trees, Journal of the Australian Mathematical Society , Volume 7 , Issue 4 , November 1967 , pp. 497-507.
• James Stirling, Methodus differentialis, sive tractatus de summatione et interpolatione serierum infinitarum, London, 1730. See Propositio XXVIII, pp. 135-139.
• Eric Weisstein's World of Mathematics, Normal Distribution.
• Wikipedia, Stirling's approximation.
• Index entries for transcendental numbers.

Crossrefs

Cf. A058293 (continued fraction), A231863 (inverse), A000796 (Pi).

Programs

Magma

R:= RealField(100); Sqrt(2*Pi(R)); // G. C. Greubel, Mar 08 2018

Mathematica

RealDigits[Sqrt[2Pi],10,120][[1]] (* Harvey P. Dale, Dec 12 2012 *)

Maxima

fpprec: 100$ ev(bfloat(sqrt(2*%pi))); /* Martin Ettl, Oct 11 2012 */

PARI

default(realprecision, 20080); x=sqrt(2*Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019727.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009

Formula

Equals lim_{n->oo} e^n*(n!)/n^n*sqrt(n).

Also equals Integral_{x >= 0} W(1/x^2) where W is the Lambert function, which is also known as ProductLog. - Jean-François Alcover, May 27 2013

Also equals the generalized Glaisher-Kinkelin constant A_0, see the Finch reference. - Jean-François Alcover, Dec 23 2014

Equals exp(-zeta'(0)). See Kimoto et al. - Michel Marcus, Jun 27 2019