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Decimal expansion of sqrt(2*Pi).
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%I #93 Nov 15 2024 12:55:20

%S 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,

%T 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,

%U 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

%N Decimal expansion of sqrt(2*Pi).

%C 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

%C Appears in the formula of the normal distribution. - _Johannes W. Meijer_, Feb 23 2013

%C 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

%C 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

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

%D Clifford A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 307.

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.

%H Harry J. Smith, <a href="/A019727/b019727.txt">Table of n, a(n) for n = 1..20000</a>

%H Mohammad K. Azarian, <a href="http://www.jstor.org/stable/27646572">An Expression for Pi, Problem #870</a>, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. <a href="http://www.jstor.org/stable/27646723">Solution</a> appeared in Vol. 40, No. 1, January 2009, pp. 62-64.

%H Abraham de Moivre, <a href="https://archive.org/details/bub_gb_TFX1165yEc4C">Miscellanea Analytica de Seriebus et Quadraturis</a>, London, England: J. Tonson & J. Watts, 1730, pp. 96-106.

%H K. Kimoto, N. Kurokawa, C. Sonoki, and M. Wakayama, <a href="https://doi.org/10.2996/kmj/1104247354">Some examples of generalized zeta regularized products</a>, Kodai Math. J. 27 (2004), 321-335.

%H Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="https://zbmath.org/?q=an:0983.00008">Zentralblatt review</a>.

%H A. Rényi and G. Szekeres, <a href="https://doi.org/10.1017/S1446788700004432">On the height of trees</a>, Journal of the Australian Mathematical Society , Volume 7 , Issue 4 , November 1967 , pp. 497-507.

%H James Stirling, <a href="https://archive.org/details/bub_gb_71ZHAAAAYAAJ/page/n145/mode/2up">Methodus differentialis, sive tractatus de summatione et interpolatione serierum infinitarum</a>, London, 1730. See Propositio XXVIII, pp. 135-139.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NormalDistribution.html">Normal Distribution</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation">Stirling's approximation</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

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

%F 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

%F Also equals the generalized Glaisher-Kinkelin constant A_0, see the Finch reference. - _Jean-François Alcover_, Dec 23 2014

%F Equals exp(-zeta'(0)). See Kimoto et al. - _Michel Marcus_, Jun 27 2019

%e 2.506628274631000502415765284811045253006986740609938316629923576342293....

%t RealDigits[Sqrt[2Pi],10,120][[1]] (* _Harvey P. Dale_, Dec 12 2012 *)

%o (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

%o (Maxima) fpprec: 100$ ev(bfloat(sqrt(2*%pi))); /* _Martin Ettl_, Oct 11 2012 */

%o (Magma) R:= RealField(100); Sqrt(2*Pi(R)); // _G. C. Greubel_, Mar 08 2018

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

%K nonn,cons,changed

%O 1,1

%A _N. J. A. Sloane_