cp's OEIS Frontend

A030178 Decimal expansion of LambertW(1): the solution to x*exp(x) = 1.

%I A030178 #136 Feb 16 2025 08:32:35
%S A030178 5,6,7,1,4,3,2,9,0,4,0,9,7,8,3,8,7,2,9,9,9,9,6,8,6,6,2,2,1,0,3,5,5,5,
%T A030178 4,9,7,5,3,8,1,5,7,8,7,1,8,6,5,1,2,5,0,8,1,3,5,1,3,1,0,7,9,2,2,3,0,4,
%U A030178 5,7,9,3,0,8,6,6,8,4,5,6,6,6,9,3,2,1,9,4,4,6,9,6,1,7,5,2,2,9,4,5,5,7,6,3,8
%N A030178 Decimal expansion of LambertW(1): the solution to x*exp(x) = 1.
%C A030178 Sometimes called the Omega constant.
%C A030178 Infinite power tower for c = 1/E, i.e., c^c^c^..., where c = 1/A068985. - _Stanislav Sykora_, Nov 03 2013
%C A030178 Notice the narrow interval exp(-gamma) < w(1) < gamma, with gamma = A001620. - _Jean-François Alcover_, Dec 18 2013
%C A030178 Also the solution to x = -log(x). - _Robert G. Wilson v_, Feb 22 2014
%D A030178 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.
%H A030178 G. C. Greubel and Stanislav Sykora, <a href="/A030178/b030178.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1999 from Stanislav Sykora)
%H A030178 Mateus Araújo, Andrew J. P. Garner, and Miguel Navascues, <a href="https://arxiv.org/abs/2408.02572">Non-commutative optimization problems with differential constraints</a>, arXiv:2408.02572 [quant-ph], 2024. See p. 13.
%H A030178 blackpenredpen, <a href="https://www.youtube.com/watch?v=EjUp_5X6io4">Finding Omega, featuring Newton's method</a>, video (2018).
%H A030178 Daniel Cummerow, <a href="https://web.archive.org/web/20091027122939/http://www.geocities.com/Vienna/9349/constants.html">Sound of Mathematics</a>, Constants.
%H A030178 István Mező, <a href="https://arxiv.org/abs/2012.02480">An integral representation for the Lambert W function</a>, arXiv:2012.02480 [math.CA], 2020.
%H A030178 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/omega.txt">Lambert W(1, 0)</a>.
%H A030178 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap70.html">The omega constant or W(1)</a>.
%H A030178 Stanislav Sykora, <a href="https://doi.org/10.3247/SL6Math16.002">Fixed points of the mappings exp(z) and -exp(z) in C</a>, Stan's Library, Vol. VI, 2016.
%H A030178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OmegaConstant.html">Omega Constant</a>.
%H A030178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%H A030178 Wikipedia, <a href="https://en.wikipedia.org/wiki/Omega_constant">Omega constant</a>.
%H A030178 Wadim Zudilin, <a href="https://arxiv.org/abs/2004.11029">Diophantine problems related to the Omega constant</a>, arXiv:2004.11029 [math.NT], 2020.
%H A030178 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A030178 Equals 1/A030797.
%F A030178 Equals (1/Pi) * Integral_{x=0..Pi} log(1 + sin(x)*exp(x*cot(x))/x) dx (Mező, 2020). - _Amiram Eldar_, Jul 04 2021
%e A030178 0.5671432904097838729999686622103555497538157871865125081351310792230457930866...
%p A030178 evalf(LambertW(1));
%t A030178 RealDigits[ ProductLog[1], 10, 111][[1]] (* _Robert G. Wilson v_, May 19 2004 *)
%o A030178 (PARI) solve(x=0,1,x*exp(x)-1) \\ _Charles R Greathouse IV_, Mar 20 2012
%o A030178 (PARI) lambertw(1) \\ _Stanislav Sykora_, Nov 03 2013
%Y A030178 Cf. A019474, A059526, A059527, A238274.
%Y A030178 Cf. A276759 (another fixed point of -exp(z)).
%K A030178 nonn,cons
%O A030178 0,1
%A A030178 _N. J. A. Sloane_