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

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, 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, 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

Offset

0,1

Comments

Sometimes called the Omega constant.

Infinite power tower for c = 1/E, i.e., c^c^c^..., where c = 1/A068985. - Stanislav Sykora, Nov 03 2013

Notice the narrow interval exp(-gamma) < w(1) < gamma, with gamma = A001620. - Jean-François Alcover, Dec 18 2013

Also the solution to x = -log(x). - Robert G. Wilson v, Feb 22 2014

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.

Links

G. C. Greubel and Stanislav Sykora, Table of n, a(n) for n = 0..10000 (terms 0..1999 from Stanislav Sykora)

Mateus Araújo, Andrew J. P. Garner, and Miguel Navascues, Non-commutative optimization problems with differential constraints, arXiv:2408.02572 [quant-ph], 2024. See p. 13.

blackpenredpen, Finding Omega, featuring Newton's method, video (2018).

Daniel Cummerow, Sound of Mathematics, Constants.

István Mező, An integral representation for the Lambert W function, arXiv:2012.02480 [math.CA], 2020.

Simon Plouffe, Lambert W(1, 0).

Simon Plouffe, The omega constant or W(1).

Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, Stan's Library, Vol. VI, 2016.

Eric Weisstein's World of Mathematics, Omega Constant.

Eric Weisstein's World of Mathematics, Lambert W-Function.

Wikipedia, Omega constant.

Wadim Zudilin, Diophantine problems related to the Omega constant, arXiv:2004.11029 [math.NT], 2020.

Index entries for transcendental numbers.

Formula

Equals 1/A030797.

Equals (1/Pi) * Integral_{x=0..Pi} log(1 + sin(x)*exp(x*cot(x))/x) dx (Mező, 2020). - Amiram Eldar, Jul 04 2021

Example

0.5671432904097838729999686622103555497538157871865125081351310792230457930866...

Maple

evalf(LambertW(1));

Mathematica

RealDigits[ ProductLog[1], 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)

Prog

(PARI) solve(x=0, 1, x*exp(x)-1) \\ Charles R Greathouse IV, Mar 20 2012
(PARI) lambertw(1) \\ Stanislav Sykora, Nov 03 2013

Crossrefs

Cf. A019474, A059526, A059527, A238274.

Cf. A276759 (another fixed point of -exp(z)).

Sequence in context: A306324 A214681 A019978 * A038458 A284361 A267017

Adjacent sequences: A030175 A030176 A030177 * A030179 A030180 A030181

Keyword

nonn,cons

Author

N. J. A. Sloane

Status

approved