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

Author

N. J. A. Sloane

Keywords

• nonn
• cons

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

Examples

			0.5671432904097838729999686622103555497538157871865125081351310792230457930866...
		

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.

Crossrefs

Cf. A019474, A059526, A059527, A238274.

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

Programs

Maple

evalf(LambertW(1));

Mathematica

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

PARI

solve(x=0,1,x*exp(x)-1) \\ Charles R Greathouse IV, Mar 20 2012

PARI

lambertw(1) \\ Stanislav Sykora, Nov 03 2013

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