A030797 Decimal expansion of the constant x such that x^x = e. Inverse of W(1), where W is Lambert's function.

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

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• cons

Comments

Decimal expansion of the solution to y*log(y) = 1. - Benoit Cloitre, Mar 30 2002

Let u(n+1) = exp(1/u(n)) then for any u(1) which is nonzero and real (positive or negative), lim n -> infinity u(n) = 1.763222834.... - Benoit Cloitre, Aug 06 2002

Conjecture: Another series can be defined as follows. Let z = a + b*i <> 0 be complex, and let z = v^v. Then log(z) + v = v*(1 + log(v)), so f(z, v) = (log(z) + v)/(1 + log(v)) = v. Suppose lim_{n -> infinity} (log(z) + v(n))/(1 + Log(v(n))) = v, for some sequence {v(n)}. Then, since v(n) -> v(n+1), similarly f_(n+1)(z, v) = v(n+1) = (log(z) + v(n))/(1 + log(v(n))). If Im(z) <> 0, recall that log(z) is multi-valued, so one might take both log(z) and log(v(n)) modulo 2*Pi*i. If Im(z) = 0 (i.e., if z is real), then one should use the recurrence f_(n+1)(z, v) = v(n+1) = (log(z) + v(n))/(1 + abs(log(v(n)))). For example, when z = e, we have lim_{n -> infinity} (1 + v(n))/(1 + abs(log(v(n)))) = 1.763222..., for v(0) <> 1/e, with apparent quadratic convergence, and most rapidly when v(0) = 1. Pathologies occur when v(0) is in the vicinity of a fixed point of f(z, v); e.g., if z = 2^(1/4), then such a fixed point is c = 0.806693797003867301..., so f_(n)(z, v) -> c, for all n, with a(0) near c. The constant c was calculated to 250 digits by Joerg Arndt. - L. Edson Jeffery, Apr 12 2011

Examples

			1.763222834351896710225201776951707080436017986667473634570456905547275847...
		

References

• S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Simon Plouffe, 1/W(1), the inverse of the omega number:W(1)
• Simon Plouffe, Plouffe's Inverter, 1/W(1), the inverse of the omega number:W(1)
• Simon Plouffe, Project Gutenberg Etext of Miscellaneous Mathematical Constants #13 in our math series

Crossrefs

Cf. A001113, A030178, A299614, A299617.

Programs

Mathematica

RealDigits[1/ProductLog[1], 10, 111][[1]] (* Robert G. Wilson v *)
RealDigits[x/.FindRoot[x^x==E,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jun 19 2024 *)

PARI

solve(x=1,2,x^x-exp(1)) \\ Charles R Greathouse IV, Apr 01 2012

PARI

solve(x=1,2,log(x)*x - 1) \\ John W. Nicholson, Apr 10 2015

PARI

1/lambertw(1) \\ G. C. Greubel, Mar 02 2018

Formula

Equals 1/A030178.

Equals e^A030178. - Colin Linzer, Nov 20 2024

Equals sqrt(A299614) = A299617/e. - Hugo Pfoertner, Nov 20 2024

Extensions

Broken URL to Project Gutenberg replaced by Georg Fischer, Jan 03 2009