|
%I #61 Mar 03 2018 09:51:05
%S 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,
%T 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,
%U 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
%N Decimal expansion of the constant x such that x^x = e. Inverse of W(1), where W is Lambert's function.
%C Decimal expansion of the solution to y*log(y) = 1. - _Benoit Cloitre_, Mar 30 2002
%C 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
%C 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
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.
%H G. C. Greubel, <a href="/A030797/b030797.txt">Table of n, a(n) for n = 1..10000</a>
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap49.html">1/W(1), the inverse of the omega number:W(1)</a>
%H Simon Plouffe, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/iomega.txt">1/W(1), the inverse of the omega number:W(1)</a>
%H Simon Plouffe, <a href="http://www.gutenberg.org/etext/634">Project Gutenberg Etext of Miscellaneous Mathematical Constants #13 in our math series</a>
%F Equals 1/A030178.
%e 1.763222834351896710225201776951707080436017986667473634570456905547275847...
%t RealDigits[1/ProductLog[1], 10, 111][[1]] (* _Robert G. Wilson v_ *)
%o (PARI) solve(x=1,2,x^x-exp(1)) \\ _Charles R Greathouse IV_, Apr 01 2012
%o (PARI) solve(x=1,2,log(x)*x - 1) \\ _John W. Nicholson_, Apr 10 2015
%o (PARI) 1/lambertw(1) \\ _G. C. Greubel_, Mar 02 2018
%Y Cf. A030178.
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_, _Simon Plouffe_
%E Broken URL to Project Gutenberg replaced by _Georg Fischer_, Jan 03 2009
|
