

A030797


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


15



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
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OFFSET

1,2


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 multivalued, 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


REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448452.


LINKS



FORMULA



EXAMPLE

1.763222834351896710225201776951707080436017986667473634570456905547275847...


MATHEMATICA

RealDigits[x/.FindRoot[x^x==E, {x, 1}, WorkingPrecision>120]][[1]] (* Harvey P. Dale, Jun 19 2024 *)


PROG



CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS

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


STATUS

approved



