|
| |
|
|
A030797
|
|
1.763222834351... raised to itself is E. Inverse of LambertW(1).
|
|
10
| |
|
|
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, 5, 6, 1, 9
(list; constant; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| Decimal expansion of the solution to y*ln(y)=1 - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (benoit7848c(AT)orange.fr), Aug 06 2002
From L. Edson Jeffery, Apr 12 2011: (Start)
(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. (End)
|
|
|
REFERENCES
| S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.
|
|
|
LINKS
| S. Plouffe, 1/W(1), the inverse of the omega number:W(1)
S. Plouffe, Plouffe's Inverter, 1/W(1), the inverse of the omega number:W(1)
S. Plouffe, Project Gutenberg Etext of Miscellaneous Mathematical Constants #13 in our math series
|
|
|
MATHEMATICA
| RealDigits[1/ProductLog[1], 10, 111][[1]] (from Robert G. Wilson v)
|
|
|
CROSSREFS
| Equals 1/A030178.
Sequence in context: A181152 A073011 A086312 * A019908 A021135 A198374
Adjacent sequences: A030794 A030795 A030796 * A030798 A030799 A030800
|
|
|
KEYWORD
| nonn,cons
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
|
|
|
EXTENSIONS
| Broken URL to Project Gutenberg replaced by Georg Fischer (Georg.Fischer(AT)T-Online.de), Jan 03 2009
|
| |
|
|
