|
| |
|
|
A196515
|
|
Decimal expansion of the number x satisfying x*e^x = 2.
|
|
7
|
|
|
|
8, 5, 2, 6, 0, 5, 5, 0, 2, 0, 1, 3, 7, 2, 5, 4, 9, 1, 3, 4, 6, 4, 7, 2, 4, 1, 4, 6, 9, 5, 3, 1, 7, 4, 6, 6, 8, 9, 8, 4, 5, 3, 3, 0, 0, 1, 5, 1, 4, 0, 3, 5, 0, 8, 7, 7, 2, 1, 0, 7, 3, 9, 4, 6, 5, 2, 5, 1, 5, 0, 6, 5, 6, 7, 4, 2, 6, 3, 0, 4, 4, 8, 9, 6, 5, 7, 7, 3, 7, 8, 3, 5, 0, 2, 4, 9, 4, 8, 4, 7
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
LINKS
|
|
|
|
FORMULA
|
Equals LambertW(2).
Consider LambertW(z), where z is a complex number: let x(0) be an arbitrary complex number; x(n+1) = z*exp(-x(n)); if lim_{n -> inf) x(n) exists (which is the case for z = 2), then LambertW(z) = lim_{n -> inf) x(n). The region in the complex plane for which this seems to work is as follows: let z = x+iy, then -1/e < x < e for y = 0 and -c < y < c, c = 1.9612... for x = 0. It is not known if the area is open or closed. (End)
|
|
|
EXAMPLE
|
0.852605502013725491346472414695317466898...
|
|
|
MATHEMATICA
|
Plot[{E^x, 1/x, 2/x, 3/x, 4/x}, {x, 0, 2}]
t = x /. FindRoot[E^x == 1/x, {x, 0.5, 1}, WorkingPrecision -> 100]
t = x /. FindRoot[E^x == 2/x, {x, 0.5, 1}, WorkingPrecision -> 100]
t = x /. FindRoot[E^x == 3/x, {x, 0.5, 2}, WorkingPrecision -> 100]
t = x /. FindRoot[E^x == 4/x, {x, 0.5, 2}, WorkingPrecision -> 100]
t = x /. FindRoot[E^x == 5/x, {x, 0.5, 2}, WorkingPrecision -> 100]
t = x /. FindRoot[E^x == 6/x, {x, 0.5, 2}, WorkingPrecision -> 100]
(* A good approximation (the first 30 digits) is given by this power series evaluated at z=2, expanded at log(z): *)
Clear[x, a, nn, b, z]
z = 2;
nn = 100;
a = Series[Exp[-x], {x, N[Log[z], 50], nn}];
b = Normal[InverseSeries[Series[x/a, {x, 0, nn}]]];
x = z;
N[b, 30]
RealDigits[LambertW[2], 10, 50][[1]] (* G. C. Greubel, Nov 16 2017 *)
|
|
|
PROG
|
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
