A196515 Decimal expansion of the number x satisfying x*e^x = 2.
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
Offset: 0
Examples
0.852605502013725491346472414695317466898...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..5000
- Eric Weisstein's World of Mathematics, Lambert W-Function
- Index entries for transcendental numbers
Programs
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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] RealDigits[t] (* A030178 *) t = x /. FindRoot[E^x == 2/x, {x, 0.5, 1}, WorkingPrecision -> 100] RealDigits[t] (* A196515 *) t = x /. FindRoot[E^x == 3/x, {x, 0.5, 2}, WorkingPrecision -> 100] RealDigits[t] (* A196516 *) t = x /. FindRoot[E^x == 4/x, {x, 0.5, 2}, WorkingPrecision -> 100] RealDigits[t] (* A196517 *) t = x /. FindRoot[E^x == 5/x, {x, 0.5, 2}, WorkingPrecision -> 100] RealDigits[t] (* A196518 *) t = x /. FindRoot[E^x == 6/x, {x, 0.5, 2}, WorkingPrecision -> 100] RealDigits[t] (* A196519 *) RealDigits[ ProductLog[2], 10, 100] // First (* Jean-François Alcover, Feb 26 2013 *) (* 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] N[LambertW[z], 30] (* Mats Granvik, Nov 29 2013 *) RealDigits[LambertW[2], 10, 50][[1]] (* G. C. Greubel, Nov 16 2017 *) -
PARI
lambertw(2) \\ G. C. Greubel, Nov 16 2017
Formula
From A.H.M. Smeets, Nov 19 2018: (Start)
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)