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A196515 Decimal expansion of the number x satisfying x*e^x = 2.

%I A196515 #47 Feb 16 2025 08:33:15
%S A196515 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,
%T A196515 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,
%U A196515 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
%N A196515 Decimal expansion of the number x satisfying x*e^x = 2.
%H A196515 G. C. Greubel, <a href="/A196515/b196515.txt">Table of n, a(n) for n = 0..5000</a>
%H A196515 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>
%H A196515 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A196515 From _A.H.M. Smeets_, Nov 19 2018: (Start)
%F A196515 Equals LambertW(2).
%F A196515 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)
%e A196515 0.852605502013725491346472414695317466898...
%t A196515 Plot[{E^x, 1/x, 2/x, 3/x, 4/x}, {x, 0, 2}]
%t A196515 t = x /. FindRoot[E^x == 1/x, {x, 0.5, 1}, WorkingPrecision -> 100]
%t A196515 RealDigits[t]  (* A030178 *)
%t A196515 t = x /. FindRoot[E^x == 2/x, {x, 0.5, 1}, WorkingPrecision -> 100]
%t A196515 RealDigits[t]  (* A196515 *)
%t A196515 t = x /. FindRoot[E^x == 3/x, {x, 0.5, 2}, WorkingPrecision -> 100]
%t A196515 RealDigits[t]  (* A196516 *)
%t A196515 t = x /. FindRoot[E^x == 4/x, {x, 0.5, 2}, WorkingPrecision -> 100]
%t A196515 RealDigits[t]  (* A196517 *)
%t A196515 t = x /. FindRoot[E^x == 5/x, {x, 0.5, 2}, WorkingPrecision -> 100]
%t A196515 RealDigits[t]  (* A196518 *)
%t A196515 t = x /. FindRoot[E^x == 6/x, {x, 0.5, 2}, WorkingPrecision -> 100]
%t A196515 RealDigits[t]  (* A196519 *)
%t A196515 RealDigits[ ProductLog[2], 10, 100] // First (* _Jean-François Alcover_, Feb 26 2013 *)
%t A196515 (* A good approximation (the first 30 digits) is given by this power series evaluated at z=2, expanded at log(z): *)
%t A196515 Clear[x, a, nn, b, z]
%t A196515 z = 2;
%t A196515 nn = 100;
%t A196515 a = Series[Exp[-x], {x, N[Log[z], 50], nn}];
%t A196515 b = Normal[InverseSeries[Series[x/a, {x, 0, nn}]]];
%t A196515 x = z;
%t A196515 N[b, 30]
%t A196515 N[LambertW[z], 30] (* _Mats Granvik_, Nov 29 2013 *)
%t A196515 RealDigits[LambertW[2], 10, 50][[1]] (* _G. C. Greubel_, Nov 16 2017 *)
%o A196515 (PARI) lambertw(2) \\ _G. C. Greubel_, Nov 16 2017
%K A196515 nonn,cons
%O A196515 0,1
%A A196515 _Clark Kimberling_, Oct 03 2011