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A202322 Decimal expansion of x satisfying x+2=exp(-x).

%I A202322 #23 Nov 21 2024 15:33:09
%S A202322 4,4,2,8,5,4,4,0,1,0,0,2,3,8,8,5,8,3,1,4,1,3,2,7,9,9,9,9,9,9,3,3,6,8,
%T A202322 1,9,7,1,6,2,6,2,1,2,9,3,7,3,4,7,9,6,8,4,7,1,7,7,3,3,0,7,6,9,8,2,0,1,
%U A202322 5,9,9,2,1,4,2,0,0,4,0,7,8,4,9,0,8,6,5,9,2,4,8,1,7,8,7,3,9,5,5
%N A202322 Decimal expansion of x satisfying x+2=exp(-x).
%C A202322 For many choices of u and v, there is just one value of x satisfying u*x+v=e^(-x).  Guide to related sequences, with graphs included in Mathematica programs:
%C A202322 u.... v.... x
%C A202322 1.... 2.... A202322
%C A202322 1.... 3.... A202323
%C A202322 2.... 2.... A202353
%C A202322 2.... e.... A202354
%C A202322 1... -1.... A202355
%C A202322 1.... 0.... A030178
%C A202322 2.... 0.... A202356
%C A202322 e.... 0.... A202357
%C A202322 3.... 0.... A202392
%C A202322 Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
%C A202322 For an example related to A202322, take f(x,u,v)=x+2-e^(-x) and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.
%H A202322 G. C. Greubel, <a href="/A202322/b202322.txt">Table of n, a(n) for n = 0..5000</a>
%H A202322 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A202322 x(u,v) = W(e^(v/u)/u) - v/u, where W = ProductLog = LambertW. - _Jean-François Alcover_, Feb 14 2013
%F A202322 Equals A226571 - 2 = LambertW(exp(2))-2. - _Vaclav Kotesovec_, Jan 09 2014
%e A202322 x=-0.442854401002388583141327999999336819716262...
%t A202322 (* Program 1:  A202322 *)
%t A202322 u = 1; v = 2;
%t A202322 f[x_] := u*x + v; g[x_] := E^-x
%t A202322 Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t A202322 r = x /. FindRoot[f[x] == g[x], {x, -.45, -.44}, WorkingPrecision -> 110]
%t A202322 RealDigits[r]  (* A202322 *)
%t A202322 (* Program 2: implicit surface of u*x+v=e^(-x) *)
%t A202322 f[{x_, u_, v_}] := u*x + v - E^-x;
%t A202322 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 2}]}, {v, 1, 3}, {u, 1, 3}];
%t A202322 ListPlot3D[Flatten[t, 1]] (* for A202322 *)
%t A202322 RealDigits[ ProductLog[E^2] - 2, 10, 99] // First (* _Jean-François Alcover_, Feb 14 2013 *)
%o A202322 (PARI) lambertw(exp(2)) - 2 \\ _G. C. Greubel_, Jun 10 2017
%Y A202322 Cf. A202320.
%K A202322 nonn,cons
%O A202322 0,1
%A A202322 _Clark Kimberling_, Dec 18 2011
%E A202322 Digits from a(84) on corrected by _Jean-François Alcover_, Feb 14 2013