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A202537 Decimal expansion of x satisfying e^x-e^(-2x)=1.

%I A202537 #15 Feb 07 2025 13:04:40
%S A202537 3,8,2,2,4,5,0,8,5,8,4,0,0,3,5,6,4,1,3,2,9,3,5,8,4,9,9,1,8,4,8,5,7,3,
%T A202537 9,3,7,5,9,4,1,6,4,2,2,4,2,0,1,9,5,4,3,0,0,2,9,2,8,3,9,3,8,3,6,1,6,5,
%U A202537 4,8,9,0,5,5,0,5,8,3,1,8,2,0,1,7,0,1,3,5,0,8,5,1,5,9,0,0,9,1,2
%N A202537 Decimal expansion of x satisfying e^x-e^(-2x)=1.
%C A202537 If u>0 and v>0, there is a unique number x satisfying e^(ux)-e^(-vx)=1.  Guide to related sequences, with graphs included in Mathematica programs:
%C A202537 u.... v.... x
%C A202537 1.... 1.... A002390
%C A202537 1.... 2.... A202537
%C A202537 1.... 3.... A202538
%C A202537 2.... 1.... A202539
%C A202537 3.... 1.... A202540
%C A202537 2.... 2.... A202541
%C A202537 3.... 3.... A202542
%C A202537 1/2..1/2... A202543
%C A202537 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.  For an example related to A202537, take f(x,u,v)=e^(ux)-e^(-vx)-1 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 A202537 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A202537 0.382245085840035641329358499184857393759416422...
%t A202537 (* Program 1:  A202537 *)
%t A202537 u = 1; v = 2;
%t A202537 f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
%t A202537 Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
%t A202537 r = x /. FindRoot[f[x] == g[x], {x, .3, .4}, WorkingPrecision -> 110]
%t A202537 RealDigits[r]  (* A202537 *)
%t A202537 (* Program 2: implicit surface for e^(ux)-e(-vx)=1 *)
%t A202537 f[{x_, u_, v_}] := E^(u*x) - E^(-v*x) - 1;
%t A202537 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .3}]}, {v, 1, 4}, {u, 2, 20}];
%t A202537 ListPlot3D[Flatten[t, 1]] (* for A202537 *)
%t A202537 First[ RealDigits[ Log[ Root[#^3 - #^2 - 1 & , 1]], 10, 99]] (* _Jean-François Alcover_, Feb 26 2013 *)
%o A202537 (PARI) solve(x=0,1,exp(x)-exp(-2*x)-1) \\ _Charles R Greathouse IV_, Feb 26 2013
%o A202537 (PARI) log(polrootsreal(x^3-x^2-1)[1]) \\ _Charles R Greathouse IV_, Feb 07 2025
%Y A202537 Cf. A002390.
%K A202537 nonn,cons
%O A202537 0,1
%A A202537 _Clark Kimberling_, Dec 21 2011
%E A202537 Digits from a(90) on corrected by _Jean-François Alcover_, Feb 26 2013