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A201946 Decimal expansion of x>0 satisfying x*sinh(x)=2.

%I A201946 #8 Feb 08 2025 10:07:51
%S A201946 1,2,4,9,3,9,4,3,3,6,6,4,6,3,2,4,4,7,2,5,1,1,2,7,4,3,2,1,2,6,1,0,0,8,
%T A201946 1,2,3,4,6,9,4,4,1,3,0,0,9,0,1,5,2,9,6,9,6,2,9,7,2,6,0,7,6,8,6,8,8,2,
%U A201946 1,2,3,9,7,3,1,0,6,2,2,9,8,3,0,0,5,3,2,5,7,7,0,8,2,8,6,5,3,9,1
%N A201946 Decimal expansion of x>0 satisfying x*sinh(x)=2.
%C A201946 For many choices of u and v, there is exactly one x>0 satisfying x*sinh(u*x)=v.  Guide to related sequences, with graphs included in Mathematica programs:
%C A201946 u.... v.... x
%C A201946 1.... 1.... A133867
%C A201946 1.... 2.... A201946
%C A201946 1.... 3.... A202243
%C A201946 2.... 1.... A202244
%C A201946 3.... 1.... A202245
%C A201946 2.... 2.... A202284
%C A201946 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 A201946 For an example related to A199597, take f(x,u,v)=x*sinh(ux)-v 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 A201946 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201946 1.2493943366463244725112743212610081234694...
%t A201946 (* Program 1:  A201946 *)
%t A201946 u = 1; v = 2;
%t A201946 f[x_] := x*Sinh[u*x]; g[x_] := v
%t A201946 Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
%t A201946 r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
%t A201946 RealDigits[r]   (* A201946 *)
%t A201946 (* Program 2: implicit surface of u*sinh(x)=v *)
%t A201946 f[{x_, u_, v_}] := x*Sinh[u*x] - v;
%t A201946 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .2}]}, {v, 0, 10}, {u, 1, 4}];
%t A201946 ListPlot3D[Flatten[t, 1]] (* for A201946 *)
%Y A201946 Cf. A201939.
%K A201946 nonn,cons
%O A201946 1,2
%A A201946 _Clark Kimberling_, Dec 15 2011