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

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, 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, 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

Offset: 1

Author

Clark Kimberling, Dec 15 2011

Keywords

• nonn
• cons

Comments

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:

u.... v.... x

1.... 1.... A133867

1.... 2.... A201946

1.... 3.... A202243

2.... 1.... A202244

3.... 1.... A202245

2.... 2.... A202284

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 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.

Examples

			1.2493943366463244725112743212610081234694...
		

Links

• Index entries for transcendental numbers.

Crossrefs

Cf. A201939.

Programs

Mathematica

(* Program 1:  A201946 *)
u = 1; v = 2;
f[x_] := x*Sinh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]   (* A201946 *)
(* Program 2: implicit surface of u*sinh(x)=v *)
f[{x_, u_, v_}] := x*Sinh[u*x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .2}]}, {v, 0, 10}, {u, 1, 4}];
ListPlot3D[Flatten[t, 1]] (* for A201946 *)