A199370 Decimal expansion of x>0 satisfying x^2+x*sin(x)=1.

7, 2, 2, 5, 8, 7, 5, 4, 9, 9, 2, 2, 5, 2, 4, 7, 6, 8, 3, 5, 5, 9, 3, 2, 8, 7, 2, 8, 7, 7, 1, 9, 6, 7, 5, 5, 1, 5, 9, 6, 4, 5, 9, 2, 1, 1, 4, 3, 9, 4, 2, 6, 9, 8, 0, 7, 7, 6, 5, 1, 4, 7, 6, 0, 2, 5, 9, 0, 9, 4, 2, 5, 0, 7, 3, 1, 6, 0, 1, 8, 3, 0, 3, 4, 3, 5, 6, 2, 9, 4, 1, 8, 7, 2, 7, 9, 8, 3, 3

Offset: 0

Author

Clark Kimberling, Nov 05 2011

Keywords

• nonn
• cons

Comments

For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*sin(x)=c.

Guide to related sequences, with graphs included in Mathematica programs:

a.... b.... c.... x

1.... 1.... 1.... A199370

1.... 1.... 2.... A199371

1.... 1.... 3.... A199372

1.... 2.... 1.... A199373

1.... 2.... 2.... A199374

1.... 2.... 3.... A199375

1.... 3.... 1.... A199376

1.... 3.... 2.... A199377

1.... 3.... 3.... A199378

2.... 1.... 1.... A199379

2.... 1.... 2.... A199180

2.... 1.... 3.... A199181

2.... 2.... 1.... A199182

2.... 2.... 3.... A199183

2.... 3.... 1.... A199184

2.... 3.... 2.... A199185

2.... 3.... 3.... A199186

2.... 1.... 1.... A199187

3.... 1.... 2.... A199188

3.... 1.... 3.... A199189

3.... 2.... 1....

3.... 2.... 2....

3.... 2.... 3....

3.... 3.... 1....

3.... 3.... 2....

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 A199370, take f(x,u,v)=x^2+u*x*sin(x)-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

			0.722587549922524768355932872877196755159...
		

Links

• Index entries for transcendental numbers.

Crossrefs

Cf. A199371, A199170, A198866, A198755, A198414, A197737.

Programs

Mathematica

(* Program 1: A199370 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x*Sin[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -1, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .72, .73}, WorkingPrecision -> 110]
RealDigits[r]  (* A199370 *)
(* Program 2: implicit surface of x^2+u*x*sin(x)=v *)
f[{x_, u_, v_}] := x^2 + u*x*Sin[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 2.9}, {v, u, 600}];
ListPlot3D[Flatten[t, 1]]  (* for A199370 *)