OFFSET

1,2

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.

EXAMPLE

x=1.2493943366463244725112743212610081234694...

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 *)

CROSSREFS

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Dec 15 2011

STATUS

approved