OFFSET
1,3
COMMENTS
For many choices of a,b,c, there are exactly two numbers x satisfying a*x^2+b*x*cos(x)=c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
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 A199170, take f(x,u,v)=x^2+u*xcos(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.
EXAMPLE
negative: -1.19835984451866026826502160343030898927268...
positive: 0.685174133854503187895211530638458709591...
MATHEMATICA
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -2 Pi, 2 Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.2, -1.1}, WorkingPrecision -> 110]
RealDigits[r] (* A199170 *)
r = x /. FindRoot[f[x] == g[x], {x, .68, .69}, WorkingPrecision -> 110]
RealDigits[r] (* A199171 *)
(* Program 2: implicit surface of x^2+u*x*cos(x)=v *)
f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0,
1.9}, {v, u, 600}];
ListPlot3D[Flatten[t, 1]] (* for A199170 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Nov 03 2011
STATUS
approved