OFFSET
0,1
COMMENTS
For many choices of a and c, there are exactly two values of x satisfying a*x^2 - c = tan(x) and 0 < x < Pi/2; for other choices, there is exactly once such value.
Guide to related sequences, with graphs included in Mathematica programs:
a.... c.... x
5... -1.... A200628
5... -2.... A200629
5... -3.... A200630
5... -4.... A200631
6... -1.... A200637
6... -5.... A200638
1... -5.... A200239
2... -5.... A200240
3... -5.... A200241
4... -5.... A200242
5.... 0.... A200618
6.... 0.... A200632
7.... 0.... A200643
8.... 0.... A200644
9.... 0.... A200645
10... 0.... A200646
-1... 1.... A200685
-1... 2.... A200686
-1... 3.... A200687
-1... 4.... A200688
-1... 5.... A200689
-1... 6.... A200690
-1... 7.... A200691
-1... 8.... A200692
-1... 9.... A200693
-1... 10... A200694
-2... 1.... A200695
-2... 3.... A200696
-3... 1.... A200697
-3... 2.... A200698
-4... 1.... A200699
-5... 1.... A200700
-6... 1.... A200701
-7... 1.... A200702
-8... 1.... A200703
-9... 1.... A200704
-10.. 1.... A200705
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 A200614, take f(x,u,v) = u*x^2 - v = tan(x) 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
lesser: 0.839582259045302941513764008863804986308...
greater: 1.350956593976519397744696294368524715373...
MATHEMATICA
a = 3; c = 1;
f[x_] := a*x^2 - c; g[x_] := Tan[x]
Plot[{f[x], g[x]}, {x, -.1, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .8, .9}, WorkingPrecision -> 110]
RealDigits[r] (* A200614 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.3, 1.4}, WorkingPrecision -> 110]
RealDigits[r] (* A200615 *)
(* Program 2: implicit surface of u*x^2-v=tan(x) *)
f[{x_, u_, v_}] := u*x^2 - v - Tan[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1.55}]}, {u, 1, 20}, {v, -20, 0}];
ListPlot3D[Flatten[t, 1]] (* for A200614 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Nov 20 2011
STATUS
approved