OFFSET
0,1
COMMENTS
If u>0 and v>0, there is a unique number x satisfying e^(ux)-e^(-vx)=1. Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 1.... A002390
1.... 2.... A202537
1.... 3.... A202538
2.... 1.... A202539
3.... 1.... A202540
2.... 2.... A202541
3.... 3.... A202542
1/2..1/2... A202543
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 A202537, take f(x,u,v)=e^(ux)-e^(-vx)-1 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=0.382245085840035641329358499184857393759416422...
MATHEMATICA
(* Program 1: A202537 *)
u = 1; v = 2;
f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .3, .4}, WorkingPrecision -> 110]
RealDigits[r] (* A202537 *)
(* Program 2: implicit surface for e^(ux)-e(-vx)=1 *)
f[{x_, u_, v_}] := E^(u*x) - E^(-v*x) - 1;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .3}]}, {v, 1, 4}, {u, 2, 20}];
ListPlot3D[Flatten[t, 1]] (* for A202537 *)
First[ RealDigits[ Log[ Root[#^3 - #^2 - 1 & , 1]], 10, 99]] (* Jean-François Alcover, Feb 26 2013 *)
PROG
(PARI) solve(x=0, 1, exp(x)-exp(-2*x)-1) \\ Charles R Greathouse IV, Feb 26 2013
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Dec 21 2011
EXTENSIONS
Digits from a(90) on corrected by Jean-François Alcover, Feb 26 2013
STATUS
approved