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A201280 Decimal expansion of x satisfying x^2 + 1 = cot(x) and 0 < x < Pi.

%I A201280 #25 Jan 30 2025 15:45:18
%S A201280 6,2,3,8,9,9,5,6,0,5,8,0,9,0,3,4,4,3,6,3,9,9,0,3,2,9,3,9,4,6,3,2,4,4,
%T A201280 2,6,4,4,2,7,6,1,7,2,0,3,1,5,6,6,7,3,6,5,2,8,8,4,4,3,7,9,0,4,7,1,8,2,
%U A201280 8,0,2,1,3,1,8,5,4,3,4,2,6,6,8,5,9,8,1,6,4,7,7,3,1,9,4,3,1,2,4
%N A201280 Decimal expansion of x satisfying x^2 + 1 = cot(x) and 0 < x < Pi.
%C A201280 For many choices of a and c, there is exactly one x satisfying a*x^2 + c = cot(x) and 0 < x < Pi.
%C A201280 Guide to related sequences, with graphs included in Mathematica programs:
%C A201280 a.... c.... x
%C A201280 1.... 1.... A201280
%C A201280 1.... 2.... A201281
%C A201280 1.... 3.... A201282
%C A201280 1.... 4.... A201283
%C A201280 1.... 5.... A201284
%C A201280 1.... 6.... A201285
%C A201280 1.... 7.... A201286
%C A201280 1.... 8.... A201287
%C A201280 1.... 9.... A201288
%C A201280 1.... 10... A201289
%C A201280 1.... 0.... A201294
%C A201280 1... -1.... A201295
%C A201280 1... -2.... A201296
%C A201280 1... -3.... A201297
%C A201280 1... -4.... A201298
%C A201280 1... -5.... A201299
%C A201280 1... -6.... A201315
%C A201280 1... -7.... A201316
%C A201280 1... -8.... A201317
%C A201280 1... -9.... A201318
%C A201280 1.. -10.... A201319
%C A201280 2.... 0.... A201329
%C A201280 3.... 0.... A201330
%C A201280 4.... 0.... A201331
%C A201280 5.... 0.... A201332
%C A201280 6.... 0.... A201333
%C A201280 7.... 0.... A201334
%C A201280 8.... 0.... A201335
%C A201280 9.... 0.... A201336
%C A201280 10... 0.... A201337
%C A201280 2... -1.... A201320
%C A201280 3... -1.... A201321
%C A201280 4... -1.... A201322
%C A201280 5... -1.... A201323
%C A201280 6... -1.... A201324
%C A201280 7... -1.... A201325
%C A201280 8... -1.... A201326
%C A201280 9... -1.... A201327
%C A201280 10.. -1.... A201328
%C A201280 2.... 1.... A201290
%C A201280 2.... 3.... A201291
%C A201280 2... -3.... A201394
%C A201280 3.... 1.... A201292
%C A201280 3.... 2.... A201293
%C A201280 3... -2.... A201395
%C A201280 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.
%C A201280 For an example related to A201280, take f(x,u,v) = u*x^2 - v - cot(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.
%H A201280 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201280 0.62389956058090344363990329394632442...
%t A201280 (* Program 1: A201280 *)
%t A201280 a = 1; c = 1;
%t A201280 f[x_] := a*x^2 + c; g[x_] := Cot[x]
%t A201280 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201280 r = x /. FindRoot[f[x] == g[x], {x, .62, .63}, WorkingPrecision -> 110]
%t A201280 RealDigits[r]   (* A201280 *)
%t A201280 (* Program 2: implicit surface of u*x^2-v=cot(x) *)
%t A201280 f[{x_, u_, v_}] := u*x^2 - v - Cot[x];
%t A201280 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .001, Pi}]}, {u, 0, 5, .1}, {v, 0, 5, .1}];
%t A201280 ListPlot3D[Flatten[t, 1]]  (* for A201280 *)
%Y A201280 Cf. A200614.
%K A201280 nonn,cons
%O A201280 0,1
%A A201280 _Clark Kimberling_, Nov 29 2011
%E A201280 Edited and a(90) onwards corrected by _Georg Fischer_, Aug 03 2021