A332525 - OEIS

%I #4 Jun 15 2020 13:38:14

%S 2,5,5,7,0,1,5,6,1,4,2,4,1,3,5,8,5,2,6,0,1,3,6,6,3,5,4,1,9,0,6,7,7,1,

%T 3,7,9,6,9,9,9,8,9,0,8,9,7,8,1,2,2,8,7,7,1,8,6,6,8,9,0,4,7,4,9,1,3,7,

%U 0,4,0,1,1,5,5,6,7,8,6,6,2,0,0,5,1,2

%N Decimal expansion of the minimal distance between (0,0) and the branch of the graph of y = tan x that passes through (Pi, 0).

%C Let T be the branch of the graph of y = tan x that passes through (Pi,0).  There is a unique point (u,v) on T that is closer to (0,0) than any other point on T. Let d = distance between (u,v) and(0,0). The first code in the Mathematica section gives

%C u = 2.319805307509200010738867057136510870483647988277... ;

%C v = -1.07556133564118881053529612226074179471679754375... ;

%C d = 2.557015614241358526013663541906771379699989089781... .

%C The second code shows (u,v) as the intersection of T and the circle centered at (0,0) with radius d.

%C The third code shows minimal distance-to-origin points on 16 branches of the tangent function.

%F u = - sin u sec^3 u.

%F v = tan u.

%F d = sqrt(u^2 + v^2).

%e minimal distance = 2.557015614241358526013663541906771379699989089781...

%t (* This code computes (x,y) coordinates and the minimal distance. *)

%t x = x /. FindRoot[FullSimplify[D[Sqrt[x^2 + Tan[x]^2], x]] == 0, {x, 2},

%t    WorkingPrecision -> 150]

%t y = Tan[x]

%t d = Sqrt[x^2 + Tan[x]^2]

%t RealDigits[x][[1]]

%t RealDigits[y][[1]]

%t RealDigits[d][[1]]

%t (* _Peter J. C. Moses_, May 04 2020 *)

%t (* This code shows the two points on the graph of y = tan x and on a circle whose radius is the minimal distance. *)

%t g1 = Plot[Tan[x], {x, -2 \[Pi], 2 \[Pi]}, AspectRatio -> 1];

%t g2 = Graphics[Circle[{0, 0}, Sqrt[Tan[#]^2 + #^2] &[x /. FindRoot[

%t        FullSimplify[D[Sqrt[x^2 + Tan[x]^2], x]] == 0, {x, 2},

%t        WorkingPrecision -> 30]]]];

%t Show[g1, g2]

%t (* _Peter J. C. Moses_, May 04 2020 *)

%t * This code shows minimal distance points on 16 branches of the tangent function. *)

%t max = 25;

%t ptX = Table[x /. FindRoot[# == 0, {x, nn}, WorkingPrecision -> 10], {nn, 2,

%t       max, Pi}] &[FullSimplify[D[Sqrt[x^2 + Tan[x]^2], x]]];

%t Show[Plot[Tan[x], {x, -#, #}, PlotRange -> {-#, #}] &[max],

%t    Map[Graphics[{Red, Circle[{0, 0}, Sqrt[Tan[#]^2 + #^2]]}] &, #],

%t    Map[Graphics[{PointSize[Large], Point[-{#, Tan[#]}], Point[{0, 0}],

%t         Point[{#, Tan[#]}]}] &, #], AspectRatio -> Automatic,

%t         ImageSize -> 600] &[ptX]

%t (* _Peter J. C. Moses_, May 05 2020 *)

%Y Cf. A332526, A332527.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Jun 15 2020