A332526 - OEIS

%I #5 Jun 15 2020 13:38:24

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

%T 4,8,7,8,6,6,2,6,7,8,0,6,5,0,8,8,4,1,7,9,2,9,2,6,9,8,5,6,4,5,9,7,5,4,

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

%N Decimal expansion of the minimal distance between distinct branches of the tangent function; see Comments.

%C Let T0 and T1 be the branches of the graph of y = tan x that passes through (0,0,) and (Pi,0), respectively. There exist points P = (p,q) on T0 and U = (u,v) on T1 such that the distance between P and U is the minimal distance, d, between points on T0 and T1.

%C u = 2.549082584017596768984130292562154758705824602711...

%C v = -0.67319711901285205370684801604861382107848678888...

%C p = Pi - u

%C q = - v

%C d = 2.375069146040176349439851558778982487866267806508...

%e minimal distance = 2.375069146040176349439851558778982487866267806508...

%t min = Quiet[FindMinimum[Sqrt[(#[[1]][[1]] - #[[2]][[1]])^2 + (#[[1]][[2]] - \

%t #[[2]][[2]])^2] &[{{#, Tan[#]} &[x /. FindRoot[# Cos[#]^2 - x Cos[#]^2 + Tan[#] == Tan[x], {x, 0}, WorkingPrecision -> 500]], {#, Tan[#]} &[#]} &[y]], {y, 2}, WorkingPrecision -> 100]]

%t Show[Plot[{Tan[x], (-# Sec[#]^2) + x Sec[#]^2 + Tan[#], {(# Cos[#]^2) - x Cos[#]^2 + Tan[#]}}, {x, 0, Pi}, AspectRatio -> Automatic, ImageSize -> 300, PlotRange -> {-2, 2}], Graphics[{PointSize[Large], Point[{Pi/2, 0}], Point[{#, Tan[#]}], Point[{Pi - #, -Tan[#]}]}]] &[y /. min[[2]][[1]]]

%t (* _Peter J. C. Moses_, May 06 2020 *)

%Y Cf. A332500, A332525, A332527.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Jun 15 2020