A275336 - OEIS

%I #22 Apr 10 2017 22:56:49

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

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

%U 1,5,1,5,5,9,2,5,6,2,0,2,3,2,9,0,8,6,8,6,9,2,3,4,3,3,6,6,8,3,6,3

%N Decimal expansion of the location of the least capacity point of a unit isosceles right triangle.

%C Considering the isosceles right triangle x>0, y>0, x+y=1; there exists a unique least [electrostatic] capacity point (0.301..., 0.301...).

%C It can be observed that this constant is close to but different from the analog for the center of the incircle, which is the inradius 1-1/sqrt(2) = 0.29289...

%H G. C. Greubel, <a href="/A275336/b275336.txt">Table of n, a(n) for n = 0..10000</a>

%H Steven R. Finch, <a href="http://arxiv.org/abs/1407.4105">Least Capacity Point of Triangles</a>, arXiv:1407.4105 [math.HO] 2014, p. 6.

%F 2*sqrt(Pi)*re(F(arcsin(sqrt(1 + sqrt(3))), 1/2))/gamma(1/4)^2, where F is the elliptic integral of the first kind.

%e 0.30112161084132208155382545585018896837090055390464695504633924...

%t t0 = 2*Sqrt[Pi]*Re[EllipticF[ArcSin[Sqrt[1 + Sqrt[3]]], 1/2]]/Gamma[1/4]^2;

%t RealDigits[t0, 10, 100][[1]]

%t (* second program: *)

%t digits = 100;

%t s[z_] := WeierstrassSigma[z, {Gamma[1/4]^8/(256 Pi^2), 0}];

%t h[x_] := Abs[(s[(2+I) - (1+I)x] s[1-(1-I)x] s[2I*x + (1-I)] s[(2+2I)x] s[1 + (1-I)x] s[(1+I)x-I] s[2x+I-1])/(s[1+I] s[1-(1+I)x] s[(2-I) - (1-I)x] s[2I*x] s[2x] s[(1-I)x + I] s[1+(1+I)x] s[(1+I)(2x-1)])]

%t t0 = x /. FindMinimum[h[x], {x, 1/3}, WorkingPrecision -> 3 digits][[2]];

%t RealDigits[t0, 10, digits][[1]]

%t RealDigits[2 Sqrt[Pi] EllipticF[ArcTan[Sqrt[1 + Sqrt[3]]], 1/2]/Gamma[1/4]^2, 10, 100][[1]] (* _Jan Mangaldan_, Jan 04 2017 *)

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Jul 23 2016