A193751 - OEIS

%I #22 Jul 03 2015 05:22:03

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

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

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

%N Decimal expansion of Pear curve length.

%C The Pear Curve is the third Mandelbrot set lemniscate.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PearCurve.html">Pear Curve</a>

%e 7.738373624...

%t f[x_, y_] = ComplexExpand[#*Conjugate[#] &[c + (c + c^2)^2] /. c -> x + I*y] - 4; sy = Solve[f[x, y] == 0, y];

%t f2[x_] = y /. sy[[4]]; x2 = 3/10; y2 = f2[x2]; sx = Solve[f[x, y] == 0, x]; g1[y_] = x /. sx[[1]]; g2[y_] = x /. sx[[2]]; sg = Solve[f[g[y], y] == 0 && D[f[g[y], y], y] == 0 , g'[y]][[1]]; dg1[y_] = g'[y] /. sg /. g -> g1;

%t dg2[y_] = g'[y] /. sg /. g -> g2; ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120];

%t i1 = ni[Sqrt[1 + dg1[y]^2], {y, 0, f2[-1]} ];

%t i2 = ni[Sqrt[1 + f2'[x]^2], {x, -1, x2}];

%t i3 = ni[Sqrt[1 + dg2[y]^2], {y, 0, y2}];

%t Take[RealDigits[2(i1 + i2 + i3)][[1]], 105]

%Y Cf. A193750 (area)

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Aug 03 2011

%E Corrected and extended by _Jean-François Alcover_, Aug 26 2011