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%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
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