OFFSET
1,1
COMMENTS
The Pear Curve is the third Mandelbrot set lemniscate.
LINKS
Eric Weisstein's World of Mathematics, Pear Curve
EXAMPLE
7.738373624...
MATHEMATICA
f[x_, y_] = ComplexExpand[#*Conjugate[#] &[c + (c + c^2)^2] /. c -> x + I*y] - 4; sy = Solve[f[x, y] == 0, y];
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;
dg2[y_] = g'[y] /. sg /. g -> g2; ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120];
i1 = ni[Sqrt[1 + dg1[y]^2], {y, 0, f2[-1]} ];
i2 = ni[Sqrt[1 + f2'[x]^2], {x, -1, x2}];
i3 = ni[Sqrt[1 + dg2[y]^2], {y, 0, y2}];
Take[RealDigits[2(i1 + i2 + i3)][[1]], 105]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Aug 03 2011
EXTENSIONS
Corrected and extended by Jean-François Alcover, Aug 26 2011
STATUS
approved