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Decimal expansion of the perimeter of the second Mandelbrot set lemniscate.
1

%I #9 Feb 16 2025 08:33:15

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

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

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

%N Decimal expansion of the perimeter of the second Mandelbrot set lemniscate.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MandelbrotSetLemniscate.html">Mandelbrot Set Lemniscate</a>

%e 8.8944968157...

%t f[x_, y_] = (x^2 + y^2)*((x + 1)^2 + y^2) - 4; f[x_] = y /. Solve[f[x, y] == 0, y][[4]]; g[y_] = x /. Solve[f[x, y] == 0, x][[4]]; Take[RealDigits[ 4*NIntegrate[Sqrt[1 + f'[x]^2], {x, -1/2, 0}, WorkingPrecision -> 120] + 4*NIntegrate[Sqrt[1 + g'[y]^2], {y, 0, f[0]}, WorkingPrecision -> 120]][[1]], 105]

%Y Cf. A193780 (area).

%K nonn,cons,changed

%O 1,1

%A _Jean-François Alcover_, Aug 05 2011