OFFSET
1,1
COMMENTS
The Cartesian equation of the dipole curve, also known as the Playfair curve, is (x^2 + y^2)^3 = a^4*x^2, where the parameter 'a' is the area and the width of one lobe. The computation of the length of one lobe is done here with a=1.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
Robert Ferréol, Courbe du dipole (in French)
Jan Wassenaar, Dipole curve
FORMULA
Equals 2*Integral_{x=0..1} sqrt(1 + f'(x)^2), where f(x) = sqrt(x^(2/3) - x^2).
Equals Integral_{t=0..Pi/2} sqrt(3*cos(t)+1/cos(t)). - Jan Mangaldan, Nov 23 2020
EXAMPLE
3.582786831852204175154707859155610666392085023475548077480462784768897327952...
MATHEMATICA
2*a*Sqrt[Pi]*Gamma[5/4]*Hypergeometric2F1[-1/2, 1/4, 3/4, -3]/Gamma[3/4] /. a -> 1 // RealDigits[#, 10, 100] & // First
RealDigits[Sqrt[2 Pi] Gamma[5/4] Hypergeometric2F1[1/4, 5/4, 3/4, 3/4]/Gamma[3/4], 10, 100][[1]] (* Jan Mangaldan, Nov 22 2020 *)
PROG
(PARI) localprec(100); intnum(t=0, Pi/2, sqrt(3*cos(t)+1/cos(t))) \\ Michel Marcus, Dec 05 2020
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, May 29 2013
STATUS
approved