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A222494
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Decimal expansion of the length of the dipole curve.
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1
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3, 5, 8, 2, 7, 8, 6, 8, 3, 1, 8, 5, 2, 2, 0, 4, 1, 7, 5, 1, 5, 4, 7, 0, 7, 8, 5, 9, 1, 5, 5, 6, 1, 0, 6, 6, 6, 3, 9, 2, 0, 8, 5, 0, 2, 3, 4, 7, 5, 5, 4, 8, 0, 7, 7, 4, 8, 0, 4, 6, 2, 7, 8, 4, 7, 6, 8, 8, 9, 7, 3, 2, 7, 9, 5, 2, 6, 5, 2, 5, 5, 4, 7, 0, 4, 2, 4, 8, 5, 1, 6, 1, 3, 1, 7, 5, 5, 2, 4, 9
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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3.582786831852204175154707859155610666392085023475548077480462784768897327952...
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MATHEMATICA
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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 *)
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PROG
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(PARI) localprec(100); intnum(t=0, Pi/2, sqrt(3*cos(t)+1/cos(t))) \\ Michel Marcus, Dec 05 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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