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A118178
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Decimal expansion of arc length of eight curve.
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0
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6, 0, 9, 7, 2, 2, 3, 4, 7, 0, 1, 0, 4, 9, 1, 6, 0, 4, 6, 4, 3, 0, 3, 7, 4, 2, 0, 5, 6, 7, 3, 9, 9, 7, 8, 3, 3, 4, 9, 2, 3, 3, 7, 8, 1, 8, 3, 8, 6, 5, 5, 5, 1, 1, 4, 8, 6, 6, 1, 7, 3, 2, 1, 0, 0, 8, 2, 0, 4, 3, 7, 5, 4, 9, 4, 4, 1, 4, 0, 9, 3, 2, 0, 1, 3, 5, 4, 9, 6, 1, 4, 3, 3, 6, 5, 9, 1, 7, 6, 1, 0, 7, 7, 7, 0
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OFFSET
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1,1
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LINKS
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FORMULA
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4*Integral_{t=0..Pi/2} (sqrt(4*sin(t)^4 - 5*sin(t)^2 + 2)) dt.
Can be expressed in terms of complete elliptic integrals. Using Mathematica notation, with m = (4 + Sqrt[2])/8, the arc length is 4*2^(1/4)*(EllipticE[m] - EllipticK[m]) + (3 + 2*Sqrt[2])*2^(-1/4)*EllipticPi[(4 - 3*Sqrt[2])/8, m]. - David W. Cantrell, Apr 22 2006
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EXAMPLE
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6.097223470104916046...
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MATHEMATICA
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RealDigits[4*NIntegrate[Sqrt[4*Sin[t]^4-5*Sin[t]^2+2], {t, 0, Pi/2}, WorkingPrecision->200], 10, 110][[1]]
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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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