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Search: id:A118178
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| A118178 |
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Decimal expansion of arc length of eight curve. |
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+0
1
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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
(list; cons; graph; listen)
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OFFSET
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1,1
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LINKS
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Eric Weisstein's World of Mathematics, Eight Curve
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FORMULA
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4*Integrate[Sqrt[4*Sin[t]^4-5*Sin[t]^2+2], {t, 0, Pi/2}].
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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Adjacent sequences: A118175 A118176 A118177 this_sequence A118179 A118180 A118181
Sequence in context: A133077 A016628 A112246 this_sequence A021168 A147709 A019622
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KEYWORD
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nonn,cons
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AUTHOR
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Eric Weisstein (eric(AT)weisstein.com), Apr 13, 2006
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