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A105419
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Decimal expansion of the arc length of the sine or cosine curve for one full period.
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0
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7, 6, 4, 0, 3, 9, 5, 5, 7, 8, 0, 5, 5, 4, 2, 4, 0, 3, 5, 8, 0, 9, 5, 2, 4, 1, 6, 4, 3, 4, 2, 8, 8, 6, 5, 8, 3, 8, 1, 9, 9, 3, 5, 2, 2, 9, 2, 9, 4, 5, 4, 9, 4, 4, 2, 1, 6, 0, 9, 9, 3, 3, 1, 3, 4, 9, 4, 3, 9, 1, 6, 0, 2, 4, 2, 8, 6, 5, 9, 8, 4, 2, 1, 3, 2, 3, 6, 2, 1, 7, 8, 9, 0, 2, 4, 4, 4, 9, 6, 5, 6, 4, 4, 0, 8
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OFFSET
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1,1
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REFERENCES
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Howard Anton, Irl C. Bivens, Stephen L. Davis, Calculus, Early Transcendentals, 7th Edition, John Wiley & Sons, Inc., NY, Section 7.4 Length of a Plane Curve, page 489.
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LINKS
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Table of n, a(n) for n=1..105.
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FORMULA
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Integral_{0, 2Pi} Sqrt(1+Cos(x)^2) dx.
Also equals 4*B+Pi/B where B is the lemniscate constant A076390, or sqrt(2/Pi)*(2*gamma(3/4)^4 + Pi^2)/gamma(3/4)^2. [Jean-François Alcover, Apr 17 2013]
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EXAMPLE
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I=7.640395578055424035809524164342886583819935229294549442160993313...
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MATHEMATICA
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RealDigits[ NIntegrate[ Sqrt[1 + Cos[x]^2, {x, 0, 2Pi}, MaxRecursion -> 12, WorkingPrecision -> 128], 10, 111][[1]]
RealDigits[ N[ 4*Sqrt[2]*EllipticE[1/2], 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)
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CROSSREFS
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Sequence in context: A013675 A198878 A132714 * A175996 A134982 A064533
Adjacent sequences: A105416 A105417 A105418 * A105420 A105421 A105422
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KEYWORD
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cons,nonn
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AUTHOR
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Robert G. Wilson v, Apr 06 2005
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STATUS
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approved
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