A105419 Decimal expansion of the arc length of the sine or cosine curve for one full period.
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
Offset: 1
Examples
I=7.640395578055424035809524164342886583819935229294549442160993313...
References
- 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.
Programs
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Maple
evalf(4*sqrt(2)*EllipticE(1/sqrt(2)), 120); # Vaclav Kotesovec, Apr 22 2015
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Mathematica
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 *)
Formula
Equals Integral_{x=0..2*Pi} 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