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

1,1

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.

Links

Table of n, a(n) for n=1..105.

Formula

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]

Example

I=7.640395578055424035809524164342886583819935229294549442160993313...

Maple

evalf(4*sqrt(2)*EllipticE(1/sqrt(2)), 120); # Vaclav Kotesovec, Apr 22 2015

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 *)

Crossrefs

Sequence in context: A288935 A132714 A230327 * A288242 A289266 A175996

Adjacent sequences: A105416 A105417 A105418 * A105420 A105421 A105422

Keyword

cons,nonn

Author

Robert G. Wilson v, Apr 06 2005

Status

approved