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A105419 Decimal expansion of the arc length of the sine or cosine curve for one full period. 1
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 (list; constant; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
cons,nonn
AUTHOR
Robert G. Wilson v, Apr 06 2005
STATUS
approved

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Last modified June 23 01:40 EDT 2024. Contains 373629 sequences. (Running on oeis4.)