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A105419 Decimal expansion of the arc length of the sine or cosine curve for one full period. 0
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

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...

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: A187799 A132714 A230327 * A175996 A248940 A134982

Adjacent sequences:  A105416 A105417 A105418 * A105420 A105421 A105422

KEYWORD

cons,nonn

AUTHOR

Robert G. Wilson v, Apr 06 2005

STATUS

approved

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Last modified November 23 13:57 EST 2014. Contains 249851 sequences.