A309893 Decimal expansion of AGM(1, sqrt(3)/2).

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Offset: 0

Author

Daniel Hoyt, Aug 21 2019

Keywords

• nonn
• cons

Comments

Related to the pendulum acceleration relation at 60 degrees. In general, the period T of a mathematical pendulum with a maximum deflection angle theta is 2*Pi*sqrt(L/g)/AGM(1, cos(theta/2)), where L is the length of the pendulum, g is the gravitational acceleration, and 0 < theta <= 90 degrees. For theta = 60 degrees, the period is T = 2*Pi*sqrt(L/g)/AGM(1, sqrt(3)/2). - Jianing Song, Nov 21 2022

Examples

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Links

• Wikipedia, Pendulum (mechanics)

Crossrefs

Cf. A310000 (AGM(1, cos(Pi/5))), A096427 (AGM(1, sqrt(2)/2)), A053004, A014549, A068521.

Programs

Mathematica

RealDigits[ArithmeticGeometricMean[1, Sqrt[3]/2], 10, 100][[1]] (* Amiram Eldar, Aug 21 2019 *)

PARI

agm(1, sqrt(3)/2) \\ Michel Marcus, Aug 22 2019

Python

import decimal
prec = int(input('Precision: '))
decimal.getcontext().prec = prec
D = decimal.Decimal
def agm(a, b):
    for x in range(prec):
        a, b = (a + b) / 2,(a * b).sqrt()
    return a
print(agm(1, D(3).sqrt()/2))

Sage

RealField(300)(1.0).agm(sqrt(3)/2) # Peter Luschny, Aug 22 2019

Formula

AGM(1, sin(Pi/3)).