A237841 Decimal expansion of Ramanujan's AGM Continued Fraction R(2) = R_1(2,2).

9, 7, 4, 9, 9, 0, 9, 8, 8, 7, 9, 8, 7, 2, 2, 0, 9, 6, 7, 1, 9, 9, 0, 0, 3, 3, 4, 5, 2, 9, 2, 1, 0, 8, 4, 4, 0, 0, 5, 9, 2, 0, 2, 1, 9, 9, 9, 4, 7, 1, 0, 6, 0, 5, 7, 4, 5, 2, 6, 8, 2, 5, 1, 2, 8, 5, 8, 7, 7, 3, 8, 7, 4, 5, 5, 7, 0, 8, 5, 9, 4, 3, 5, 2, 3, 2, 5, 3, 2, 0, 9, 1, 1, 1, 2, 9, 3, 6, 2, 5

Offset: 0

Author

Jean-François Alcover, Feb 14 2014

Keywords

• nonn
• cons

Comments

Other closed form evaluations of R(p/q):

R(1/4) = Pi/2 - 4/3,

R(1/3) = 1 - log(2),

R(1/2) = 2 - Pi/2,

R(2/3) = 4 - Pi/sqrt(2),

R(1) = log(2),

R(3/2) = Pi + sqrt(3)*log(2 - sqrt(3)),

R(3) = Pi/sqrt(3) - log(2).

Examples

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Links

• D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, Ten Problems in Experimental Mathematics, page 13.
• Jonathan M. Borwein, Ramanujan's Arithmetic-Geometric Mean Continued Fractions and Dynamics

Crossrefs

Cf. A002162: R(1) = log(2); A180434: R(1/2) = 2-Pi/2.

Programs

Mathematica

RealDigits[Sqrt[2]*(Pi/2 - Log[1 + Sqrt[2]]), 10, 100] // First

PARI

(psi(7/8)-psi(3/8))/2 \\ Charles R Greathouse IV, Mar 03 2016

Formula

Equivalent formulas:

sqrt(2)*(Pi/2 - log(1 + sqrt(2))),

(Pi - 2*arccoth(sqrt(2)))/sqrt(2),

Integral_{x >= 0} sech(Pi*x/4)/(1 + x^2) dx,

2*Integral_{x = 0..1} sqrt(x)/(1 + x^2) dx,

Integral_{x >= 0} exp(-x/2)*sech(x) dx,

4*Sum_{k >= 1} (-1)^(k+1)/(4*k - 1),

1/2*(-psi(3/8) + psi(7/8)), where psi is the digamma function,

4/3 * 2F1(3/4, 1, 7/4, -1), where 2F1 is the hypergeometric function,

(H(-1/8) - H(-5/8))/2, where H(n) is the n-th harmonic number.

General formula:

The Borwein's closed form formula for R(n) with n integer simplifies to:

R(n) = Pi/2*sec(Pi/(2n)) - 2*sum( cos((k*(n+1)*Pi)/(2*n))*log(2*sin((k*Pi)/(4*n))), {k, 1, 2n-1, 2} ).

Equals 4*A181049. - Peter Bala, Apr 02 2024