|
|
A175574
|
|
Decimal expansion of sqrt(Pi) / (Gamma(3/4))^2 .
|
|
6
|
|
|
1, 1, 8, 0, 3, 4, 0, 5, 9, 9, 0, 1, 6, 0, 9, 6, 2, 2, 6, 0, 4, 5, 3, 3, 7, 9, 4, 0, 5, 5, 8, 4, 8, 8, 5, 8, 7, 2, 3, 3, 7, 1, 6, 6, 3, 4, 8, 8, 1, 4, 4, 7, 2, 9, 9, 5, 1, 5, 8, 6, 4, 3, 9, 9, 4, 0, 4, 3, 0, 4, 1, 8, 0, 7, 2, 0, 7, 1, 5, 7, 9, 4, 9, 7, 8, 4, 5, 8, 6, 1, 6, 1, 9, 5, 8, 0, 7, 9, 5, 4, 2, 0, 9, 4, 5
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Entry 34 c of chapter 11 of Ramanujan's second notebook.
This constant is also the ratio T(Pi/2)/T(0), where T(Pi/2) is the exact pendulum period for an amplitude of Pi/2 and T(0) the approximate period 2*Pi*sqrt(L/g) for small angles. - Jean-François Alcover, Aug 05 2014
|
|
LINKS
|
|
|
FORMULA
|
Equals 2F1([1/2,1/2],[1],1/2) = 1/agm(1, sqrt(1/2)) = gamma(1/4)^2/(2*Pi^(3/2)).
Equals 2*sqrt(2)*K(-1)/Pi, where K is the complete elliptic integral of the first kind, K(-1) being A085565. - Jean-François Alcover, Jun 03 2014
Equals Product_{k>=1} (1-(-1)^k/(2*k)) = 3/2 * 3/4 * 7/6 * 7/8 * 11/10 * 11/12 * ... . - Richard R. Forberg, Dec 05 2015
Reciprocal of A096427. Equals ( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2, a rapidly converging series. For example, summing from n = -5 to n = 5 gives the constant correct to 49 decimal places. - Peter Bala, Mar 06 2019
Equals Sum_{k>=0) binomial(2*k,k)^2/2^(5*k). - Amiram Eldar, Aug 26 2020
Equals (3/2)*hypergeom([-1/4, 3/4], [3/2], 1). - Peter Bala, Mar 04 2022
|
|
EXAMPLE
|
1.18034059901609622604533794..
|
|
MAPLE
|
sqrt(Pi)/GAMMA(3/4)^2 ; evalf(%) ;
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) sqrt(Pi)/gamma(3/4)^2 \\ Altug Alkan, Dec 05 2015
(MATLAB) sqrt(pi)/gamma(3/4)^2 \\ Altug Alkan, Dec 05 2015
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
A-number typo for sqrt(Pi) corrected by R. J. Mathar, Aug 01 2010
|
|
STATUS
|
approved
|
|
|
|