A197374 Pi(3): fundamental real period of the Dixonian elliptic functions sm(z) and cm(z).

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

Offset

1,1

Comments

Pi(3) = 5.29991 62508 56349 87194 ... is the real period of the doubly-periodic Dixonian elliptic functions sm(z) (A104133) and cm(z) (A104134): sm(z+Pi(3)) = sm(z); cm(z+Pi(3)) = cm(z). The other period equals Pi(3)*w, where w = exp(2*I*Pi/3).

References

A. C. Dixon, On the doubly periodic functions arising out of the curve x^3 + y^3 - 3 alpha xy = 1, Quarterly J. Pure Appl. Math. 24 (1890), 167-233.

Links

Table of n, a(n) for n=1..102.

E. van Fossen Conrad and P. Flajolet The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion, arXiv:math/0507268v1 [math.CO], 2005; Sem. Lothar. Combin. 54 (2005/06), Art. B54g, 44 pp.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 330.

Formula

Pi(3) = 3*int {0..1} 1/(1-t^3)^(2/3) dt = B(1/3,1/3) = Gamma(1/3)^2/Gamma(2/3) = sqrt(3)/(2*Pi)*Gamma(1/3)^3.

Equals Beta(1/3,1/3) (see Shamos). - Stefano Spezia, Jun 03 2025

Example

5.2999162508563498719410684989453161077156056146...

Mathematica

Sqrt[3]/(2*Pi)*Gamma[1/3]^3 // N[#, 103]& // RealDigits // First (* Jean-François Alcover, Jan 21 2013 *)

Prog

(PARI) sqrt(3)/(2*Pi)*gamma(1/3)^3 \\ Charles R Greathouse IV, Mar 04 2012

Crossrefs

Cf. A104133, A104134.

Sequence in context: A348408 A019841 A064582 * A119946 A276610 A065270

Adjacent sequences: A197371 A197372 A197373 * A197375 A197376 A197377

Keyword

easy,nonn,cons

Author

Peter Bala, Mar 04 2012

Status

approved