A197374 - OEIS

%I #11 May 13 2013 01:49:57

%S 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,

%T 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,

%U 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

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

%C 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).

%D 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.

%H E. van Fossen Conrad and P. Flajolet <a href="http://arxiv.org/abs/math/0507268">The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion</a>, arXiv:math/0507268v1 [math.CO], Sem. Lothar. Combin. 54 (2005/06), Art. B54g, 44 pp.

%F 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.

%e 5.299916250856349...

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

%o (PARI) sqrt(3)/(2*Pi)*gamma(1/3)^3 \\ _Charles R Greathouse IV_, Mar 04, 2012

%Y Cf. A104133, A104134.

%K easy,nonn,cons

%O 1,1

%A _Peter Bala_, Mar 04 2012