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Weierstrass Constant

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The Weierstrass constant is defined as the value sigma(1|1,i)/2, where sigma(z|omega_1,omega_2) is the Weierstrass sigma function with half-periods omega_1 and omega_2. Amazingly, it has the closed form

1/2sigma(1|1,i)=(2^(5/4)sqrt(pi)e^(pi/8))/(Gamma^2(1/4))
(1)
=0.4749493799...
(2)

(OEIS A094692), where Gamma(z) is the gamma function.

See also

Lemniscate Case, omega2 Constant, Weierstrass Sigma Function

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References

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 26, 1983.Plouffe, S. "Weierstrass Constant." http://pi.lacim.uqam.ca/piDATA/weier.txt.Sloane, N. J. A. Sequence A094692 in "The On-Line Encyclopedia of Integer Sequences."Waldschmidt, M. "Fonctions entières et nombres transcendants." Cong. Nat. Soc. Sav. Nancy 5, 1978.Waldschmidt, M. "Nombres transcendants et fonctions sigma de Weierstrass." C. R. Math. Rep. Acad. Sci. Canada 1, 111-114, 1978/79.

Referenced on Wolfram|Alpha

Weierstrass Constant

Cite this as:

Weisstein, Eric W. "Weierstrass Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrassConstant.html

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