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Equianharmonic Case

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The case of the Weierstrass elliptic function with invariants g_2=0 and g_3=1. The corresponding real half-period is given by

omega_2=(Gamma^3(1/3))/(4pi)
(1)
=1.529954037...
(2)

(OEIS A064582), known as the omega2-constant, where Gamma(z) is the gamma function. The other half-period is then given by

omega_1=1/2omega_2(1+isqrt(3))
(3)
=((1+isqrt(3))Gamma^3(1/3))/(8pi)
(4)
=0.764977...+1.324979...i
(5)

(OEIS A094961 and A094962).

See also

Lemniscate Case, Pseudolemniscate Case, omega2-Constant

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Equianharmonic Case (g_2=0, g_3=1)." §18.13 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 652-653, 1972.Finch, S. R. "Gauss' Lemniscate Constant." §6.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 420-423, 2003.Sloane, N. J. A. Sequences A064582, A094961, and A094962 in "The On-Line Encyclopedia of Integer Sequences."

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Equianharmonic Case

Cite this as:

Weisstein, Eric W. "Equianharmonic Case." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EquianharmonicCase.html

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