Equianharmonic

Last updated January 04, 2024 • 1 min readFrom Wikipedia, The Free Encyclopedia

In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g₂ = 0 and g₃ = 1.^[1] This page follows the terminology of Abramowitz and Stegun; see also the lemniscatic case. (These are special examples of complex multiplication.)

In the equianharmonic case, the minimal half period ω₂ is real and equal to

{\frac {\Gamma ^{3}(1/3)}{4\pi }}

where $\Gamma$ is the Gamma function. The half period is

\omega _{1}={\tfrac {1}{2}}(-1+{\sqrt {3}}i)\omega _{2}.

Here the period lattice is a real multiple of the Eisenstein integers.

The constants e₁, e₂ and e₃ are given by

e_{1}=4^{-1/3}e^{(2/3)\pi i},\qquad e_{2}=4^{-1/3},\qquad e_{3}=4^{-1/3}e^{-(2/3)\pi i}.

The case g₂ = 0, g₃ = a may be handled by a scaling transformation.

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References

↑ Abramowitz, Milton; Stegun, Irene A. (June 1964). "Pocketbook of Mathematical Functions--Abridged Edition of Handbook of Mathematical Functions, Milton Abramowitz and Irene A. Stegun". Mathematics of Computation. 50 (182): 652–657. doi:10.2307/2008636. ISSN 0025-5718.

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[1] Abramowitz, Milton; Stegun, Irene A. (June 1964). "Pocketbook of Mathematical Functions--Abridged Edition of Handbook of Mathematical Functions, Milton Abramowitz and Irene A. Stegun". Mathematics of Computation. 50 (182): 652–657. doi:10.2307/2008636. ISSN 0025-5718.

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