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Equianharmonic

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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. 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

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

where Γ {\displaystyle \Gamma } is the Gamma function. The half period is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \omega_1=\tfrac{1}{2}(-1+\sqrt3i)\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 ) π i , e 2 = 4 − 1 / 3 , e 3 = 4 − 1 / 3 e − ( 2 / 3 ) π i . {\displaystyle 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.

References

  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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