Equianharmonic Explained

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

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

where

\Gamma is the Gamma function. The half period is\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 / 3} _{1=4} e ^{(2/3)\pi} , e ^{-1 / 3} _{2=4} , e ^{-1 / 3} _{3=4} e ^{-(2/3)\pi} .

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 . 10.2307/2008636 . 0025-5718.