In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g<sub>2</sub> = 0 and g<sub>3</sub> = 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 ω<sub>2</sub> is real and equal to
:<math>\frac{\Gamma^3(1/3)}{4\pi}</math>
where <math>\Gamma</math> is the Gamma function. The half period is
:<math>\omega_1=\tfrac{1}{2}(-1+\sqrt3i)\omega_2.</math>
Here the period lattice is a real multiple of the Eisenstein integers.
The constants e<sub>1</sub>, e<sub>2</sub> and e<sub>3</sub> are given by
:<math>
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}.
</math>
The case g<sub>2</sub> = 0, g<sub>3</sub> = a may be handled by a scaling transformation.