mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, and in particular the study of

Weierstrass elliptic function In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred to as ℘-functions and they are usually denoted by the s ...

s, the equianharmonic case occurs when the Weierstrass invariants satisfy ''g''₂ = 0 and ''g''₃ = 1. This page follows the terminology of

Abramowitz and Stegun ''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the National Institute of Standards and T ...

; see also the lemniscatic case. (These are special examples of

complex multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...

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

\frac

where

\Gamma

is the

Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

. The half period is :

\omega_1=\tfrac(-1+\sqrt3i)\omega_2.

Here the

period lattice In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. Definitio ...

is a real multiple of the

Eisenstein integer In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form : z = a + b\omega , where and are integers and : \omega = \frac ...

s. The constants ''e''₁, ''e''₂ and ''e''₃ are given by :

e_1=4^e^,\qquad
e_2=4^,\qquad
e_3=4^e^{-(2/3)\pi i}.

The case ''g''₂ = 0, ''g''₃ = ''a'' may be handled by a scaling transformation.

References

Modular forms Elliptic curves Elliptic functions