Equianharmonic
Encyclopedia
In mathematics
, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy
and 
;
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
is real and equal to
where
is the Gamma function
. The half period is
Here the period lattice is a real multiple of the Eisenstein integer
s.
The constant
s
, 
and 
are given by
The case g2=0, g3=a may be handled by a scaling transformation.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy
and ;This page follows the terminology of Abramowitz and Stegun
Abramowitz and Stegun
Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards...
; see also the lemniscatic case. (These are special examples of complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
).
In the equianharmonic case, the minimal half period
is real and equal towhere
is the Gamma functionGamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
. The half period is
Here the period lattice is a real multiple of the Eisenstein integer
Eisenstein integer
In mathematics, Eisenstein integers , also known as Eulerian integers , are complex numbers of the formz = a + b\omega \,\!where a and b are integers and...
s.
The constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
s
, and are given byThe case g2=0, g3=a may be handled by a scaling transformation.