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Concept
Elliptic function
Summary
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse.
Important elliptic functions are Jacobi elliptic functions and the Weierstrass -function.
Further development of this theory led to hyperelliptic functions and modular forms.
A meromorphic function is called an elliptic function, if there are two -linear independent complex numbers such that
and .
So elliptic functions have two periods and are therefore doubly periodic functions.
If is an elliptic function with periods it also holds that
for every linear combination with .
The abelian group
is called the period lattice.
The parallelogram generated by and
is a fundamental domain of acting on .
Geometrically the complex plane is tiled with parallelograms. Everything that happens in one fundamental domain repeats in all the others. For that reason we can view elliptic function as functions with the quotient group as their domain. This quotient group, called an elliptic curve, can be visualised as a parallelogram where opposite sides are identified, which topologically is a torus.
The following three theorems are known as Liouville's theorems (1847).
A holomorphic elliptic function is constant.
This is the original form of Liouville's theorem and can be derived from it. A holomorphic elliptic function is bounded since it takes on all of its values on the fundamental domain which is compact. So it is constant by Liouville's theorem.
Every elliptic function has finitely many poles in and the sum of its residues is zero.
This theorem implies that there is no elliptic function not equal to zero with exactly one pole of order one or exactly one zero of order one in the fundamental domain.
A non-constant elliptic function takes on every value the same number of times in counted with multiplicity.
Official source
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