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 are also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice. Symbol for Weierstrass -function
Definition
Let be two complex numbers that are linearly independent over and let be the lattice generated by those numbers. Then the -function is defined as follows:
This series converges locally uniformly absolutely in . Oftentimes instead of only is written.
The Weierstrass -function is constructed exactly in such a way that it has a pole of the order two at each lattice point.
Because the sum alone would not converge it is necessary to add the term .
It is common to use and in the upper half-plane as generators of the lattice. Dividing by maps the lattice isomorphically onto the lattice with . Because can be substituted for , without loss of generality we can assume , and then define .
Motivation
A cubic of the form , where are complex numbers with , cannot be rationally parameterized. Yet one still wants to find a way to parameterize it.
For the quadric , the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:
Because of the periodicity of the sine and cosine is chosen to be the domain, so the function is bijective.
In a similar way one can get a parameterization of by means of the doubly periodic -function (see in the section "Relation to elliptic curves"). This parameterization has the domain , which is topologically equivalent to a torus.
There is another analogy to the trigonometric functions. Consider the integral function
It can be simplified by substituting and :
That means . So the sine function is an inverse function of an integral function.
Elliptic functions are also inverse functions of integral functions, namely of elliptic integrals. In particular the -function is obtained in the following way:
Let
Then can be extended to the complex plane and this extension equals the -function.
Properties
is an even function. That means for all , which can be seen in the following way: The second last equality holds because . Since the sum converges absolutely this rearrangement does not change the limit. is meromorphic and its derivative is and are doubly periodic with the periods and . This means:
It follows that and for all . Functions which are meromorphic and doubly periodic are also called elliptic functions.
Laurent expansion
Let . Then for the -function has the following Laurent expansion
where
for are so called Eisenstein series.
Differential equation
Set and . Then the -function satisfies the differential equation
This relation can be verified by forming a linear combination of powers of and to eliminate the pole at . This yields an entire elliptic function that has to be constant by Liouville's theorem.
Invariants
The coefficients of the above differential equation g2 and g3 are known as the invariants. Because they depend on the lattice they can be viewed as functions in and .
The series expansion suggests that g2 and g3 are homogeneous functions of degree −4 and −6. That is
for .
If and are chosen in such a way that , g2 and g3 can be interpreted as functions on the upper half-plane .
Let . One has:
That means g2 and g3 are only scaled by doing this. Set
and
As functions of are so called modular forms.
The Fourier series for and are given as follows:
where
is the divisor function and is the nome.
Modular discriminant
The modular discriminant Δ is defined as the discriminant of the polynomial on the right-hand side of the above differential equation:
The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as
where with ad − bc = 1.
Note that where is the Dedekind eta function.
For the Fourier coefficients of , see Ramanujan tau function.
The constants e1, e2 and e3
, and are usually used to denote the values of the -function at the half-periods.
They are pairwise distinct and only depend on the lattice and not on its generators.