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, i.e., meromorphic functions that are doubly periodic. 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 \wp-function

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 \wp-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 y=\sin t and s=\arcsin x: That means. So the sine function is an inverse function of an integral function. Elliptic functions are the inverse functions of elliptic integrals. In particular, let: Then the extension of u^{-1} to the complex plane equals the \wp-function. This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equations satisfying the Painlevé property, i.e., those equations that admit poles as their only movable singularities.

Definition

Let be two complex numbers that are linearly independent over \mathbb{R} and let be the period lattice generated by those numbers. Then the \wp-function is defined as follows: This series converges locally uniformly absolutely in the complex torus. It is common to use 1 and \tau in the upper half-plane as generators of the lattice. Dividing by \omega_1 maps the lattice isomorphically onto the lattice with. Because -\tau can be substituted for \tau, without loss of generality we can assume, and then define.

Properties

Laurent expansion

Let. Then for 0<|z|<r the \wp-function has the following Laurent expansion where for n \geq 3 are so called Eisenstein series.

Differential equation

Set g_2=60G_4 and g_3=140G_6. Then the \wp-function satisfies the differential equation This relation can be verified by forming a linear combination of powers of \wp and \wp' to eliminate the pole at z=0. 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 \Lambda they can be viewed as functions in \omega_1 and \omega_2. The series expansion suggests that g2 and g3 are homogeneous functions of degree −4 and −6. That is for. If \omega_1 and \omega_2 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 g_2,g_3 are so called modular forms. The Fourier series for g_2 and g_3 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 characteristic polynomial of the differential equation as follows: 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 \eta is the Dedekind eta function. For the Fourier coefficients of \Delta, see Ramanujan tau function.

The constants e1, e2 and e3

e_1, e_2 and e_3 are usually used to denote the values of the \wp-function at the half-periods. They are pairwise distinct and only depend on the lattice \Lambda and not on its generators. e_1, e_2 and e_3 are the roots of the cubic polynomial and are related by the equation: Because those roots are distinct the discriminant \Delta does not vanish on the upper half plane. Now we can rewrite the differential equation: That means the half-periods are zeros of \wp'. The invariants g_2 and g_3 can be expressed in terms of these constants in the following way: e_1, e_2 and e_3 are related to the modular lambda function:

Relation to Jacobi's elliptic functions

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions. The basic relations are: where e_1,e_2 and e_3 are the three roots described above and where the modulus k of the Jacobi functions equals and their argument w equals

Relation to Jacobi's theta functions

The function can be represented by Jacobi's theta functions: where is the nome and \tau is the period ratio. This also provides a very rapid algorithm for computing.

Relation to elliptic curves

Consider the embedding of the cubic curve in the complex projective plane For this cubic there exists no rational parameterization, if. In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the \wp-function and its derivative \wp': Now the map \varphi is bijective and parameterizes the elliptic curve. is an abelian group and a topological space, equipped with the quotient topology. It can be shown that every Weierstrass cubic is given in such a way. That is to say that for every pair with there exists a lattice, such that and. The statement that elliptic curves over \mathbb{Q} can be parameterized over \mathbb{Q}, is known as the modularity theorem. This is an important theorem in number theory. It was part of Andrew Wiles' proof (1995) of Fermat's Last Theorem.

Addition theorems

Let, so that. Then one has: As well as the duplication formula: These formulas also have a geometric interpretation, if one looks at the elliptic curve together with the mapping as in the previous section. The group structure of translates to the curve and can be geometrically interpreted there: The sum of three pairwise different points is zero if and only if they lie on the same line in. This is equivalent to: where \wp(u) = a, \wp(v)=b and.

Typography

The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘, which was Weierstrass's own notation introduced in his lectures of 1862–1863. It should not be confused with the normal mathematical script letters P, 𝒫 and 𝓅. In computing, the letter ℘ is available as in TeX. In Unicode the code point is, with the more correct alias. In HTML, it can be escaped as.

Footnotes