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 scriptp. 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.
A cubic of the form , where are complex numbers with , cannot be rationally parameterized.[1] 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.[2]
There is another analogy to the trigonometric functions. Consider the integral functionIt can be simplified by substituting and :That means . So the sine function is an inverse function of an integral function.[3]
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 .
Set and . Then the -function satisfies the differential equation[6]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.[6]
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[7] 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:[8]That means g2 and g3 are only scaled by doing this. Set and As functions of are so called modular forms.
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 aswhere with ad − bc = 1.[10]
, 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.[12]
, and are the roots of the cubic polynomial and are related by the equation:Because those roots are distinct the discriminant does not vanish on the upper half plane.[13] Now we can rewrite the differential equation:That means the half-periods are zeros of .
The invariants and can be expressed in terms of these constants in the following way:[14], and 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:[15]where and are the three roots described above and where the modulus k of the Jacobi functions equalsand 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 is the period ratio .[16] This also provides a very rapid algorithm for computing .
For this cubic there exists no rational parameterization, if .[1] In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the -function and its derivative :[17]
Now the map is bijective and parameterizes the elliptic curve .
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.[footnote 1]
In computing, the letter ℘ is available as \wp in TeX. In Unicode the code point is U+2118℘SCRIPT CAPITAL P (℘, ℘), with the more correct alias weierstrass elliptic function.[footnote 2] In HTML, it can be escaped as ℘.
N. I. Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN0-8218-4532-2
Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN0-387-97127-0 (See chapter 1.)
K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag ISBN0-387-15295-4
Konrad Knopp, Funktionentheorie II (1947), Dover Publications; Republished in English translation as Theory of Functions (1996), Dover Publications ISBN0-486-69219-1