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. 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.
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 .[1]
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 , can not be rationally parameterized.[2] 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.[3]
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.[4]
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.[5]
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.
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 real part of the invariant g3 as a function of the nome q on the unit disk.
The imaginary part of the invariant g3 as a function of the nome q on the unit disk.
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.
, 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]
Relation to elliptic curves[]
Consider the projective cubic curve
.
For this cubic, also called Weierstrass cubic, there exists no rational parameterization, if .[2] In this case it is also called an elliptic curve. Nevertheless there is a parameterization that uses the -function and its derivative :[15]
Now the map is bijective and parameterizes the elliptic curve .
where and are the three roots described above and where the modulus k of the Jacobi functions equals
and their argument w equals
Typography[]
The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘.[footnote 1]
In computing, the letter ℘ is available as \wp in TeX. In Unicode the code point is U+2118℘SCRIPT CAPITAL P (HTML ℘·℘, ℘), with the more correct alias weierstrass elliptic function.[footnote 2] In HTML, it can be escaped as ℘.
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The Unicode Consortium has acknowledged two problems with the letter's name: the letter is in fact lowercase, and it is not a "script" class letter, like U+1D4C5