Weierstrass elliptic function

This class of functions is 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.

; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:

[4] 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.

be two complex numbers that are linearly independent over

be the period lattice generated by those numbers.

-function is defined as follows: This series converges locally uniformly absolutely in the complex torus

This relation can be verified by forming a linear combination of powers of

This yields an entire elliptic function that has to be constant by Liouville's theorem.

[6] The coefficients of the above differential equation g2 and g3 are known as the invariants.

The series expansion suggests that g2 and g3 are homogeneous functions of degree −4 and −6.

The discriminant is a modular form of weight 12.

That is, under the action of the modular group, it transforms as

They are pairwise distinct and only depend on the lattice

does not vanish on the upper half plane.

are related to the modular lambda function:

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions.

are the three roots described above and where the modulus k of the Jacobi functions equals

[16] This also provides a very rapid algorithm for computing

Consider the embedding of the cubic curve in the complex projective plane For this cubic there exists no rational parameterization, if

[1] In this case it is also called an elliptic curve.

is bijective and parameterizes the elliptic curve

This is an important theorem in number theory.

It was part of Andrew Wiles' proof (1995) of Fermat's Last Theorem.

These formulas also have a geometric interpretation, if one looks at the elliptic curve

and can be geometrically interpreted there: The sum of three pairwise different points

[21] 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] It should not be confused with the normal mathematical script letters P: 𝒫 and 𝓅.

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 ℘.