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Wikipedia / Weierstrass elliptic function

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 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. -function

Motivation

A cubic, cannot be rationally parameterized. Yet one still wants to find a way to parameterize it.

For the quadric; the unit circle is chosen to be the domain, so the function is bijective.

, which is topologically equivalent to a torus.

There is another analogy to the trigonometric functions. So the sine function is an inverse function of an integral function.

Elliptic functions are the inverse functions of elliptic integrals-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

be two complex numbers that are linearly independent be the period lattice-function is defined as follows:

This series converges locally uniformly absolutely in the complex torus.

in the upper half-plane as generators of the lattice, without loss of generality.

Properties

Laurent expansion

-function has the following Laurent expansion are so called Eisenstein series.

Differential equation

-function satisfies the differential equation This relation can be verified by forming a linear combination. This yields an entire elliptic function that has to be constant by Liouville's theorem.

Invariants

are known as the invariants.

are homogeneous functions. That is.

can be interpreted as functions on the upper half-plane.

. One has: That means g2 and g3 are so called modular forms.

The Fourier series are given as follows: is the divisor function is the nome.

Modular discriminant

The modular discriminant is defined as the discriminant of the characteristic polynomial. That is, under the action of the modular group.

is the Dedekind eta function.

, see Ramanujan tau function.

The constants e1, e2 and e3

and not on its generators.

does not vanish on the upper half plane..

can be expressed in terms of these constants in the following way: 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: are the three roots described above and where the modulus k and their argument w

Relation to Jacobi's theta functions

can be represented by Jacobi's theta functions..

Relation to elliptic curves

Consider the embedding of the cubic curve in the complex projective plane

is a point lying on the line at infinity. In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates:

is bijective.

is an abelian group and a topological space, equipped with the quotient topology.

, such that

.

Addition theorem

The addition theorem states are collinear, the geometric form of the group law of an elliptic curve.

This can be proven.

Alternative form

:

As well as the duplication formula:

Proofs

A direct proof is as follows. is the Weierstrass sigma function are the respective zeros and poles in the period parallelogram

By definition the Weierstrass zeta function

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 \wp in TeX. In Unicode the code point is , with the more correct alias weierstrass elliptic function. In HTML, it can be escaped as weierp; or wp;.

See also

Footnotes

References

  • 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

  • Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York (See chapter 1.)

  • K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag

  • Konrad Knopp, Funktionentheorie II (1947), Dover Publications; Republished in English translation as Theory of Functions (1996), Dover Publications

  • Serge Lang, Elliptic Functions (1973), Addison-Wesley,

  • E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 1952, chapters 20 and 21