In mathematics, **Weierstrass's 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 **P-functions** and generally written using the symbol ℘ (or

## Contents

## Definitions

The **Weierstrass elliptic function** can be defined in three closely related ways, each of which possesses certain advantages. One is as a function of a complex variable *z* and a lattice Λ in the complex plane. Another is in terms of *z* and two complex numbers ω_{1} and ω_{2} defining a pair of generators, or periods, for the lattice. The third is in terms of *z* and a modulus τ in the upper half-plane. This is related to the previous definition by τ = ω_{2}/ω_{1}, which by the conventional choice on the pair of periods is in the upper half-plane. Using this approach, for fixed *z* the Weierstrass functions become modular functions of τ.

In terms of the two periods, **Weierstrass's elliptic function** is an elliptic function with periods ω_{1} and ω_{2} defined as

Then
**period lattice**, so that

for any pair of generators of the lattice defines the Weierstrass function as a function of a complex variable and a lattice.

If

The above sum is homogeneous of degree minus two, from which we may define the Weierstrass ℘ function for any pair of periods, as

We may compute ℘ very rapidly in terms of theta functions; because these converge so quickly, this is a more expeditious way of computing ℘ than the series we used to define it. The formula here is

There is a second-order pole at each point of the period lattice (including the origin). With these definitions,
*z*, ℘′, is an odd function.

Further development of the theory of elliptic functions shows that Weierstrass's function is determined up to addition of a constant and multiplication by a non-zero constant by the position and type of the poles alone, amongst all meromorphic functions with the given period lattice.

## Invariants

In a deleted neighborhood of the origin, the Laurent series expansion of

where

The numbers *g*_{2} and *g*_{3} are known as the *invariants*. The summations after the coefficients 60 and 140 are the first two Eisenstein series, which are modular forms when considered as functions G_{4}(τ) and G_{6}(τ), respectively, of τ = ω_{2}/ω_{1} with Im(τ) > 0.

Note that *g*_{2} and *g*_{3} are homogeneous functions of degree −4 and −6; that is,

Thus, by convention, one frequently writes

The Fourier series for

where

The invariants may be expressed in terms of Jacobi's theta functions. This method is very convenient for numerical calculation: the theta functions converge very quickly. In the notation of Abramowitz and Stegun, but denoting the primitive periods by

where

and

## Special cases

If the invariants are *g*_{2} = 0, *g*_{3} = 1, then this is known as the equianharmonic case; *g*_{2} = 1, *g*_{3} = 0 is the lemniscatic case.

## Differential equation

With this notation, the ℘ function satisfies the following differential equation:

where dependence on

This relation can be quickly verified by comparing the poles of both sides, for example, the pole at *z* = 0 of lhs is

while the pole at *z* = 0 of

Comparing these two yields the relation above.

## Integral equation

The Weierstrass elliptic function can be given as the inverse of an elliptic integral. Let

Here, *g*_{2} and *g*_{3} are taken as constants. Then one has

The above follows directly by integrating the differential equation.

## Modular discriminant

The *modular discriminant* Δ is defined as the quotient by 16 of the discriminant of the right-hand side of the above differential equation:

This is studied in its own right, as a cusp form, in modular form theory (that is, as a *function of the period lattice*).

Note that

The presence of 24 can be understood by connection with other occurrences, as in the eta function and the Leech lattice.

The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as

with τ being the half-period ratio, and *a*,*b*,*c* and *d* being integers, with *ad* − *bc* = 1.

For the Fourier coefficients of

## The constants e1, e2 and e3

Consider the cubic polynomial equation 4*t*^{3} − *g*_{2}*t* − *g*_{3} = 0 with roots *e*_{1}, *e*_{2}, and *e*_{3}. Its discriminant is 16 times the modular discriminant Δ = *g*_{2}^{3} − 27*g*_{3}^{2}. If it is not zero, no two of these roots are equal. Since the quadratic term of this cubic polynomial is zero, the roots are related by the equation

The linear and constant coefficients (*g*_{2} and *g*_{3}, respectively) are related to the roots by the equations (see Elementary symmetric polynomial).

The roots *e*_{1}, *e*_{2}, and *e*_{3} of the equation
*τ* and can be expressed in terms of theta functions. As before, let,

then

Since

Where
**Δ** = *g*_{2}^{3} − 27*g*_{3}^{2} determines the nature of the roots. If

The half-periods ω_{1}/2 and ω_{2}/2 of Weierstrass' elliptic function are related to the roots

where

If *g*_{2} and *g*_{3} are real and Δ > 0, the *e*_{i} are all real, and
_{3}, ω_{1} + ω_{3}, and ω_{1}. If the roots are ordered as above (*e*_{1} > *e*_{2} > *e*_{3}), then the first half-period is completely real

whereas the third half-period is completely imaginary

## Addition theorems

The Weierstrass elliptic functions have several properties that may be proved:

A symmetrical version of the same identity is

Also

and the *duplication formula*

unless 2*z* is a period.

## The case with 1 a basic half-period

If
**Weierstrass ℘ function** by

The sum extends over the lattice {*n*+*m*τ : *n* and *m* in **Z**} with the origin omitted. Here we regard τ as fixed and ℘ as a function of *z*; fixing *z* and letting τ vary leads into the area of elliptic modular functions.

## General theory

℘ is a meromorphic function in the complex plane with a double pole at each lattice point. It is doubly periodic with periods 1 and τ; this means that ℘ satisfies

The above sum is homogeneous of degree minus two, and if *c* is any non-zero complex number,

from which we may define the Weierstrass ℘ function for any pair of periods. We also may take the derivative (of course, with respect to *z*) and obtain a function algebraically related to ℘ by

where

defines an elliptic curve, and we see that

so that all such functions are rational functions in the Weierstrass function and its derivative.

One can wrap a single period parallelogram into a torus, or donut-shaped Riemann surface, and regard the elliptic functions associated to a given pair of periods to be functions defined on that Riemann surface.

℘ can also be expressed in terms of theta functions; because these converge very rapidly, this is a more expeditious way of computing ℘ than the series used to define it.

The function ℘ has two zeros (modulo periods) and the function ℘′ has three. The zeros of ℘′ are easy to find: since ℘′ is an odd function they must be at the half-period points. On the other hand, it is very difficult to express the zeros of ℘ by closed formula, except for special values of the modulus (e.g. when the period lattice is the Gaussian integers). An expression was found, by Zagier and Eichler.

The Weierstrass theory also includes the Weierstrass zeta function, which is an indefinite integral of ℘ and not doubly periodic, and a theta function called the Weierstrass sigma function, of which his zeta-function is the log-derivative. The sigma-function has zeros at all the period points (only), and can be expressed in terms of Jacobi's functions. This gives one way to convert between Weierstrass and Jacobi notations.

The Weierstrass sigma-function is an entire function; it played the role of 'typical' function in a theory of *random entire functions* of J. E. Littlewood.

## Relation to Jacobi 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*_{1–3} are the three roots described above and where the modulus *k* of the Jacobi functions equals

and their argument *w* equals