In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used.

Definitions

Consider two complex numbers ω₁ and ω₂, with ω₁ / ω₂ not purely real; the definition depends on these. There are some significant choices of convention, and the literature is not consistent in its usage. It is common to name these constants so that ω₁ / ω₂ has a negative imaginary part. As defined below, the two numbers serve as half-periods. Compare the trigonometric usage of 2π.

Then Weierstrass's elliptic function is an elliptic function with periods 2ω₁ and 2ω₂ is defined as

where represents the sum over all pairs of integers m and n except m = n = 0. It is usual to write Ω_m,n = 2mω₁ + 2nω₁, the points of the period lattice, so that

.

There is thus a second order pole at each point of the period lattice (including the origin). With these definitions, is an even function and its derivative an odd function.

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

It can be shown that

which converges faster than the other formula given above.

Invariants

The real part of the invariant g_3 as a function of the nome q on the unit disk.

The imaginary part of the invariant g_3 as a function of the nome q on the unit disk.

If points close to the origin are considered the appropriate Laurent series is

where

The numbers g₂ and g₃ are known as the invariants — they are special cases of Eisenstein series. (Abramowitz and Stegun restrict themselves to the case of real g₂ and g₃, stating that this case "seems to cover most applications"; this may be true from the point of view of applied mathematics. If ω₁ is real and ω₂ pure imaginary, or if , the invariants are real).

Note that g₂ and g₃ are homogeneous functions of degree -4 and -6; that is,

g₂(λω₁,λω₂) = λ ^{- 4}g₂(ω₁,ω₂)

and

g₃(λω₁,λω₂) = λ ^{- 6}g₃(ω₁,ω₂).

Thus, by convention, one frequently writes g₂ and g₃ in terms of the half-period ratio τ = ω₂ / ω₁ and take τ to lie in the upper half plane. Thus, g₂(τ) = g₂(1,ω₂ / ω₁) and g₃(τ) = g₃(1,ω₂ / ω₁).

The Fourier series for g₂ and g₃ can be written in terms of the square of the nome q = exp(iπτ) as

and

where σ_a(k) is the divisor function. In pratical calculations, these are best re-written as Lambert series.

Differntial equation

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

where dependence on ω₁ and ω₂ is suppressed.

Modular discriminant

The modular discriminant Δ is defined as

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

The constants e₁, e₂ and e₃

Consider the algebraic equation 4t³ - g₂t - g₃ = 0, and name its roots e₁, e₂, and e₃. It can be shown from the non-vanishing of the discriminant that no two of these three are equal. (When all three are real, it is conventional to name them so that e₁ > e₂ > e₃; from the reality of the invariants it follows only that one is real.) We also have

where ω₃ = - ω₁ - ω₂. Also, for i = 1,2,3.

If g₂ and g₃ are real and Δ > 0, the e_i are all real, and is real on the perimeter of the rectangle with corners 0, ω₃, ω₁ + ω₃, and ω₁.

Addition theorems

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

(a symmetrical version would be

where u + v + w = 0).

Also

and

unless 2z is a period.

The case with 1 a basic half-period

If ω₁ = 1, much of the above theory becomes simpler; it is then conventional to write τ for ω₂. For a fixed τ in the upper half plane, so that the imaginary part of τ is positive, we define the Weierstrass function by:

The sum extends over the lattice {n+mτ : 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 poles at the lattice points. 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 g₂ and g₃ depend only on τ, being modular forms. The equation

Y² = X³ - g₂X - g₃

defines an elliptic curve, and we see that (,) is a parametrization of that curve.

The totality of meromorphic doubly periodic functions with given periods defines an algebraic function field, associated to that curve. It can be shown that this field is

,

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

We can also 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.

The roots e₁, e₂, and e₃ of the equation X³ - g₂X - g₃ depend on τ and can be expressed in terms of theta functions; we have

Since g₂ = - 4(e₁e₂ + e₂e₃ + e₃e₁) and g₃ = 4e₁e₂e₃ we have these in terms of theta functions also.

We may also express in terms of theta functions; because these converge very rapidly, this is a more expeditious way of computing than the series we used to define it.

The function has two zeroes (modulo periods) and the function has three. The zeroes 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 zeroes 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 zeroes 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.

References

• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0 (See chapter 1.)
• K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag ISBN 0-387-15295-4
• Abramowitz and Stegun, chapter 18