Weierstrass's elliptic functions

Definitions

Consider two complex numbers and defining a lattice. 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 has a positive imaginary part. As defined below, the two numbers serve as half-periodss. Compare the trigonometric usage of 2π.

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

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

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

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

Note that and are homogeneous functions of degree -4 and -6; that is,
:
and
:.

Thus, by convention, one frequently writes and in terms of the half-period ratio and take to lie in the upper half plane. Thus, and .

The Fourier series for and can be written in terms of the square of the nome as
:
and
:
where is the divisor function. In practical calculations, these are best re-written as Lambert series.

Special cases

If the invariants are , , then this is known as the Equianharmonic case; , is the Lemniscatic case.

Differential equation

Integral equation

The above follows directly by integrating the differential equation.

Modular discriminant

The modular discriminant 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).

Note that where is the Dedekind eta function.

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.

The constants e₁, e₂ and e₃

Consider the algebraic equation , and name its roots , , and . It can be shown from the non-vanishing of the discriminant that no two of these three are equal.

Algebraic considerations show that .

In the case of real invariants, the sign of determines the nature of the roots. If , all three are real and it is conventional to name them so that . If , it is conventional to write (where , ), whence and is real and non-negative. We also have
:
where . Also, for .

If and are real and , the are all real, and is real on the perimeter of the rectangle with corners , , , and .

Addition theorems

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

Also
:

and the duplication formula
:
unless is a period.

The case with 1 a basic half-period

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

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 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

,

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 , , and of the equation depend on τ and can be expressed in terms of theta functions; we have

:
:

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 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