Weierstrass's elliptic functions

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 ω₁ and ω₂ 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 = ω₂/ω₁, 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 ω₁ and ω₂ defined as

Then Λ = { m ω 1 + n ω 2 : m , n ∈ Z } are the points of the 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 τ is a complex number in the upper half-plane, then

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 even function and its derivative with respect to 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 ℘ is

where

The numbers g₂ and g₃ 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 G₄(τ) and G₆(τ), respectively, of τ = ω₂/ω₁ with Im(τ) > 0.

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

Thus, by convention, one frequently writes g 2 and g 3 in terms of the period ratio τ = ω 2 / ω 1 and take τ to lie in the upper half-plane. Thus, g 2 ( τ ) = g 2 ( 1 , ω 2 / ω 1 ) and g 3 ( τ ) = g 3 ( 1 , ω 2 / ω 1 ) .

The Fourier series for g 2 and g 3 can be written in terms of the square of the nome q = exp ⁡ ( i π τ ) as

where σ a ( k ) is the divisor function. This formula may be rewritten in terms of Lambert series.

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 ω 1 , ω 2 , the invariants satisfy

where

and τ = ω 2 / ω 1 is the period ratio, q = e π i τ is the nome, and θ m and ϑ m n are alternative notations.

Special cases

If the invariants are g₂ = 0, g₃ = 1, then this is known as the equianharmonic case; g₂ = 1, g₃ = 0 is the lemniscatic case.

Differential equation

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

where dependence on ω 1 and ω 2 is suppressed.

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, g₂ and g₃ 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 Δ = ( 2 π ) 12 η 24 where η is the Dedekind eta function.

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 Δ , see Ramanujan tau function.

The constants e1, e2 and e3

Consider the cubic polynomial equation 4t³ − g₂t − g₃ = 0 with roots e₁, e₂, and e₃. Its discriminant is 16 times the modular discriminant Δ = g₂³ − 27g₃². 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 (g₂ and g₃, respectively) are related to the roots by the equations (see Elementary symmetric polynomial).

The roots e₁, e₂, and e₃ of the equation 4 x 3 − g 2 x − g 3 = 0 depend on τ and can be expressed in terms of theta functions. As before, let,

then

Since g 2 = 2 ( e 1 2 + e 2 2 + e 3 2 ) and g 3 = 4 e 1 e 2 e 3 , then these can also be expressed as theta functions. In simplified form,

Where η is the Dedekind eta function. In the case of real invariants, the sign of Δ = g₂³ − 27g₃² determines the nature of the roots. If Δ > 0 , all three are real and it is conventional to name them so that e 1 > e 2 > e 3 . If Δ < 0 , it is conventional to write e 1 = − α + β i (where α ≥ 0 , β > 0 ), whence e 3 = e 1 ¯ , and e 2 is real and non-negative.

The half-periods ω₁/2 and ω₂/2 of Weierstrass' elliptic function are related to the roots

where ω 3 = − ( ω 1 + ω 2 ) . Since the square of the derivative of Weierstrass' elliptic function equals the above cubic polynomial of the function's value, ℘ ′ ( ω i 2 ) 2 = ℘ ′ ( ω i 2 ) = 0 for i = 1 , 2 , 3 . Conversely, if the function's value equals a root of the polynomial, the derivative is zero.

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. If the roots are ordered as above (e₁ > e₂ > e₃), 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 2z is a period.

The case with 1 a basic half-period

If ω 1 = 1 , much of the above theory becomes simpler; it is then conventional to write τ for ω 2 . 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 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 g 2 and g 3 depend only on τ, being modular forms. The equation

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.

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