Weierstrass s elliptic functions

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

For a fixed τ in the upper half plane, so that the imaginary part of τ is positive, we define the Weierstrass $\wp$ function by:

$\wp(z;\tau) =\frac{1}{z^2} + \sum_{n^2+m^2 \ne 0}{1 \over (z-n-m\tau)^2} - {1 \over (n+m\tau)^2}$

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

$\wp$ 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 $\wp$ satisfies

$\wp(z+1) = \wp(z+\tau) = \wp(z)$

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

$\wp(cz;c\tau) = \wp(z;\tau)/c^2$

from which we may define the Weierstrass $\wp$ 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 $\wp$ by

$\wp'^2 = \wp^3 - g_2 \wp - g_3$

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 ( $\wp$ , $\wp'$ ) 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

$\Bbb{C}(\wp, \wp')$ ,

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

$e_1(\tau) = {\pi^2 \over {3}}(\vartheta^4(0;\tau) + \vartheta_{01}^4(0;\tau))$

$e_2(\tau) = -{\pi^2 \over {3}}(\vartheta^4(0;\tau) + \vartheta_{10}^4(0;\tau))$

$e_3(\tau) = {\pi^2 \over {3}}(\vartheta_{10}^4(0;\tau) - \vartheta_{01}^4(0;\tau))$

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

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

$\wp(z; \tau) = \pi^2 \vartheta^2(0;\tau) \vartheta_{10}^2(0;\tau){\vartheta_{01}^2(z;\tau) \over \vartheta_{11}^2(z;\tau)} + e_2(\tau)$

The function $\wp$ has two zeroes (modulo periods) and the function $\wp'$ has three. The zeroes of $\wp'$ are easy to find: since $\wp'$ 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 $\wp$ 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 $\wp$ 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.

Categories: Mathematics | Complex analysis | Special functions | Elliptic functions

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