The best-known properties and formulas for Weierstrass utilities

Specific values

The Weierstrass invariants have the following values at infinities:

The Weierstrass function values at half-periods can be evaluated at closed forms for some values of arguments , :

The Weierstrass zeta function values at half-periods can also be evaluated at closed forms for some values of arguments , :

Analyticity

The Weierstrass half‐periods , the Weierstrass function values at half-periods , and the Weierstrass zeta function values at half-periods are vector‐valued functions of and that are analytic in each vector component, and they are defined over .

The Weierstrass invariants is a vector‐valued function of and that is analytic in each vector component, and it is defined over (for ).

Periodicity

The Weierstrass invariants with is a periodic function with period :

The other Weierstrass utility functions , , and are not periodic functions.

Parity and symmetry

The Weierstrass half‐periods and Weierstrass zeta function values at half-periods have mirror symmetry:

The Weierstrass invariants and the Weierstrass function values at half-periods have standard mirror symmetry:

The Weierstrass invariants have permutation symmetry and are homogeneous:

The Weierstrass invariants are the invariants under the change of variables and with integers , , , and , satisfying the restriction (modular transformations):

This property leads to similar properties of the Weierstrass function values at half-periods and the Weierstrass zeta function values at half-periods :

Series representations

The Weierstrass half‐periods and invariants have the following double series expansions:

where is a Klein invariant modular function.

The last double series can be rewritten in the following forms:

q-series representations

The Weierstrass invariants , the Weierstrass function values at half-periods , and the Weierstrass zeta function values at half-periods have numerous q‐series representations, for example:

where .

The following rational function of and is a modular function if considered as a function of :

Other series representations

The Weierstrass utilities have some other forms of series expansions, for example:

where is the divisor sigma function.

Integral representations

The Weierstrass half‐periods and invariants have the following integral representations:

Product representations

The Weierstrass utilities can have product representations. For example, the Weierstrass function values at half-periods can be expressed through the following products:

where .

Identities

The Weierstrass utilities satisfy numerous identities, for example:

Representations of derivatives

The first derivatives of Weierstrass half‐periods and the Weierstrass and zeta function values at half-periods and with respect to variable and have the following representations:

where are the values of the derivative of the Weierstrass elliptic function at half-period points .

The first derivatives of Weierstrass invariants with respect to the variables and can be represented in different forms:

The -order derivatives of Weierstrass invariants with respect to the variables and have the following representations:

Integration

The indefinite integrals of Weierstrass invariants with respect to the variable have the following representations:

Differential equations

The Weierstrass half‐periods satisfy the following differential equations:

The Weierstrass invariants satisfy the following differential equations:

The Weierstrass zeta function values at half-periods satisfy the following differential equations: