The bestknown properties and formulas for Weierstrass utilities
Specific values
The Weierstrass invariants have the following values at infinities:
The Weierstrass function values at halfperiods can be evaluated at closed forms for some values of arguments , :
The Weierstrass zeta function values at halfperiods can also be evaluated at closed forms for some values of arguments , :
Analyticity
The Weierstrass half‐periods , the Weierstrass function values at halfperiods , and the Weierstrass zeta function values at halfperiods 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 halfperiods have mirror symmetry:
The Weierstrass invariants and the Weierstrass function values at halfperiods 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 halfperiods and the Weierstrass zeta function values at halfperiods :
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:
qseries representations
The Weierstrass invariants , the Weierstrass function values at halfperiods , and the Weierstrass zeta function values at halfperiods 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 halfperiods 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 halfperiods and with respect to variable and have the following representations:
where are the values of the derivative of the Weierstrass elliptic function at halfperiod 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 halfperiods satisfy the following differential equations:
