Introduction to the Weierstrass utility functions

General

The creation and development of the elliptic functions' theory in the 18th century required the introduction of special supporting utility functions, which were frequently used for description of the properties of the elliptic functions. Among such utilities the basic role is played by so-called Weierstrass invariants and Weierstrass half-periods. These were given the unusual notations and {} instead of a consecutive numbering. The Weierstrass utility functions are a pair of bivariate functions that are inverses of each other:

Half-periods and (and ) were mentioned in the works of C. G. J. Jacobi (1835), K. Weierstrass (1862), and A. Hurwitz (1905). The invariants and were mentioned in the works of A. Cayley and G. Boole (1845).

Numerous formulas of Weierstrass elliptic functions include values of the Weierstrass function and the Weierstrass zeta functions and at the points . These values have the following widely used notations:

Definitions of Weierstrass utilities

The Weierstrass half‐periods and the Weierstrass invariants , the Weierstrass function values at half-periods , and the Weierstrass zeta function values at half-periods are defined by the following formulas:

where is the Klein invariant modular function, is Weierstrass elliptic function, and is theWeierstrass zeta function.

The previous four vector functions are sometimes called the Weierstrass utilities because they are the basic elements of the Weierstrass theory of elliptic functions.

A quick look at the Weierstrass utilities

The following graphics show the values of the real and imaginary parts of the Weierstrass half-periods over the real ‐ plane.

The following graphics show the values of the real and imaginary parts of the Weierstrass invarants over the ‐ plane.

The following graphics show the values of the real and imaginary parts of the Weierstrass function at the half-periods over the real ‐ plane.

The following graphics show the values of the real and imaginary parts of the Weierstrass zeta function at the half-periods over the real ‐ plane.

Connections within the group of Weierstrass utilities and inverses and with other function groups

Representations through related equivalent functions

The Weierstrass half‐periods can be represented through the complete elliptic integral and the inverse elliptic nome by the formula:

The Weierstrass invariants can be represented through the complete elliptic integral , the inverse elliptic nome , the modular lambda function , and the theta functions by the following formulas:

The Weierstrass function values at half-periods can be represented through the complete elliptic integral , the modular lambda function , the Weierstrass sigma function , and the theta functions by the following formulas:

The Weierstrass zeta function values at half-periods can be represented through the complete elliptic integrals and , the modular lambda function , and the theta functions by the following formulas:

Relations to inverse functions

The following formula shows that the Weierstrass half‐periods play the role of inverse functions to the Weierstrass invariants :

Representations through other Weierstrass utilities

The Weierstrass half‐periods , the invariants , and the Weierstrass function values at half-periods are connected by the following formulas:

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:

Applications of Weierstrass utilities

Applications of Weierstrass utilities include the application areas of the Weierstrass elliptic functions, such as integrable nonlinear differential equations, motion in cubic and quartic potentials, description of the movement of a spherical pendulum, and construction of minimal surfaces.