Introduction to the Weierstrass functions and inverses

General

Historical remarks

The Weierstrass elliptic functions are identified with the famous mathematicians N. H. Abel (1827) and K. Weierstrass (1855, 1862). In the year 1849, C. Hermite first used the notation ℘123 for the basic Weierstrass doubly periodic function with only one double pole. The sigma and zeta Weierstrass functions were introduced in the works of F. G. Eisenstein (1847) and K. Weierstrass (1855, 1862, 1895).

The Weierstrass elliptic and related functions can be defined as inversions of elliptic integrals like and . Such integrals were investigated in the works of L. Euler (1761) and J.‐L. Lagrange (1769), who basically introduced the functions that are known today as the inverse Weierstrass functions.

Periodic functions

An analytic function is called periodic if there exists a complex constant such that . The number (with a minimal possible value of ) is called the period of the function .

Examples of well‐known singly periodic functions are the exponential functions, all the trigonometric and hyperbolic functions: , sin(z), cos(z), csc(z), sec(z), tan(z), cot(z), sinh(z), cosh(z), csch(z), sech(z), tanh(z), and coth(z), which have periods , , , , and . The study of such functions can be restricted to any period‐strip , because outside this strip, the values of these functions coincide with their corresponding values inside the strip.

Nonconstant analytic functions over the field of complex numbers cannot have more than two independent periods. So, generically, periodic functions can satisfy the following relations:

where , , and are periods (basic primitive periods). The condition for doubly periodic functions implies the existence of a period‐parallelogram , which is the analog of the period‐strip for singly periodic functions with period .

In the case , this parallelogram is called the basic fundamental period‐parallelogram: . The two line segments lying on the boundary of the period-parallelogram and beginning from the origin belong to . The region includes only one corner point from four points lying at the boundary of the parallelogram with corners in . Sometimes the convention is used.

The set of all such period‐parallelograms:

covers all complex planes: .

Any doubly periodic function is called an elliptic function. The set of numbers is called the period‐lattice for elliptic function .

An elliptic function , which does not have poles in the period‐parallelogram, is equal to a constant (Liouville's theorem).

Nonconstant elliptic (doubly periodic) functions cannot be entire functions. This is not the case for singly periodic functions, for example, is entire function.

Any nonconstant elliptic function has at least two simple poles or at least one double pole in any period‐parallelogram. The sum of all its residues at the poles inside a period‐parallelogram is zero.

The numbers of the zeros and poles of a nonconstant elliptic function in a fundamental period‐parallelogram P are finite.

The number of the zeros of , where is any complex number, in a fundamental period‐parallelogram does not depend on the value and coincides with number of the poles counted according to their multiplicity ( is called the order of the elliptic function ).

The simplest elliptic function has order 2.

Let (and ) be the zeros (and poles) of a nonconstant elliptic function in a fundamental period‐parallelogram , both listed one or more times according to their multiplicity. Then the following hold:

So, the number of zeros of a nonconstant elliptic function in the fundamental period‐parallelogram is equal to the number of poles there and counted according to their multiplicity. The sum of zeros of a nonconstant elliptic function in the fundamental period‐parallelogram differs from the sum of its poles by a period , where and the values of , depend on the function .

All elliptic functions satisfy a common fundamental property, which generalizes addition, duplication, and multiple angle properties for trigonometric and hyperbolic functions (like , ). It can be formulated as the following:

.

It can also be expressed as an algebraic function of .

In other words, there exists an irreducible polynomial in variables with constant coefficients, for which the following relation holds:

And conversely, among all smooth functions, only elliptic functions and their degenerations have algebraic addition theorems.

The simplest elliptic functions (with order 2) can be divided into two classes:

(1) Functions that at the period‐parallelogram have only a double pole with residue zero (e.g., the Weierstrass elliptic functions ).

(2) Functions that in the period‐parallelogram have only two simple poles with residues, which are equal in absolute value but opposite in sign (e.g., Jacobian elliptic functions etc.).

Any elliptic function with periods and can be expressed as a rational function of the Weierstrassian elliptic functions and their derivative with the same periods .

The Weierstrass elliptic function arises as a solution to the following ordinary nonlinear differential equation:

Definitions of Weierstrass functions and inverses

The Weierstrass elliptic function , its derivative , the Weierstrass sigma function , associated Weierstrass sigma functions , Weierstrass zeta function , inverse elliptic Weierstrass function , and generalized inverse Weierstrass function are defined by the following formulas:

The function is the unique value of for which and . For the existence of , the values and must be related by .

The previous nine functions are typically called Weierstrass elliptic functions. The last two functions are called inverse elliptic Weierstrass functions.

Despite the commonly used naming convention, only the Weierstrass function and its derivative are elliptic functions because only these functions are doubly periodic. The other Weierstrass functions , , and are not elliptic functions because they are only quasi‐periodic functions with respect to . But historically they are also placed into the class of elliptic functions.

The Weierstrass half‐periods and the invariants , the Weierstrass function values at half-periods , and the Weierstrass zeta function values at half-periods are defined by the following formulas. The description of the Weierstrass functions follows the notations used throughout. The left‐hand sides indicate that and are either independent variables or depend on and , or vice versa:

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

A quick look at the Weierstrass functions and inverses

Here is a quick look at the graphics for the Weierstrass functions and inverses. All of the following graphics use the half-periods .

The next pair of graphics shows the Weierstrass function over the complex ‐plane. The double periodicity of the function and the poles of order 2 are clearly visible.

The next pair of graphics shows the derivative of the Weierstrass function over the complex ‐plane. The double periodicity of the function and the poles of order 3 are clearly visible.

The next pair of graphics shows the Weierstrass zeta function over the complex ‐plane. The pseudo‐double periodicity of the function and the poles of order 1 are clearly visible.

The next pair of graphics shows the Weierstrass sigma function over the complex ‐plane.

The next three pairs of graphics show the associated Weierstrass sigma functions over the complex ‐plane.

The last pair of graphics shows the inverse of the Weierstrass function over the complex ‐plane. Compared with the direct function, it is relatively structureless.

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

Representations through more general functions

The Weierstrass elliptic function and its inverse can be represented through the more general hypergeometric Appell function of two variables by the following formulas:

Representations through related equivalent functions

The Weierstrass functions , , , , , , and can be represented through some related equivalent functions, for example, through Jacobi functions:

where is modular lambda function, or through theta functions:

or through elliptic integrals and the inverse elliptic nome:

Relations to inverse functions

The Weierstrass function and its derivative are interconnected with the inverse functions and by the following formulas:

Representations through other Weierstrass functions

Each of the Weierstrass functions , , , , and can be expressed through the other Weierstrass functions using numerous formulas, for example:

Note that the Weierstrass functions , , , , and form a chain with respect to differentiation:

The best-known properties and formulas for Weierstrass functions and inverses

Simple values at zero

The Weierstrass functions , , , and have the following simple values at the origin point:

Specific values for specialized parameter

The Weierstrass functions , , , , and can be represented through elementary functions, when or :

At points , all Weierstrass functions , , , , and can be equal to zero or can have poles and be equal to :

The values of Weierstrass functions , , , , and at the points can sometimes be evaluated in closed form:

The Weierstrass functions , , and have rather simple values, when and or :

The Weierstrass functions , , , and can be represented through elementary functions, when :

Analyticity

The Weierstrass functions , , , , , and are analytical functions of , , and , which are defined in . The inverse Weierstrass function is an analytical function of , , , , which is also defined in , because is not an independent variable.

Poles and essential singularities

For fixed , , the Weierstrass functions , , and have an infinite set of singular points:

(a) are the poles of order 2 with residues 0 (for ), of order 3 with residues 0 (for ) and simple poles with residues 1 (for ).

(b) is an essential singular point.

For fixed , , the Weierstrass functions and have only one singular point at . It is an essential singular point.

The Weierstrass functions and do not have poles and essential singularities with respect to their variables.

Branch points and branch cuts

For fixed , , the Weierstrass functions , , , , and do not have branch points and branch cuts.

For fixed , , the inverse Weierstrass function has two branch points: .

For fixed , , the inverse Weierstrass function has two branch points: .

For fixed , , the inverse Weierstrass function has four branch points: .

Periodicity

The Weierstrass functions and are doubly periodic functions with respect to with periods and :

The Weierstrass functions , , and are quasi‐periodic functions with respect to :

The inverse Weierstrass functions and do not have periodicity and symmetry.

Transformation of half-periods

The Weierstrass functions , , , , and are the invariant functions under the linear transformation of the half‐periods , with integer coefficients , , , and , satisfying restrictions (modular transformations):

Homogeneity

The Weierstrass functions , , , , and satisfy the following homogeneity type relations:

Parity and symmetry

The Weierstrass functions , , , , , and have mirror symmetry:

The Weierstrass functions and are even functions with respect to :

The Weierstrass functions , , and are odd functions with respect to :

Series representations

The Weierstrass functions , , , and have the following series expansions at the point :

The inverse Weierstrass function has the following series expansion at the point :

q-series representations

The Weierstrass functions , , , , and have the following so-called ‐series representations:

Other series representations

The Weierstrass functions , , , , and with can be represented through series of different forms, for example:

Integral representations

The Weierstrass functions and their inverses , , , , , and can be represented through the following integrals from elementary or Weierstrass functions:

Product representations

The Weierstrass functions , , and have the following product representations:

q-product representations

The Weierstrass functions , , and can be represented as so-called ‐products by the following formulas:

Transformations

The Weierstrass functions , , , , and satisfy numerous relations that can provide transformations of its arguments. One of these transformations simplifies argument to , for example:

Other transformations are described by so-called addition formulas:

Half‐angle formulas provide one more type of transformation, for example:

The Weierstrass functions , , , and satisfy the following double-angle formulas:

These formulas can be expanded on triple angle formulas, for example:

Generally the following multiple angle formulas take place:

Sometimes transformations have a symmetrical character, which includes operations like determinate, for example:

A special class of transformation includes the simplification of Weierstrass functions , , , , and with invariants , where , for example:

Identities

The Weierstrass functions satisfy numerous functional identities, for example:

Representations of derivatives

The first two derivatives of all Weierstrass functions , , , , and , and their inverses and with respect to variable can also be expressed through Weierstrass functions:

The first derivatives of Weierstrass functions , , , and with respect to parameter can also be expressed through Weierstrass functions by the following formulas:

The first derivatives of Weierstrass functions , , , and with respect to parameter can also be expressed through Weierstrass functions by the following formulas:

Weierstrass invariants and can be expressed as functions of half-periods and . This property allows obtaining the following formulas for the first derivatives of Weierstrass functions , , , and with respect to half-period :

Similar formulas take place for the first derivatives of Weierstrass functions , , , and with respect to half-period :

The derivatives of all Weierstrass functions , , , , , and their inverses and with respect to variable can be represented by the following formulas:

Integration

The indefinite integrals of Weierstrass functions , , , and with respect to variable can be expressed by the following formulas:

Summation

Finite and infinite sums including Weierstrass functions can sometimes be evaluated in closed forms, for example:

Differential equations

The Weierstrass functions , , , , and their inverses and satisfy the following nonlinear differential equations:

The Weierstrass functions , , , and are the special solutions of the corresponding partial differential equations:

Applications of Weierstrass functions and inverses

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