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: