The bestknown 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 halfperiods
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 :
qseries representations
The Weierstrass functions , , , , and have the following socalled ‐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:
qproduct representations
The Weierstrass functions , , and can be represented as socalled ‐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 socalled addition formulas:
Half‐angle formulas provide one more type of transformation, for example:
The Weierstrass functions , , , and satisfy the following doubleangle 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 halfperiods and . This property allows obtaining the following formulas for the first derivatives of Weierstrass functions , , , and with respect to halfperiod :
Similar formulas take place for the first derivatives of Weierstrass functions , , , and with respect to halfperiod :
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:
