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The Weierstrass functions  ,  ,  , and  have the following simple values at the origin point:
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  :
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.
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.
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:  .
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.
The Weierstrass functions  ,  ,  ,  , and  are the invariant functions under the linear transformation of the half‐periods  ,  with integer coefficients  ,  ,  , and  , satisfying restrictions  (modular transformations):
The Weierstrass functions  ,  ,  ,  , and  satisfy the following homogeneity type relations:
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  :
The Weierstrass functions  ,  ,  , and  have the following series expansions at the point  :
The inverse Weierstrass function  has the following series expansion at the point  :
The Weierstrass functions  ,  ,  ,  , and  have the following so-called  ‐series representations:
The Weierstrass functions  ,  ,  ,  , and  with  can be represented through series of different forms, for example:
The Weierstrass functions and their inverses  ,  ,  ,  ,  , and  can be represented through the following integrals from elementary or Weierstrass functions:
The Weierstrass functions  ,  , and  have the following product representations:
The Weierstrass functions  ,  , and  can be represented as so-called  ‐products by the following formulas:
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:
The Weierstrass functions satisfy numerous functional identities, for example:
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:
The indefinite integrals of Weierstrass functions  ,  ,  , and  with respect to variable  can be expressed by the following formulas:
Finite and infinite sums including Weierstrass functions can sometimes be evaluated in closed forms, for example:
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:
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