Introductions to WeierstrassPPrime

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 ℘ 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 ℐ=(∫_∞^z)(1/(Sqrt[4 (t^3)-(Subscript[g, 2]) t-(Subscript[g, 3])]))dt and z=z(ℐ;(Subscript[g, 2]),(Subscript[g, 3])). 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 f(z) is called periodic if there exists a complex constant ρ≠0, such that f(z+ρ)==f(z)/;z∈ℂ. The number ρ (with a minimal possible value of |ρ|) is called the period of the function f(z).

Examples of well-known singly periodic functions are the 
exponential functions, all the trigonometric and hyperbolic functions: (e^z), 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  ρ=2 π i,  ρ=2 π,  ρ=π,  ρ=2 π i, and ρ=π i. The study of such functions can be restricted to any period-strip 
{(Subscript[z, 0])+α ρ/;0≤α<1∧(Subscript[z, 0])∈ℂ}, 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 ρ, (Subscript[ρ, 1]), and (Subscript[ρ, 2]) are periods (basic primitive periods). The condition Im(((Subscript[ρ, 1])/(Subscript[ρ, 2])))≠0 for doubly periodic functions implies the existence of a 
period-parallelogram {(Subscript[z, 0])+(Subscript[α, 1]) (Subscript[ρ, 1])+(Subscript[α, 2])(Subscript[ρ, 2])/;0≤(Subscript[α, 1])<1∧0≤(Subscript[α, 1])<1∧(Subscript[z, 0])∈ℂ}, which is the analog of the period-strip {(Subscript[z, 0])+α ρ/;0≤α<1∧(Subscript[z, 0])∈ℂ} for singly periodic functions with period ρ.

In the case (Subscript[z, 0])==0∧Im(((Subscript[ρ, 1])/(Subscript[ρ, 2])))>0, this parallelogram is called the basic fundamental 
period-parallelogram: (Subscript[Ρ, 0,0])=={(Subscript[α, 1]) (Subscript[ρ, 1])+(Subscript[α, 2])(Subscript[ρ, 2])/;0≤(Subscript[α, 1])<1∧0≤(Subscript[α, 2])<1}. The two line segments {(Subscript[α, i]) (Subscript[ρ, i])/;0≤(Subscript[α, i])<1}/;i==1,2 lying on the boundary of the period-parallelogram and beginning from the 
origin 0 belong to (Subscript[Ρ, 0,0]). The region (Subscript[Ρ, 0,0]) includes only one corner point 0 from four points lying at the boundary of the parallelogram with corners 
in {0, (Subscript[ρ, 1]), (Subscript[ρ, 1])+(Subscript[ρ, 2]),(Subscript[ρ, 2])}.
Sometimes the convention (Subscript[Ρ, 0,0])=={(Subscript[α, 1]) (Subscript[ρ, 1])+(Subscript[α, 2])(Subscript[ρ, 2])/;-1/2≤(Subscript[α, 1])<1/2∧-1/2≤(Subscript[α, 2])<1/2} is used.

The set of all such period-parallelograms:

covers all complex planes: ℂ=={(Subscript[Ρ, m,n])|-∞<m,n<∞∧m∈ℤ∧n∈ℤ}.

Any doubly periodic function ℰ(z) is called an elliptic function. The set of numbers m (Subscript[ρ, 1])+n (Subscript[ρ, 2])/;{m,n}∈ℤ is called the period-lattice for elliptic function ℰ(z).

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

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

Any nonconstant elliptic function ℰ(z) 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 ℰ(z) in a fundamental period-parallelogram P are finite.

The number of the zeros of ℰ(z)-A, where A is any complex number, in a fundamental period-parallelogram (Subscript[Ρ, 0,0]) does not depend on the value A and coincides with number s of the poles (Subscript[b, 1]),(Subscript[b, 2]),…,(Subscript[b, s]) counted according to their multiplicity (s is called the order of the elliptic function ℰ(z)).

The simplest elliptic function has order 2.

Let (Subscript[a, 1]),(Subscript[a, 2]),…,(Subscript[a, r]) (and (Subscript[b, 1]),(Subscript[b, 2]),…,(Subscript[b, s])) be the zeros (and poles) of a nonconstant elliptic function ℰ(z) in a fundamental period-parallelogram (Subscript[Ρ, 0,0]), both listed one or more times according to their multiplicity. Then the 
following hold:

So, the number of zeros of a nonconstant elliptic function ℰ(z) in the fundamental period-parallelogram (Subscript[Ρ, 0,0]) is equal to the number of poles there and counted according to their 
multiplicity. The sum of zeros of a nonconstant elliptic function ℰ(z) in  the fundamental period-parallelogram (Subscript[Ρ, 0,0]) differs from the sum of its poles by a period μ  (Subscript[ρ, 1])+ν (Subscript[ρ, 2]), where μ∈ℤ∧ν∈ℤ and the values of μ, ν depend on the function ℰ(z).

All elliptic functions ℰ(z) satisfy a common fundamental property, which generalizes addition, 
duplication, and multiple angle properties for trigonometric and hyperbolic 
functions (like sin((Subscript[z, 1])+(Subscript[z, 2])), sin(n z)/;n∈(SuperPlus[ℕ])). It can be formulated as the following:

ℰ((∑_(k=1)^n)(Subscript[z, k])).

It can also be expressed as an algebraic function of ℰ((Subscript[z, k]))/;1≤k≤n.

In other words, there exists an irreducible polynomial C((Subscript[t, 1]),(Subscript[t, 2]),…,(Subscript[t, n+1])) in n+1 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 (Subscript[Ρ, 0,0]) have only a double pole with residue zero (e.g., the Weierstrass elliptic 
functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3]))).

(2) Functions that in the period-parallelogram (Subscript[Ρ, 0,0]) have only two simple poles with residues, which are equal in absolute 
value but opposite in sign (e.g., Jacobian elliptic functions sn(z|m) etc.).

Any elliptic function ℰ(z) with periods (Subscript[ρ, 1]) and (Subscript[ρ, 2]) can be expressed as a rational function of the Weierstrassian elliptic 
functions ℘(z;(Subscript[g, 2])((Subscript[ω, 1]),(Subscript[ω, 3])),(Subscript[g, 3])((Subscript[ω, 1]),(Subscript[ω, 3]))) and their derivative ℘'(z;(Subscript[g, 2])((Subscript[ω, 1]),(Subscript[ω, 3])),(Subscript[g, 3])((Subscript[ω, 1]),(Subscript[ω, 3]))) with the same periods (Subscript[ρ, 1])==2(Subscript[ω, 1]),(Subscript[ρ, 2])==2(Subscript[ω, 3]) .

The Weierstrass elliptic function ℘(z;(Subscript[g, 2]),(Subscript[g, 3])) arises as a solution to the following ordinary nonlinear differential 
equation:

Definitions of Weierstrass functions and inverses

The Weierstrass elliptic ℘ function ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), its derivative ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), the Weierstrass sigma function σ(z;(Subscript[g, 2]),(Subscript[g, 3])),  associated Weierstrass sigma functions (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, Weierstrass zeta function ζ(z;(Subscript[g, 2]),(Subscript[g, 3])), inverse elliptic Weierstrass ℘ function (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])), and generalized inverse Weierstrass ℘ function (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) are defined by the following formulas:

The function (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) is the unique value of u for which (Subscript[z, 1])=℘(u;(Subscript[g, 2]),(Subscript[g, 3])) and (Subscript[z, 2])=℘'(u;(Subscript[g, 2]),(Subscript[g, 3])). For the existence of  (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])), the values  (Subscript[z, 1]) and (Subscript[z, 2]) must be related by (Subscript[z, 2]^2)=4 (Subscript[z, 1]^3)-(Subscript[g, 2]) (Subscript[z, 1])-(Subscript[g, 3]).

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 
℘(z;(Subscript[g, 2]),(Subscript[g, 3])) and its derivative ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])) are elliptic functions because only these functions are doubly periodic. 
The other Weierstrass functions σ(z;(Subscript[g, 2]),(Subscript[g, 3])),  (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) are not elliptic functions because they are only quasi-periodic 
functions with respect to z. But historically they are also placed into the class of elliptic 
functions.

The Weierstrass half-periods {(Subscript[ω, 1]),(Subscript[ω, 3])}=={(Subscript[ω, 1])((Subscript[g, 2]),(Subscript[g, 3])),(Subscript[ω, 3])((Subscript[g, 2]),(Subscript[g, 3]))} and the invariants {(Subscript[g, 2]),(Subscript[g, 3])}=={(Subscript[g, 2])((Subscript[ω, 1]),(Subscript[ω, 3])),(Subscript[g, 3])((Subscript[ω, 1]),(Subscript[ω, 3]))}, the Weierstrass ℘ function values at half-periods {(Subscript[e, 1]),(Subscript[e, 2]),(Subscript[e, 3])}=={(Subscript[e, 1])((Subscript[g, 2]),(Subscript[g, 3])),(Subscript[e, 2])((Subscript[g, 2]),(Subscript[g, 3])),(Subscript[e, 3])((Subscript[g, 2]),(Subscript[g, 3]))}, and the Weierstrass zeta function values at half-periods {(Subscript[η, 1]),(Subscript[η, 2]),(Subscript[η, 3])}=={(Subscript[η, 1])((Subscript[g, 2]),(Subscript[g, 3])),(Subscript[η, 2])((Subscript[g, 2]),(Subscript[g, 3])),(Subscript[η, 3])((Subscript[g, 2]),(Subscript[g, 3]))} are defined by the following formulas. The description of the Weierstrass 
functions follows the notations used throughout. The left-hand sides 
indicate that (Subscript[ω, 1]) and (Subscript[ω, 3]) are either independent variables or depend on (Subscript[g, 2]) and (Subscript[g, 3]), or vice versa:

where J(t) is the Klein invariant modular function, ℘(z;(Subscript[g, 2]),(Subscript[g, 3])) is the Weierstrass elliptic ℘ function, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) 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 {(Subscript[ω, 1]),(Subscript[ω, 3])}={1,(1+i)/(Sqrt[2])}.

The next pair of graphics shows the Weierstrass ℘ function over the complex z-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 z-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 z-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 z-plane.

Show[GraphicsArray[

ParametricPlot3D[{Re[s ω1 + t ω3], Im[s ω1 + t 
ω3], #[ WeierstrassSigma[s ω1 + t ω3, {g2, g3}] ],

                  {EdgeForm[]}}, {s, -3.5, 3.5}, {t, -3.5, 3.5}, 

                  PlotPoints -> 72, BoxRatios -> {1, 1, 0.6}, PlotLabel -> #,

                  DisplayFunction -> Identity,

                 PlotRange -> {All, All, {-16, 16}}, AxesLabel -> {Re[z], 
Im[z], None}]& /@

                 {Re, Im}],

                 PlotLabel -> StyleForm[TraditionalForm @

                               WeierstrassSigma[z, {Subscript[g, 2], 
Subscript[g, 3]}]]]

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

The last pair of graphics shows the inverse of the Weierstrass ℘ function over the complex z-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 ℘(z;(Subscript[g, 2]),(Subscript[g, 3])) and its inverse (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) can be represented through the more general hypergeometric Appell (Subscript[F, 1]) function of two variables by the following formulas:

Representations through related equivalent functions

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, ζ(z;(Subscript[g, 2]),(Subscript[g, 3])), (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])), and (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) can be represented through some related equivalent functions, for example, 
through Jacobi functions:

where λ(((Subscript[ω, 3])/(Subscript[ω, 1]))) is modular lambda function, or through theta functions:

or through elliptic integrals and the inverse elliptic nome:

Relations to inverse functions

The Weierstrass function ℘(z;(Subscript[g, 2]),(Subscript[g, 3])) and its derivative ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])) are interconnected with the inverse functions (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) and (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) by the following formulas:

Representations through other Weierstrass functions

Each of the Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) can be expressed through the other Weierstrass functions using numerous 
formulas, for example:

Note that the Weierstrass functions σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3])), ζ(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), and ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])) 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 ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) have the following simple values at the origin point:

Specific values for specialized parameter

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) can be represented through elementary functions, when {(Subscript[g, 2]),(Subscript[g, 3])}=={0,0} or {(Subscript[g, 2]),(Subscript[g, 3])}=={3,1}:

At points z==m (Subscript[ω, 1])+n (Subscript[ω, 3])/;{m,n}∈ℤ, all Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) can be equal to zero or can have poles and be equal to (OverTilde[∞]):

The values of Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) at the points z=r (Subscript[ω, j])/;r∈ℚ can sometimes be evaluated in closed form:

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) have rather simple values, when z==(Subscript[ω, j])==(Subscript[ω, j])((Subscript[g, 2]),(Subscript[g, 3]))/;j∈{1,2,3} and {(Subscript[g, 2]),(Subscript[g, 3])}=={0,1} or {(Subscript[g, 2]),(Subscript[g, 3])}=={1,0}:

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) can be represented through elementary functions, when (Subscript[ω, 3])==(OverTilde[∞]):

Analyticity

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, ζ(z;(Subscript[g, 2]),(Subscript[g, 3])), and (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) are analytical functions of z, (Subscript[g, 2]), and (Subscript[g, 3]), which are defined in (ℂ^3). The inverse Weierstrass function (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) is an analytical function of (Subscript[z, 1]), (Subscript[z, 2]), (Subscript[g, 2]), (Subscript[g, 3]), which is also defined in (ℂ^3), because (Subscript[z, 2]) is not an independent variable.

Poles and essential singularities

For fixed (Subscript[g, 2]),  (Subscript[g, 3]), the Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) have an infinite set of singular points:

(a) z==2m (Subscript[ω, 1])((Subscript[g, 2]),(Subscript[g, 3]))+2n (Subscript[ω, 3])((Subscript[g, 2]),(Subscript[g, 3])),{m,n}∈ℤ are the poles of order 2 with residues 0 (for ℘(z;(Subscript[g, 2]),(Subscript[g, 3]))), of order 3 with residues 0 (for ℘'(z;(Subscript[g, 2]),(Subscript[g, 3]))) and simple poles with residues 1 (for ζ(z;(Subscript[g, 2]),(Subscript[g, 3]))).

(b) z== (OverTilde[∞]) is an essential singular point.

For fixed (Subscript[g, 2]), (Subscript[g, 3]), the Weierstrass functions σ(z;(Subscript[g, 2]),(Subscript[g, 3])) and (Subscript[σ, j])(z;(Subscript[g, 2]),(Subscript[g, 3])),j∈{1,2,3} have only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

The Weierstrass functions (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) and (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) do not have poles and essential singularities with respect to their 
variables.

Branch points and branch cuts

For fixed (Subscript[g, 2]), (Subscript[g, 3]), the Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, j])(z;(Subscript[g, 2]),(Subscript[g, 3])),j∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) do not have branch points and branch cuts.

Periodicity

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])) and ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])) are doubly periodic functions with respect to z with periods 2(Subscript[ω, 1]) and 2(Subscript[ω, 3]):

The Weierstrass functions σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3])),n∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) are quasi-periodic functions with respect to z:

Transformation of half-periods

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, j])(z;(Subscript[g, 2]),(Subscript[g, 3])),j∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) are the invariant functions under the linear transformation of the half
-periods (Subscript[ω, 1])->a (Subscript[ω, 1])+b (Subscript[ω, 3]), (Subscript[ω, 3])->c (Subscript[ω, 1])+d (Subscript[ω, 3]) with integer coefficients a, b, c, and d, satisfying restrictions a d-b c==±1 (modular transformations):

Homogeneity

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, j])(z;(Subscript[g, 2]),(Subscript[g, 3])),j∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) satisfy the following homogeneity type relations:

Parity and symmetry

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, j])(z;(Subscript[g, 2]),(Subscript[g, 3])),j∈{1,2,3}, ζ(z;(Subscript[g, 2]),(Subscript[g, 3])), and (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3]))  have mirror symmetry:

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])) and (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3])),n∈{1,2,3} are even functions with respect to z:

The Weierstrass functions ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) are odd functions with respect to z:

Series representations

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) have the following series expansions at the point z==0:

The inverse Weierstrass function (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) has the following series expansion at the point z==(OverTilde[∞]):

q-series representations

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), log(σ(z;(Subscript[g, 2]),(Subscript[g, 3]))), log((Subscript[σ, j])(z;(Subscript[g, 2]),(Subscript[g, 3]))),j∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3]))  have the following so-called q-series representations:

Other series representations

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, j])(z;(Subscript[g, 2]),(Subscript[g, 3])),j∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) with {(Subscript[g, 2]),(Subscript[g, 3])}=={(Subscript[g, 2])((Subscript[ω, 1]),(Subscript[ω, 3])),(Subscript[g, 3])((Subscript[ω, 1]),(Subscript[ω, 3]))} can be represented through series of different forms, for example:

Integral representations

The Weierstrass functions and their inverses ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), ζ(z;(Subscript[g, 2]),(Subscript[g, 3])), (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])), and (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) can be represented through the following integrals from elementary or 
Weierstrass functions:

Product representations

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and (Subscript[σ, j])(z;(Subscript[g, 2]),(Subscript[g, 3])),j∈{1,2,3} have the following product representations:

q-product representations

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and (Subscript[σ, j])(z;(Subscript[g, 2]),(Subscript[g, 3])),j∈{1,2,3} can be represented as so-called q-products by the following formulas:

Transformations

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) satisfy numerous relations that can provide transformations of its 
arguments. One of these transformations simplifies argument i z to z, 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 ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) 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 ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) with invariants (Subscript[g, 2])(((Subscript[ω, 1])/n),(Subscript[ω, 3]))  and (Subscript[g, 3])(((Subscript[ω, 1])/n),(Subscript[ω, 3])), where n∈(SuperPlus[ℕ]), for example:

Identities

The Weierstrass functions satisfy numerous functional identities, for 
example:

Representations of derivatives

The first two derivatives of all Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])), and their inverses (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) and (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) with respect to variable z can also be expressed through Weierstrass functions:

The first derivatives of Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) with respect to parameter (Subscript[g, 2]) can also be expressed through Weierstrass functions by the following 
formulas:

The first derivatives of Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) with respect to parameter (Subscript[g, 3]) can also be expressed through Weierstrass functions by the following 
formulas:

Weierstrass invariants (Subscript[g, 2]) and (Subscript[g, 3]) can be expressed as functions of half-periods (Subscript[ω, 1]) and (Subscript[ω, 3]). This property allows obtaining the following formulas for the first 
derivatives of Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) with respect to half-period (Subscript[ω, 1]):

Similar formulas take place for the first derivatives of Weierstrass 
functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) with respect to half-period (Subscript[ω, 3]):

The (n^th) derivatives of all Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), (Subscript[σ, n])(z,(Subscript[g, 2]),(Subscript[g, 3]))/;n∈{1,2,3}, ζ(z;(Subscript[g, 2]),(Subscript[g, 3])), and their inverses (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) and (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) with respect to variable z can be represented by the following formulas:

Integration

The indefinite integrals of Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) with respect to variable z 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 ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), ζ(z;(Subscript[g, 2]),(Subscript[g, 3])), and their inverses (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) and (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) satisfy the following nonlinear differential equations:

The Weierstrass functions ℘(z;(Subscript[g, 2]),(Subscript[g, 3])), ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])), σ(z;(Subscript[g, 2]),(Subscript[g, 3])), and ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) 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.

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