WeierstrassP

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Special notations for this file:

Specific values

Specialized values

For fixed z

For fixed {(Subscript[g, 2]),(Subscript[g, 3])}

Values at fixed points

Equianharmonic case {(Subscript[g, 2]),(Subscript[g, 3])}=={0,1}

Lemniscatic case {(Subscript[g, 2]),(Subscript[g, 3])}=={1,0}

Values at infinities

General characteristics

Domain and analyticity

℘(z;(Subscript[g, 2]),(Subscript[g, 3])) is an analytical function  of z, (Subscript[g, 2]), and (Subscript[g, 3]), which is defined in (ℂ^3).

Symmetries and periodicities

Parity

℘(z;(Subscript[g, 2]),(Subscript[g, 3])) is an even function with respect to z.

Mirror symmetry

Periodicity

℘(z;(Subscript[g, 2]),(Subscript[g, 3])) is a doubly periodic function with respect to z with periods 2(Subscript[ω, 1]) and 2(Subscript[ω, 3]).

Transformation of half-periods

Homogeneity

Poles and essential singularities

With respect to z

For fixed (Subscript[g, 2]), (Subscript[g, 3]), the function ℘(z;(Subscript[g, 2]),(Subscript[g, 3])) has 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 two with residues 0;
b)  z== (OverTilde[∞]) is an essential singular point.

Branch points

With respect to z

For fixed (Subscript[g, 2]), (Subscript[g, 3]), the function ℘(z;(Subscript[g, 2]),(Subscript[g, 3])) does not have branch points.

Branch cuts

With respect to z

For fixed (Subscript[g, 2]), (Subscript[g, 3]), the function ℘(z;(Subscript[g, 2]),(Subscript[g, 3])) does not have branch cuts.

Series representations

Generalized power series

Expansions at z==0

q-series

Other series representations

Integral representations

On the real axis

Of the direct function

Involving related functions

Product representations

Differential equations

Ordinary nonlinear differential equations

Partial differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Translation by half-periods

Half-angle formulas

Multiple arguments

Argument involving numeric multiples of variable

Argument involving symbolic multiples of variable

Multiple angle formulas:

Determinants involving derivatives

Related transformations

Halving half-period

Third of half-period

General fractions of half-periods

Other constructions

Identities

Functional identities

Expressions involving  a+b+c==0 mod(2(Subscript[ω, 1]),2(Subscript[ω, 3]))

Generalization for arbitrary a, b, c, d

Differentiation

Low-order differentiation

With respect to z

With respect to (Subscript[g, 2])

With respect to (Subscript[g, 3])

With respect to (Subscript[ω, 1])

With respect to (Subscript[ω, 3])

Symbolic differentiation

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving functions of the direct function

Summation

Finite summation

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With inverse function

With related functions

Involving other Weierstrass functions

Involving Jacobi functions

Involving theta functions

Involving other related functions

Theorems

Rational representation of double periodic functions

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]) .

Solution of Korteweg-de Vries equation

The elongation of a water wave in a shallow channel is described by the Korteweg-de Vries equation

((∂w(z,t))/(∂t))==3w(z,t)((∂w(z,t))/(∂z))-(1/2)(((∂^3)w(z,t))/(∂(z^3))).

A solution to this nonlinear partial differential equation is given by

w(z,t)=2(Subscript[c, 1])℘((Subscript[c, 1])(z-(Subscript[c, 2])t)+ω'((Subscript[g, 2]),(Subscript[g, 3]));(Subscript[g, 2]),(Subscript[g, 3]))-((Subscript[c, 2])/3)/;ω((Subscript[g, 2]),(Subscript[g, 3]))>0,i ω'((Subscript[g, 2]),(Subscript[g, 3]))>0.

Algebraic independence of some Weierstrass functions

If the algebraic numbers (Subscript[z, 1]) , (Subscript[z, 2]), (Subscript[z, 3]) are linearly independent over ℚ and (Subscript[g, 2]), (Subscript[g, 3]) are algebraic numbers, then two of the three numbers ℘((Subscript[z, 1]);(Subscript[g, 2]),(Subscript[g, 3])),  ℘((Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])), ℘((Subscript[z, 3]);(Subscript[g, 2]),(Subscript[g, 3])) are algebraically independent over ℚ.

The Picard solution of a special case of the Painlevé VI- equation

The Picard solution of a special case of the Painlevé VI– equation,

w''(z)==(1/2) ((1/(w(z)-z))+(1/(w(z)))+(1/(w(z)-1))) ((w'(z))^2)-((1/(w(z)-z))+(1/z)+(1/(z-1))) w'(z)+((w(z) (w(z)-1) (w(z)-z))/(2 ((z^2) (((z-1))^2)))) (((z-1) z)/(((w(z)-z))^2)),

is given by

w(z)==℘((Subscript[c, 1]) K(z)+(Subscript[c, 2]) K(1-z),{(Subscript[ω, 1])(z),(Subscript[ω, 2])(z)})+((z+1)/3)/;{(Subscript[ω, 1])(z),(Subscript[ω, 2])(z)}={ω( K(z), K(1-z)),ω'( K(z), K(1-z))}.

Uniformization of elliptic curves

The general elliptic curve (y^2)==f(x)==(Subscript[a, 0])(x^4)+4(Subscript[a, 1])(x^3)+6(Subscript[a, 2])(x^2)+4(Subscript[a, 3])x+(Subscript[a, 4]) can be uniformized (meaning x and y can both be parameterized as a single-valued analytic function) through

x(z)==(Subscript[x, 0])+((f'((Subscript[x, 0])))/(4 (℘(z;(Subscript[g, 2]),(Subscript[g, 3]))-(1/24) f''((Subscript[x, 0])))))
y(z)==-((f'((Subscript[x, 0])) ℘'(z;(Subscript[g, 2]),(Subscript[g, 3])))/(4 (((℘(z;(Subscript[g, 2]),(Subscript[g, 3]))-(1/24) f''((Subscript[x, 0]))))^2))).

Here (Subscript[x, 0]) is any zero of f(x), and (Subscript[g, 2])=(Subscript[a, 0])(Subscript[a, 4])-4(Subscript[a, 1])(Subscript[a, 3])+3 (Subscript[a, 2]^2), (Subscript[g, 3])=(Subscript[a, 0])(Subscript[a, 2])(Subscript[a, 4])+2(Subscript[a, 1])(Subscript[a, 2])(Subscript[a, 3])- (Subscript[a, 2]^3)- (Subscript[a, 0])(Subscript[a, 3]^2)- (Subscript[a, 1]^2)(Subscript[a, 4]).

History

– N. H. Abel (1827)
– K. Weierstrass (1855, 1862)
– C. Hermite (1849) first used the notation ℘

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