WeierstrassInvariants

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at infinities

General characteristics

Domain and analyticity

{(Subscript[g, 2])((Subscript[ω, 1]),(Subscript[ω, 3])),(Subscript[g, 3])((Subscript[ω, 1]),(Subscript[ω, 3]))} is an vector-valued function of (Subscript[ω, 1]) and (Subscript[ω, 3]) that is analytic in each component and it is defined over (ℂ^2)(for (Subscript[ω, 1])≠a (Subscript[ω, 3]),a∈ℝ).

Symmetries and periodicities

Mirror symmetry

Permutation symmetry

Periodicity

{(Subscript[g, 2])(1,(Subscript[ω, 3])),(Subscript[g, 3])(1,(Subscript[ω, 3]))} is a periodic vector-valued function with period 1 .

Transformation of half-periods

Homogeneity

Series representations

Generalized power series

q-series

Other series representations

Integral representations

On the real axis

Of the direct function

Differential equations

Partial differential equations

Transformations

Transformations and argument simplifications

Identities

Functional identities

Differentiation

Low-order differentiation

With respect to (Subscript[ω, 1])

With respect to (Subscript[ω, 3])

Symbolic differentiation

With respect to (Subscript[ω, 1])

With respect to (Subscript[ω, 3])

Fractional integro-differentiation

With respect to (Subscript[ω, 1])

With respect to (Subscript[ω, 3])

Integration

Indefinite integration

Involving only one direct function with respect to (Subscript[ω, 1])

Representations through equivalent functions

With inverse function

With related functions

Involving Weierstrass functions

Involving theta functions

Involving elliptic integrals and modular functions

Theorems

Transcendentality of one expression with Weierstrass functions

Any (Subscript[α, 1]) (Subscript[ω, 1])((Subscript[g, 2]),(Subscript[g, 3]))+(Subscript[α, 2]) (Subscript[ω, 3])((Subscript[g, 2]),(Subscript[g, 3]))+(Subscript[α, 3])ζ((Subscript[ω, 1])((Subscript[g, 2]),(Subscript[g, 3]));(Subscript[g, 2]),(Subscript[g, 3]))+(Subscript[α, 4])ζ((Subscript[ω, 3])((Subscript[g, 2]),(Subscript[g, 3]));(Subscript[g, 2]),(Subscript[g, 3]))+(Subscript[α, 5])2π i≠0 , where (Subscript[α, 1]), (Subscript[α, 2]), (Subscript[α, 3]), (Subscript[α, 4]), (Subscript[α, 5]) are algebraic numbers, is transcendental.

History

- A. Cayley
– G. Boole (1845)

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