WeierstrassHalfPeriods

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

General characteristics

Domain and analyticity

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

Symmetries and periodicities

Mirror symmetry

Series representations

Generalized power series

Integral representations

On the real axis

Of the direct function

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Differentiation

Low-order differentiation

With respect to (Subscript[g, 2])

With respect to (Subscript[g, 3])

Representations through equivalent functions

With inverse function

With related functions

Involving elliptic integrals

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

– C. G. J. Jacobi (1835)
– K. Weierstrass (1862)
– A. Hurwitz (1905)

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