WeierstrassPHalfPeriodValues

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Special notations for this file:

Specific values

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

{{(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]))}} 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

Transformation of half-periods

Series representations

q-series

Other series representations

Product representations

Identities

Functional identities

Differentiation

Low-order differentiation

With respect to (Subscript[g, 2])

With respect to (Subscript[g, 3])

Representations through more general functions

Through other functions

Representations through equivalent functions

With related functions

Involving other Weierstrass functions

Involving Weierstrass invariants

Involving theta functions

Involving elliptic integrals and modular functions

History

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

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