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lanxiyu

Member

Registered: 2022-05-10

Posts: 64

General form of Dixon elliptic functions

Specific values:

With Λ=nλ(α)+n'λ'(α)+n''λ''(α):

Differential equations:

Indefinite integrals:

Definite integrals:

Identities:

Hypergeometric series representation:

Limit on branch cut:

Opposite angle formula:

Periodicity: with Λ=nλ(α)+n'λ'(α)+n''λ''(α):

Translation by one-third periods:

Addition theorems:

Double angle formulas:

Transformations:

Legendre's relation:

Weierstrass elliptic function representations:

Dedekind eta function representations:

Equianharmonic (α=0) case:

Degenerate cases:

See On the doubly periodic functions arising out of the curve x³+y³-3αxy=1, Cubic theta functions, Cubics in Desmos

Last edited by lanxiyu (2025-06-25 15:53:32)

lanxiyu

Member

Registered: 2022-05-10

Posts: 64

Re: General form of Dixon elliptic functions

Specific values for equianharmonic (α=0) case:

Last edited by lanxiyu (2025-05-30 20:33:38)