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#1 2023-10-12 17:02:44

lanxiyu
Member
Registered: 2022-05-10
Posts: 53

Weierstrass functions

Definitions:

Conditions:

Parity:

Periodicity:

Homogeneity:

Transformation of half-periods equation?tex=%5Cleft(%7B%5Crm%20if%7D%5C%20%5Cleft(m_1%2Cm_2%2Cm_3%2Cn_1%2Cn_2%2Cn_3%5Cright)%5Cin%5Cmathbb%7BZ%7D%5E6%5C%20%7B%5Crm%20and%7D%5C%20%5Cleft(m_1n_2-m_2n_1%2Bm_2n_3-m_3n_2%2Bm_3n_1-m_1n_3%5Cright)%5E2%3D1%5Cright):

Power series representations:

Recurrence relations of Eisenstein series:

Differential equations:

Integrals:

equation?tex=%5Cint%5Cfrac%7B%5Cmathrm%7Bd%7Dz%7D%7B%5Cleft(%5Cwp%5Cleft(z%5Cright)%2Ba%5Cright)%5En%7D%3D%5Cfrac%7B1%7D%7B%5Cleft(n-1%5Cright)%5Cleft(4a%5E3-g_2a%2Bg_3%5Cright)%7D%5Cleft(%5Cfrac%7B%5Cwp%27%5Cleft(z%5Cright)%7D%7B%5Cleft(%5Cwp%5Cleft(z%5Cright)%2Ba%5Cright)%5E%7Bn-1%7D%7D%2B(2n-3)%5Cleft(6a%5E2-%5Cfrac%7B1%7D%7B2%7Dg_2%5Cright)%5Cint%5Cfrac%7B%5Cmathrm%7Bd%7Dz%7D%7B%5Cleft(%5Cwp%5Cleft(z%5Cright)%2Ba%5Cright)%5E%7Bn-1%7D%7D-6(2n-4)a%5Cint%5Cfrac%7B%5Cmathrm%7Bd%7Dz%7D%7B%5Cleft(%5Cwp%5Cleft(z%5Cright)%2Ba%5Cright)%5E%7Bn-2%7D%7D%2B2(2n-5)%5Cint%5Cfrac%7B%5Cmathrm%7Bd%7Dz%7D%7B%5Cleft(%5Cwp%5Cleft(z%5Cright)%2Ba%5Cright)%5E%7Bn-3%7D%7D%5Cright)

equation?tex=%5Cint%5Cfrac%7B%5Cmathrm%7Bd%7Dz%7D%7B%5Cleft(%5Cwp%5Cleft(z%5Cright)-e_j%5Cright)%5En%7D%3D-%5Cfrac%7B1%7D%7B%5Cleft(2n-1%5Cright)%5Cwp%27%27%5Cleft(%5Comega_j%5Cright)%7D%5Cleft(%5Cfrac%7B%5Cwp%27%5Cleft(z%5Cright)%7D%7B%5Cleft(%5Cwp%5Cleft(z%5Cright)-e_j%5Cright)%5En%7D%2B12(n-1)e_j%5Cint%5Cfrac%7B%5Cmathrm%7Bd%7Dz%7D%7B%5Cleft(%5Cwp%5Cleft(z%5Cright)-e_j%5Cright)%5E%7Bn-1%7D%7D%2B2(2n-3)%5Cint%5Cfrac%7B%5Cmathrm%7Bd%7Dz%7D%7B%5Cleft(%5Cwp%5Cleft(z%5Cright)-e_j%5Cright)%5E%7Bn-2%7D%7D%5Cright)

For definite integral, see https://www.mathisfunforum.com/viewtopic.php?id=30259

Values at lattice points:

Half period values:

One-third period values:

Quarter period values:

Translation by half-periods:

Translation by quarter-periods:

Addition theorems:

equation?tex=%5Cbegin%7Balign%7D%26%5Csigma%5Cleft(z_1%2Bz_2%5Cright)%5Csigma%5Cleft(z_1-z_2%5Cright)%5Csigma%5Cleft(z_3%2Bz_4%5Cright)%5Csigma%5Cleft(z_3-z_4%5Cright)%5C%5C%2B%26%5Csigma%5Cleft(z_1%2Bz_3%5Cright)%5Csigma%5Cleft(z_1-z_3%5Cright)%5Csigma%5Cleft(z_4%2Bz_2%5Cright)%5Csigma%5Cleft(z_4-z_2%5Cright)%5C%5C%2B%26%5Csigma%5Cleft(z_1%2Bz_4%5Cright)%5Csigma%5Cleft(z_1-z_4%5Cright)%5Csigma%5Cleft(z_2%2Bz_3%5Cright)%5Csigma%5Cleft(z_2-z_3%5Cright)%5C%5C%3D%260%5Cend%7Balign%7D

Triple addition formulas:
equation?tex=%5Cbegin%7Balign%7D%26%5Csigma_j%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma%5Cleft(z_1%5Cright)%5Csigma_k%5Cleft(z_3%5Cright)%5Csigma_%5Cell%5Cleft(z_2%5Cright)%5C%5C%2B%26%5Csigma_k%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma%5Cleft(z_2%5Cright)%5Csigma_%5Cell%5Cleft(z_1%5Cright)%5Csigma_j%5Cleft(z_3%5Cright)%5C%5C%2B%26%5Csigma_%5Cell%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma%5Cleft(z_3%5Cright)%5Csigma_j%5Cleft(z_2%5Cright)%5Csigma_k%5Cleft(z_1%5Cright)%5C%5C%3D%26%5Csigma%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma_j%5Cleft(z_1%5Cright)%5Csigma_k%5Cleft(z_2%5Cright)%5Csigma_%5Cell%5Cleft(z_3%5Cright)%5Cend%7Balign%7D

equation?tex=%5Cbegin%7Balign%7D%26%5Csigma%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma%5Cleft(z_1%5Cright)%5Csigma%5Cleft(z_2%5Cright)%5Csigma%5Cleft(z_3%5Cright)%2F2%5C%5C%2B%26%5Csigma_j%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma_j%5Cleft(z_1%5Cright)%5Csigma_j%5Cleft(z_2%5Cright)%5Csigma_j%5Cleft(z_3%5Cright)%2F%5Cwp%27%27%5Cleft(%5Comega_j%5Cright)%5C%5C%2B%26%5Csigma_k%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma_k%5Cleft(z_1%5Cright)%5Csigma_k%5Cleft(z_2%5Cright)%5Csigma_k%5Cleft(z_3%5Cright)%2F%5Cwp%27%27%5Cleft(%5Comega_k%5Cright)%5C%5C%2B%26%5Csigma_%5Cell%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma_%5Cell%5Cleft(z_1%5Cright)%5Csigma_%5Cell%5Cleft(z_2%5Cright)%5Csigma_%5Cell%5Cleft(z_3%5Cright)%2F%5Cwp%27%27%5Cleft(%5Comega_%5Cell%5Cright)%5C%5C%3D%260%5Cend%7Balign%7D

equation?tex=%5Cbegin%7Balign%7D%26%5Cleft(e_k-e_%5Cell%5Cright)%5Csigma%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma%5Cleft(z_1%5Cright)%5Csigma_j%5Cleft(z_2%5Cright)%5Csigma_j%5Cleft(z_3%5Cright)%5C%5C%2B%26%5Cleft(e_k-e_%5Cell%5Cright)%5Csigma_j%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma_j%5Cleft(z_1%5Cright)%5Csigma%5Cleft(z_2%5Cright)%5Csigma%5Cleft(z_3%5Cright)%5C%5C%3D%26%5Csigma_k%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma_k%5Cleft(z_1%5Cright)%5Csigma_%5Cell%5Cleft(z_2%5Cright)%5Csigma_%5Cell%5Cleft(z_3%5Cright)%5C%5C-%26%5Csigma_%5Cell%5Cleft(z_1%2Bz_2%2Bz_3%5Cright)%5Csigma_%5Cell%5Cleft(z_1%5Cright)%5Csigma_k%5Cleft(z_2%5Cright)%5Csigma_k%5Cleft(z_3%5Cright)%5Cend%7Balign%7D

Double angle formulas:

Triple angle formulas:

Multiple angle formulas:

Half-angle formulas:

Quadratic transformations:

Relations involving squares:

Relations involving cubes:

Relations involving quartic powers:

Relations involving roots:

Sigma function representations:

Inverse functions:

Relations to Carlson elliptic integrals:

(see https://fungrim.org/topic/Carlson_symme … integrals/)

Relations to Jacobi elliptic functions:

Legendre's relation:

Lemniscatic cases:

Equianharmonic cases:

Degenerate cases:

Last edited by lanxiyu (2024-11-24 15:33:16)

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