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#1 2023-10-09 03:33:59

lanxiyu
Member
Registered: 2022-05-10
Posts: 52

Jacobi elliptic functions identities

Notations:














Specific values:




























Differential equations:









































Pythagoras-like identities:






Opposite angle identities:










Double angle identities:










Triple angle identities:










Half angle identities:
































Angle sum and difference identities:
















Translation by real one eighth-period:


Translation by imaginary one eighth-period:


Translation by real quarter-period:






Translation by imaginary quarter-period:








Translation by real half-period:









Translation by imaginary half-period:








Jacobi imaginary transformations:










Reciprocal modulus transformations:





Imaginary modulus transformations:





Quadratic transformations:





































Indefinite integrals:






















Theta function representations:






Value of Jacobi amplitude:


Degenerate cases:
































Approximation for small k, k':























Conversion (q = nome, τ = half period ratio, α = modular angle, k = elliptic modulus, k' = complementary elliptic modulus, m = elliptic parameter):









Differential function solution:


Numerical computation of Jacobi elliptic functions
Notation for the theta functions

Last edited by lanxiyu (2024-11-03 16:35:47)

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#2 2024-09-08 22:10:38

lanxiyu
Member
Registered: 2022-05-10
Posts: 52

Re: Jacobi elliptic functions identities

Fourier series (with q = exp(-π K(k')/K(k))):






















Hyperbolic series:












Last edited by lanxiyu (2024-09-13 13:53:44)

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