Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

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

Notations:

Specific values:

Differential equations:

Pythagoras-like identities:

Opposite angle identities:

Double angle identities:

Half angle identities:

Angle sum and difference identities:

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 (2024-09-26 02:51:09)*

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

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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