Hyperbolic series:

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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]]>