Jacobi elliptic functions

The Jacobi elliptic functions include

sn (u, k)

,

cn (u, k)

and

dn (u, k)

, with

sn^{2} + cn^{2} = 1

and

dn^{2} + k^{2} sn^{2} = 1

. Their ratios give the other standard members. The Jacobi elliptic sine is the inverse of the elliptic integral of the first kind.

Table of contents
- Jacobi elliptic sine Jacobi elliptic functions
  - Period lattice of the Jacobi elliptic sine Jacobi elliptic sine

Jacobi elliptic sine ( $sn (u, k)$ )

Near zero, the Jacobi elliptic sine is the inverse of the elliptic integral of the first kind:

u = \int_{0}^{z} \frac{d t}{( 1 - t ^{2} ) ( 1 - k ^{2} t ^{2} )}, z = sn (u, k) .

Choose both square roots with value one at zero. The inverse then has

sn (0, k) = 0

and derivative one there. Differentiation yields

(\partial_{u} sn)^{2} = (1 - sn^{2}) (1 - k^{2} sn^{2}) .

For nondegenerate modulus

k \in / {0, 1, - 1}

, continuation gives a meromorphic doubly periodic elliptic function.

The excluded limits are meaningful:

sn (u, 0) = sin u

and

sn (u, 1) = tanh u

. Their inverse integrals and differential equations verify these identities. These elementary limiting functions are not doubly periodic elliptic functions, so the nondegeneracy qualification cannot be dropped.^[ref]

For real

0 < k < 1

, its periods are generated by

4 K (k)

and

2 i K^{'} (k)

, where

K^{'} (k) = K (1 - k^{2})

. The limiting loss of a period is another way to understand degeneration to trigonometric or hyperbolic functions.

Period lattice of the Jacobi elliptic sine

For real

0 < k < 1

, the period lattice of

sn (u, k)

is generated by

4 K (k) and 2 i K^{'} (k) .

The periods arise by composing the monodromy reflections of its inverse elliptic integral of the first kind.

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Jacobi elliptic functions by

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Jacobi elliptic functions are a set of basic elliptic functions that generalize trigonometric functions and are used in many areas of mathematics, including number theory, algebraic geometry, and physics. They are particularly useful in the study of elliptic curves and in solving problems involving periodic phenomena. The Jacobi elliptic functions are defined in terms of a parameter, typically denoted as \(k\) (or \(m\)), which is called the elliptic modulus.

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