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JacobiND

gives the Jacobi elliptic function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
, where .
is a doubly periodic function in with periods and , where is the elliptic integral EllipticK.
JacobiND is a meromorphic function in both arguments.
For certain special arguments, JacobiND automatically evaluates to exact values.
JacobiND can be evaluated to arbitrary numerical precision.
JacobiND automatically threads over lists.

Examples

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Basic Examples (4)

Evaluate numerically:

Plot the function over a subset of the reals:

Plot over a subset of the complexes:

Series expansions about the origin:

Scope (34)

Numerical Evaluation (5)

Evaluate numerically to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate JacobiND efficiently at high precision:

Compute average case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix JacobiND function using MatrixFunction:

Specific Values (3)

Simple exact values are generated automatically:

Some poles of JacobiND:

Find a local minimum of JacobiND as a root of :

Visualization (3)

Plot the JacobiND functions for various parameter values:

Plot JacobiND as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (8)

JacobiND is -periodic along the real axis:

JacobiND is -periodic along the imaginary axis:

JacobiND is an even function in its first argument:

is an analytic function of for :

It is not, in general, an analytic function:

It has both singularities and discontinuities:

is neither nondecreasing nor nonincreasing:

is not injective for :

is not surjective for any fixed :

is non-negative for :

In general, it has indeterminate sign:

JacobiND is neither convex nor concave:

Differentiation (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Integration (3)

Indefinite integral of JacobiND:

Definite integral of the even function over the interval centered at the origin:

This is twice the integral over half the interval:

More integrals:

Series Expansions (3)

Taylor expansion for :

Plot the first three approximations for around :

Taylor expansion for :

Plot the first three approximations for around :

JacobiND can be applied to a power series:

Function Identities and Simplifications (3)

Parity transformations and periodicity relations are automatically applied:

Identity involving JacobiSD:

Argument simplifications:

Function Representations (3)

Primary definition:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting:

Applications (4)

Cartesian coordinates of a pendulum:

Plot the time‐dependence of the coordinates:

Plot the trajectory:

Periodic solution of the nonlinear Schrödinger equation :

Check the solution numerically:

Plot the solution:

Parametrize a lemniscate by arc length:

Show arc length parametrization and classical parametrization:

Zero modes of the periodic supersymmetric partner potentials:

Check the solutions:

Plot the zero modes:

Properties & Relations (3)

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the inverse function:

Solve a transcendental equation:

JacobiND can be represented with related elliptic functions:

Possible Issues (2)

Machine-precision input is insufficient to give the correct answer:

Currently only simple simplification rules are built in for Jacobi functions:

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