# JacobiND

JacobiND[u,m]

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:

## Scope(33)

### Numerical Evaluation(4)

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:

### 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:

## Applications(4)

Cartesian coordinates of a pendulum:

Plot the timedependence 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(2)

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the inverse function:

Solve a transcendental equation:

## Possible Issues(2)

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

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