# JacobiNS

JacobiNS[u,m]

gives the Jacobi elliptic function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• , where .
• is a doubly periodic function in u with periods and , where is the elliptic integral EllipticK.
• JacobiNS is a meromorphic function in both arguments.
• For certain special arguments, JacobiNS automatically evaluates to exact values.
• JacobiNS can be evaluated to arbitrary numerical precision.
• JacobiNS 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 JacobiNS efficiently at high precision:

### Specific Values(3)

Simple exact values are generated automatically:

Some poles of JacobiNS:

Find a local minimum of JacobiNS as a root of :

### Visualization(3)

Plot the JacobiNS functions for various parameter values:

Plot JacobiNS as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(8)

JacobiNS is -periodic along the real axis:

JacobiNS is -periodic along the imaginary axis:

JacobiNS is an odd function in its first argument:

JacobiNS is not an analytic function:

It has both singularities and discontinuities: is neither nondecreasing nor nonincreasing: is not injective for any fixed :

It is injective for :

JacobiNS is not surjective for any fixed :

JacobiNS is neither non-negative nor non-positive:

JacobiNS is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

### Integration(3)

Indefinite integral of JacobiNS:

Definite integral of an odd function over the interval centered at the origin is 0:

More integrals:

### Series Expansions(3)

Series expansion for :

Plot the first three approximations for around :

Taylor expansion for :

Plot the first three approximations for around :

JacobiNS can be applied to power series:

### Function Identities and Simplifications(3)

Parity transformation and periodicity relations are automatically applied:

Identity involving JacobiCS:

Argument simplifications:

### Function Representations(3)

Representation in terms of Csc of JacobiAmplitude:

Relation to other Jacobi elliptic functions:

## Applications(4)

Map a rectangle conformally onto the lower halfplane:

Solution of the pendulum equation:

Check the solution:

Plot solutions:

Closed form of iterates of the KatsuraFukuda map:

Compare the closed form with explicit iterations:

Plot a few hundred iterates:

Solution of the sinhGordon equation :

Check the solution:

Plot the solution:

## 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 may be insufficient to give the correct answer:

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