# JacobiCS

JacobiCS[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.
• JacobiCS is a meromorphic function in both arguments.
• For certain special arguments, JacobiCS automatically evaluates to exact values.
• JacobiCS can be evaluated to arbitrary numerical precision.
• JacobiCS automatically threads over lists.

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot the function over a subset of the reals:

Plot over a subset of the complexes:

Series expansions at the origin:

Series expansion at a singular point:

## 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 JacobiCS efficiently at high precision:

### Specific Values(3)

Simple exact values are generated automatically:

Some poles of JacobiCS:

Find a zero of :

### Visualization(3)

Plot the JacobiCS functions for various parameter values:

Plot JacobiCS as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(8)

JacobiCS is -periodic along the real axis:

JacobiCS is -periodic along the imaginary axis:

JacobiCS is an odd function in its first argument:

JacobiCS 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 : is not surjective for fixed :

It is surjective for :

JacobiCS is neither non-negative nor non-positive:

JacobiCS is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

### Integration(3)

Indefinite integral of JacobiCS:

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 :

JacobiCS can be applied to a power series:

### Function Identities and Simplifications(3)

Primary definition:

Parity transformation and periodicity relations are automatically applied:

Automatic argument simplifications:

### Function Representations(3)

Representation in terms of Cot of JacobiAmplitude:

Relation to other Jacobi elliptic functions:

## Applications(4)

Hierarchy of solutions of the nonlinear diffusion equation :

Check:

Flow lines in a rectangular region with a current flowing from the lowerright to the upperleft corner:

Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

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 is insufficient to give the correct answer:

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