# JacobiCN

JacobiCN[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.
• JacobiCN is a meromorphic function in both arguments.
• For certain special arguments, JacobiCN automatically evaluates to exact values.
• JacobiCN can be evaluated to arbitrary numerical precision.
• JacobiCN 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 expansion at a singular point:

## Scope(32)

### Numerical Evaluation(4)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiCN efficiently at high precision:

### Specific Values(3)

Simple exact values are generated automatically:

Some poles of JacobiCN:

Find a zero of :

### Visualization(3)

Plot the JacobiCN functions for various parameter values:

Plot JacobiCN as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(8)

JacobiCN is -periodic along the real axis:

JacobiCN is -periodic along the imaginary axis:

JacobiCN is an even function in its first argument:

JacobiCN is an analytic function of x:

It has no singularities or discontinuities: is neither nondecreasing nor nonincreasing: is not injective for any fixed : is not surjective for any fixed : is non-negative for positive integer values of :

In general, it is neither non-negative nor non-positive:

JacobiCN is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

### Integration(3)

Indefinite integral of JacobiCN:

Definite integral of an even integrand 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 :

JacobiCN can be applied to power series:

### Function Identities and Simplifications(2)

Parity transformations and periodicity relations are automatically applied:

Identity involving JacobiSN:

### Function Representations(3)

Representation in terms of Cos of JacobiAmplitude:

Relation to other Jacobi elliptic functions:

## Applications(8)

Cnoidal solution of the KdV equation:

Verify the solution:

Plot the solution:

Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

Solution of an anharmonic oscillator :

Plot various solutions:

Elliptic parametrization of an ellipse:

Plot using elliptic parametrization and circular parametrization:

Solution of Nahm equations:

Check that the solutions fulfill the Nahm equations:

Parametrization of a mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):

Plot the inflated balloon:

Use JacobiCN to map a point from a 2D simplex to a 3by3 correlation matrix:

Visualize the determinant over the simplex:

Verify that the determinant is maximal at the centroid of the simplex:

Plot the maximum value of the determinant as a function of the elliptic parameter :

Use the correlation matrix to define a 3D T copula distribution

Draw a sample:

Parameterization of an algebraic lemniscate:

Verify that the parametric function solves the algebraic equation for the lemniscate:

Periodic motion of 3 bodies, where the bodies chase each other on a common orbit with equal timespacing on a lemniscate:

The elliptic parameter is fixed by requiring that the center of mass of the 3-body system remains fixed at the origin:

Visualize the 3body configuration:

A 5-body configuration:

The 5-body configuration admits two elliptic parameters that fix the center of mass:

## Properties & Relations(4)

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the inverse function:

Evaluate as a result of applying Cos to JacobiAmplitude:

Solve a transcendental equation:

Integral:

## Possible Issues(2)

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

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