# JacobiDC

JacobiDC[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.
• JacobiDC is a meromorphic function in both arguments.
• For certain special arguments, JacobiDC automatically evaluates to exact values.
• JacobiDC can be evaluated to arbitrary numerical precision.
• JacobiDC 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 at the origin:

## Scope(34)

### Numerical Evaluation(4)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiDC efficiently at high precision:

### Specific Values(3)

Simple exact answers are generated automatically:

Some poles of JacobiDC:

Find a local maximum of JacobiDC as a root of :

### Visualization(3)

Plot the JacobiDC functions for various parameter values:

Plot JacobiDC as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(8)

JacobiDC is -periodic along the real axis:

JacobiDC is -periodic along the imaginary axis:

JacobiDC is an even function: is an analytic function of for :

It is not, in general, analytic:

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

JacobiDC is neither non-negative nor non-positive:

JacobiDC is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

### Integration(3)

Indefinite integral of JacobiDC:

Definite integral of JacobiDC:

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 :

JacobiDC can be applied to a power series:

### Function Identities and Simplifications(4)

Primary definition:

Identity involving JacobiNC:

Parity transformations and periodicity relations are automatically applied:

Automatic argument simplifications:

### Function Representations(3)

Representation in terms of trigonometric functions and JacobiAmplitude:

Relation to other Jacobi elliptic functions:

## Applications(2)

Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

Solution of the PoissonBoltzmann equation :

Check solution using series expansion:

## 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: