# JacobiSD

JacobiSD[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.
• JacobiSD is a meromorphic function in both arguments.
• For certain special arguments, JacobiSD automatically evaluates to exact values.
• JacobiSD can be evaluated to arbitrary numerical precision.
• JacobiSD 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(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 JacobiSD efficiently at high precision:

### Specific Values(3)

Simple exact values are generated automatically:

Some poles of JacobiSD:

Find a local maximum of JacobiSD as a root of :

### Visualization(3)

Plot the JacobiSD functions for various values of parameter:

Plot JacobiSD as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(8)

JacobiSD is -periodic along the real axis:

JacobiSD is -periodic along the imaginary axis:

JacobiSD is an odd function in its first argument: is an analytic function for :

It is not in general analytic:

It has both singularities and discontinuities for : is neither nondecreasing nor nonincreasing:

JacobiSD is not injective for any fixed :

It is injective for : is not surjective for :

It is surjective for :

JacobiSD is neither non-negative nor non-positive:

JacobiSD is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

### Integration(3)

Indefinite integral of JacobiSD:

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

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 :

JacobiSD can be applied to a power series:

### Function Identities and Simplifications(3)

Parity transformation and periodicity relations are automatically applied:

Identity involving JacobiCD:

Automatic argument simplifications:

### Function Representations(3)

Representation in terms of Csc and JacobiAmplitude:

Relation to other Jacobi elliptic functions:

## Applications(5)

Conformal map from a rectangle to the unit disk:

Visualize the map:

Generator for the hierarchy of solutions of the nonlinear diffusion equation :

Numerical check of the solutions:

Conformal map from an ellipse to the unit disk:

Visualize the map:

Cartesian coordinates of a pendulum:

Plot the time dependence of the coordinates:

Plot the trajectory:

Parameterization of Costa's minimal surface [MathWorld]:

## 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: