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JacobiSD

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 (34)

Numerical Evaluation (5)

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:

Compute average case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix JacobiSD function using MatrixFunction:

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:

TraditionalForm formatting:

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:

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