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JacobiDC

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 (35)

Numerical Evaluation (5)

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:

Compute average case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix JacobiDC function using MatrixFunction:

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:

TraditionalForm formatting:

Applications (2)

Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

Solution of the Poisson–Boltzmann 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:

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