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JacobiNC

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

Compute average case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix JacobiNC function using MatrixFunction:

Specific Values (3)

Simple exact values are generated automatically:

Some poles of JacobiNC:

Find a local maximum of JacobiNC as a root of :

Visualization (3)

Plot the JacobiNC functions for various parameter values:

Plot JacobiNC as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (8)

JacobiNC is -periodic along the real axis:

JacobiNC is -periodic along the imaginary axis:

JacobiNC is an even function in its first argument:

is an analytic function of for :

It has both singularities and discontinuities for :

is neither nondecreasing nor nonincreasing:

is not injective for any fixed :

is not surjective for any fixed :

JacobiNC is non-negative nor for :

In general, it has indeterminate sign:

JacobiNC is neither convex nor concave:

Differentiation (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Integration (3)

Indefinite integral of JacobiNC:

Definite integral of JacobiNC:

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 :

JacobiNC can be applied to power series:

Function Identities and Simplifications (3)

Parity transformations and periodicity relations are automatically applied:

Identity involving JacobiSC:

Argument simplifications:

Function Representations (3)

Representation in terms of Sec of JacobiAmplitude:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting:

Applications (5)

Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

Parametrize a lemniscate by arc length:

Show arc length parametrization and classical parametrization:

Solution of an anharmonic oscillator :

Plot various solutions:

Solution of the field theory wave equation :

Plot a solution:

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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