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JacobiNS

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

Compute average case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix JacobiNS function using MatrixFunction:

Specific Values (3)

Simple exact values are generated automatically:

Some poles of JacobiNS:

Find a local minimum of JacobiNS as a root of :

Visualization (3)

Plot the JacobiNS functions for various parameter values:

Plot JacobiNS as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (8)

JacobiNS is -periodic along the real axis:

JacobiNS is -periodic along the imaginary axis:

JacobiNS is an odd function in its first argument:

JacobiNS is not an analytic function:

It has both singularities and discontinuities:

is neither nondecreasing nor nonincreasing:

is not injective for any fixed :

It is injective for :

JacobiNS is not surjective for any fixed :

JacobiNS is neither non-negative nor non-positive:

JacobiNS is neither convex nor concave:

Differentiation (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Integration (3)

Indefinite integral of JacobiNS:

Definite integral of an odd function over the interval centered at the origin is 0:

More integrals:

Series Expansions (3)

Series expansion for :

Plot the first three approximations for around :

Taylor expansion for :

Plot the first three approximations for around :

JacobiNS can be applied to power series:

Function Identities and Simplifications (3)

Parity transformation and periodicity relations are automatically applied:

Identity involving JacobiCS:

Argument simplifications:

Function Representations (3)

Representation in terms of Csc of JacobiAmplitude:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting:

Applications (5)

Map a rectangle conformally onto the lower half‐plane:

Solution of the pendulum equation:

Check the solution:

Plot solutions:

Closed form of iterates of the Katsura–Fukuda map:

Compare the closed form with explicit iterations:

Plot a few hundred iterates:

Solution of the sinh‐Gordon equation :

Check the solution:

Plot the solution:

Hierarchy of solutions of the nonlinear diffusion equation :

Verify these functions:

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 may be insufficient to give the correct answer:

Currently only simple simplification rules are built in for Jacobi functions:

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