Built-in Mathematica Symbol

JacobiDN

JacobiDN[u, m]
gives the Jacobi elliptic function

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

, where .

a is doubly periodic function in u with periods and , where is the elliptic integral EllipticK.

JacobiDN is a meromorphic function in both arguments.

For certain special arguments, JacobiDN automatically evaluates to exact values.

JacobiDN can be evaluated to arbitrary numerical precision.

JacobiDN automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

Series expansions about the origin:

Evaluate numerically:

Series expansions about the origin:

Scope (6)

Evaluate for complex arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

JacobiDN threads element-wise over lists:

Simple exact values are generated automatically:

Parity transformations and periodicity relations are automatically applied:

TraditionalForm formatting:

Generalizations & Extensions (1)

JacobiDN can be applied to a power series:

Applications (8)

Cartesian coordinates of a pendulum:

Plot the time dependence of the coordinates:

Plot the trajectory:

Uniformization of a Fermat cubic

:

Check:

Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

Solution of Nahm equations:

Check that the solutions fulfill the Nahm equations:

Periodic solution of the nonlinear Schrödinger equation

:

Plot the solution:

Parametrize a lemniscate by arc length [more info]:

Show arc length parametrization and classical parametrization:

Zero modes of the periodic supersymmetric partner potentials:

Plot the zero modes:

Complex parametrization of a "sphere":

Plot real and imaginary parts:

Properties & Relations (4)

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the inverse function:

Evaluate as a result of applying D to JacobiAmplitude:

Solve a transcendental equation:

Integrals:

Possible Issues (2)

Machine-precision input is insufficient to give the correct answer:

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