JacobiND

JacobiND[u,m]

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.
  • JacobiND is a meromorphic function in both arguments.
  • For certain special arguments, JacobiND automatically evaluates to exact values.
  • JacobiND can be evaluated to arbitrary numerical precision.
  • JacobiND automatically threads over lists.

Examples

open allclose all

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  (27)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiND efficiently at high precision:

JacobiND threads elementwise over lists:

Specific Values  (3)

Simple exact values are generated automatically:

Some poles of JacobiND:

Find a local minimum of JacobiND as a root of (d)/(dx)TemplateBox[{x, {2, /, 3}}, JacobiND]=0:

Visualization  (3)

Plot the JacobiND functions for various parameter values:

Plot JacobiND as a function of its parameter :

Plot the real part of TemplateBox[{{x, +, {ⅈ,  , y}}, {1, /, 2}}, JacobiND]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}, {1, /, 2}}, JacobiND]:

Function Properties  (2)

JacobiND is 2TemplateBox[{m}, EllipticK]-periodic along the real axis:

JacobiND is 4ⅈTemplateBox[{{1, -, m}}, EllipticK]-periodic along the imaginary axis:

JacobiND is an even function:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Integration  (3)

Indefinite integral of JacobiND:

Definite integral of the even function over the interval centered at the origin:

This is twice the integral over half the interval:

More integrals:

Series Expansions  (3)

Taylor expansion for TemplateBox[{x, {1, /, 3}}, JacobiND]:

Plot the first three approximations for TemplateBox[{x, {1, /, 3}}, JacobiND] around :

Taylor expansion for TemplateBox[{1, m}, JacobiND]:

Plot the first three approximations for TemplateBox[{1, m}, JacobiND] around :

JacobiND can be applied to a power series:

Function Identities and Simplifications  (3)

Parity transformations and periodicity relations are automatically applied:

Identity involving JacobiSD:

Argument simplifications:

Function Representations  (3)

Primary definition:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting:

Applications  (4)

Cartesian coordinates of a pendulum:

Plot the timedependence of the coordinates:

Plot the trajectory:

Periodic solution of the nonlinear Schrödinger equation :

Check the solution numerically:

Plot the solution:

Parametrize a lemniscate by arc length:

Show arc length parametrization and classical parametrization:

Zero modes of the periodic supersymmetric partner potentials:

Check the solutions:

Plot the zero modes:

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:

Introduced in 1988
 (1.0)