JacobiNC

JacobiNC[u,m]

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  (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 JacobiNC efficiently at high precision:

JacobiNC threads elementwise over lists:

Specific Values  (3)

Simple exact values are generated automatically:

Some poles of JacobiNC:

Find a local maximum of JacobiNC as a root of (d)/(dx)TemplateBox[{x, {1, /, 3}}, JacobiNC]=0:

Visualization  (3)

Plot the JacobiNC functions for various parameter values:

Plot JacobiNC as a function of its parameter :

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

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

Function Properties  (2)

JacobiNC is 4TemplateBox[{m}, EllipticK]-periodic along the real axis:

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

JacobiNC is an even function:

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 TemplateBox[{x, {1, /, 3}}, JacobiNC]:

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

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

Plot the first three approximations for TemplateBox[{1, m}, JacobiNC] 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  (4)

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:

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)