JacobiSN

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

Examples

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Basic Examples  (3)

Evaluate numerically:

Series expansions about the origin:

Scope  (27)

Numerical Evaluation  (4)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiSN efficiently at high precision:

JacobiSN threads elementwise over lists:

Specific Values  (3)

Simple exact values are generated automatically:

Some poles of JacobiSN:

Find a zero of TemplateBox[{x, {1, /, 3}}, JacobiSN]:

Visualization  (3)

Plot the JacobiSN functions for various values of parameter:

Plot JacobiSN as a function of its parameter :

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

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

Function Properties  (2)

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

JacobiSN is 2ⅈTemplateBox[{{1, -, m}}, EllipticK]-periodic along the imaginary axis:

JacobiSN is an odd function:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Integration  (3)

Indefinite integral of JacobiSN:

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

More integrals:

Series Expansions  (3)

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

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

Taylor expansion for TemplateBox[{2, m}, JacobiSN]:

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

JacobiSN can be applied to power series:

Function Identities and Simplifications  (3)

Parity transformation and periodicity relations are automatically applied:

Automatic argument simplifications:

Identity involving JacobiCN:

Function Representations  (3)

Representation in terms of Sin of JacobiAmplitude:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting:

Applications  (7)

Map a rectangle conformally onto the upper half-plane:

Solution of the pendulum equation:

Check the solution:

Plot solutions:

Cnoidal solution of the KdV equation:

A numerical check of the solution:

Plot the solution:

Closed form of iterates of the KatsuraFukuda map:

Compare the closed form with explicit iterations:

Plot a few hundred iterates:

Implicitly defined periodic maximal surface in Minkowski space:

Calculate partial derivatives:

Check numerically the equation for a maximal surface:

Plot the maximal surface in Euclidean space:

Solution of the Euler top equations for :

Check the solutions numerically:

Plot the solutions:

Define a compacton solution of the nonlinear differential equation :

Verify the solution:

Plot the compacton:

Properties & Relations  (5)

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the inverse function:

Evaluate as a result of applying Sin to JacobiAmplitude:

Solve a transcendental equation:

Numerically find a root of a transcendental equation:

Solve the Painlevé-VIII differential 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:

Introduced in 1988
 (1.0)