Built-in Mathematica Symbol

JacobiSN

JacobiSN[u, m]
gives the Jacobi elliptic function

.

MORE INFORMATION

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:

Evaluate numerically:

Series expansions about the origin:

Scope (5)

Evaluate for complex arguments:

Evaluate to high precision:

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

JacobiSN threads element-wise over lists:

Simple exact values are generated automatically:

Parity transformation and periodicity relations are automatically applied:

TraditionalForm formatting:

Generalizations & Extensions (1)

JacobiSN can be applied to power series:

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 Katsura-Fukuda map:

Compare the closed form with explicit iterations:

Plot 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 (6)

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:

Integrals:

Solve the Painlevé-VIII differential 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: