Built-in Mathematica Symbol

JacobiSD

JacobiSD[u, m]
gives the Jacobi elliptic function

.

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

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

Evaluate numerically:

Series expansions:

Evaluate for complex arguments:

Evaluate to high precision:

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

JacobiSD threads element-wise over lists:

Simple exact values are generated automatically:

Parity transformation and periodicity relations are automatically applied:

JacobiSD can be applied to a power series:

Conformal map from a rectangle to the unit disk:

Visualize the map:

Generator for the hierarchy of solutions of the nonlinear diffusion equation

:

Numerical check of the solutions:

Conformal map from an ellipse to the unit disk:

Visualize the map:

Cartesian coordinates of a pendulum:

Plot the time dependence of the coordinates:

Plot the trajectory:

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the inverse function:

Solve a transcendental equation:

Integrals:

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

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