Built-in Mathematica Symbol

JacobiCS

JacobiCS[u, m]
gives the Jacobi elliptic function

.

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

For certain special arguments, JacobiCS 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:

JacobiCS threads element-wise over lists:

Simple exact results are generated automatically:

Parity transformation and periodicity relations are automatically applied:

JacobiCS can be applied to a power series:

Hierarchy of solutions of the nonlinear diffusion equation

:

Check:

Flow lines in a rectangular region with a current flowing from the lower-right to the upper-left corner:

Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

Solution of the sinh-Gordon equation

:

Check the solution:

Plot the solution:

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: