Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how

InverseJacobiCD

InverseJacobiCD[v,m]

gives the inverse Jacobi elliptic function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
gives the value of for which .
InverseJacobiCD has branch cut discontinuities in the complex v plane with branch points at and infinity, and in the complex m plane with branch points at and infinity.
The inverse Jacobi elliptic functions are related to elliptic integrals.
For certain special arguments, InverseJacobiCD automatically evaluates to exact values.
InverseJacobiCD can be evaluated to arbitrary numerical precision.
InverseJacobiCD automatically threads over lists.

Examples

open allclose all

Basic Examples (5)

Evaluate numerically:

Plot the function over a subset of the reals:

Plot over a subset of the complexes:

Series expansions at the origin:

Series expansion at Infinity:

Scope (28)

Numerical Evaluation (6)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiCD efficiently at high precision:

InverseJacobiCD threads elementwise over lists:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix InverseJacobiCD function using MatrixFunction:

Specific Values (3)

Simple exact results are generated automatically:

Value at infinity:

Find a real root of the equation :

Visualization (3)

Plot InverseJacobiCD for various values of the second parameter :

Plot InverseJacobiCD as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (6)

InverseJacobiCD is not an analytic function:

It has both singularities and discontinuities:

is nonincreasing on its real domain:

is injective:

is not surjective:

is non-negative on its real domain:

InverseJacobiCD is neither convex nor concave on its real domain:

Differentiation (4)

First derivative:

Higher derivatives:

Differentiate InverseJacobiCD with respect to the second argument :

Higher derivatives:

Series Expansions (2)

Taylor expansion for :

Plot the first three approximations for around :

Taylor expansion for :

Plot the first three approximations for around :

Function Identities and Simplifications (2)

InverseJacobiCD is the inverse function of JacobiCD:

Compose with the inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features (2)

InverseJacobiCD can be applied to a power series:

TraditionalForm formatting:

Generalizations & Extensions (1)

InverseJacobiCD can be applied to a power series:

Applications (1)

Plot contours of constant real and imaginary parts in the complex plane:

Properties & Relations (1)

Obtain InverseJacobiCD from solving equations containing elliptic functions:

Top