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

InverseJacobiDN

InverseJacobiDN[v,m]

gives the inverse Jacobi elliptic function .

Details

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

Examples

open allclose all

Basic Examples (4)

Evaluate numerically:

Plot the real part of the function:

Series expansion at the origin:

Series expansion at Infinity:

Scope (26)

Numerical Evaluation (5)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiDN efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix InverseJacobiDN 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 InverseJacobiDN for various values of the second parameter :

Plot InverseJacobiDN as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (4)

InverseJacobiDN is not an analytic function:

It has both singularities and discontinuities:

is injective:

InverseJacobiDN is neither non-negative nor non-positive:

InverseJacobiDN is neither convex nor concave:

Differentiation (4)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiDN with respect to the second argument :

Higher derivatives:

Series Expansions (3)

Taylor expansion for around and :

Plot the first three approximations for around :

Series expansion for around :

InverseJacobiDN can be applied to a power series:

Function Identities and Simplifications (2)

InverseJacobiDN is the inverse function of JacobiDN:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features (2)

InverseJacobiDN threads elementwise over lists:

TraditionalForm formatting:

Applications (1)

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

Properties & Relations (1)

Obtain InverseJacobiDN from solving equations containing elliptic functions:

Top