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InverseJacobiND

InverseJacobiND[v,m]

gives the inverse Jacobi elliptic function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
gives the value of for which .
InverseJacobiND 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, InverseJacobiND automatically evaluates to exact values.
InverseJacobiND can be evaluated to arbitrary numerical precision.
InverseJacobiND automatically threads over lists.

Examples

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Basic Examples (5)

Evaluate numerically:

Plot the function over a subset of the reals:

Plot the imaginary part of the function:

Plot over a subset of the complexes:

Series expansions at the origin:

Series expansion at Infinity:

Scope (28)

Numerical Evaluation (5)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiND efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix InverseJacobiND function using MatrixFunction:

Specific Values (3)

Simple exact results are generated automatically:

Values at infinity:

Find a real root of the equation :

Visualization (3)

Plot InverseJacobiND for various values of the second parameter :

Plot InverseJacobiND as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (6)

InverseJacobiND is not an analytic function:

It has both singularities and discontinuities:

is nondecreasing on its real domain:

is injective:

is not surjective:

is non-negative on its real domain:

is concave on its real domain:

Differentiation (4)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiND with respect to the second argument :

Plot higher derivatives for :

Series Expansions (3)

Taylor expansion for around :

Plot the first three approximations for around :

Series expansion for around :

InverseJacobiND can be applied to a power series:

Function Identities and Simplifications (2)

InverseJacobiND is the inverse function of JacobiND:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features (2)

InverseJacobiND threads element‐wise over lists:

TraditionalForm formatting:

Applications (1)

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

Properties & Relations (1)

Obtain InverseJacobiND from solving equations containing elliptic functions:

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