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InverseJacobiNC

InverseJacobiNC[v,m]

gives the inverse Jacobi elliptic function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
gives the value of for which .
InverseJacobiNC has branch cut discontinuities in the complex 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, InverseJacobiNC automatically evaluates to exact values.
InverseJacobiNC can be evaluated to arbitrary numerical precision.
InverseJacobiNC automatically threads over lists.

Examples

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Basic Examples (5)

Evaluate numerically:

Check defining property:

Plot the function over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope (28)

Numerical Evaluation (5)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiNC efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix InverseJacobiNC function using MatrixFunction:

Specific Values (3)

Simple exact values are generated automatically:

Value at infinity:

Find a real root of the equation :

Visualization (3)

Plot InverseJacobiNC for various values of the second parameter :

Plot InverseJacobiNC as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (6)

InverseJacobiNC 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 InverseJacobiNC with respect to the second argument :

Plot higher derivatives for :

Series Expansions (3)

Taylor expansion for around :

Plot the first three approximations for around :

Taylor expansion for around :

Plot the first three approximations for around :

InverseJacobiNC can be applied to a power series:

Function Identities and Simplifications (2)

InverseJacobiNC is the inverse function of JacobiNC:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features (2)

InverseJacobiNC threads elementwise over lists:

TraditionalForm formatting:

Applications (1)

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

Properties & Relations (1)

Obtain InverseJacobiNC from solving equations containing elliptic functions:

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