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InverseJacobiNS

InverseJacobiNS[v,m]

gives the inverse Jacobi elliptic function .

Details

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

Examples

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Basic Examples (5)

Evaluate numerically:

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 (30)

Numerical Evaluation (5)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiNS efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix InverseJacobiNS function using MatrixFunction:

Specific Values (4)

Simple exact results are generated automatically:

Values at infinity:

Find a real root of the equation :

Parity transformation is automatically applied:

Visualization (3)

Plot InverseJacobiNS for various values of the second parameter :

Plot InverseJacobiNS as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (6)

InverseJacobiNS is not an analytic function:

It has both singularities and discontinuities:

is neither nondecreasing nor nonincreasing on its real domain:

is injective:

is not surjective:

is neither non-negative nor non-positive on its real domain:

is neither convex nor concave on its real domain:

Differentiation and Integration (5)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiNS with respect to the second argument :

Higher derivatives:

Definite integral of an odd function over an interval centered at the origin is 0:

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 :

InverseJacobiNS can be applied to a power series:

Function Identities and Simplifications (2)

InverseJacobiNS is the inverse function of JacobiNS:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features (2)

InverseJacobiNS threads elementwise over lists:

TraditionalForm formatting:

Applications (1)

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

Properties & Relations (1)

Obtain InverseJacobiNS from solving equations containing elliptic functions:

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