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InverseJacobiDS

InverseJacobiDS[v,m]

gives the inverse Jacobi elliptic function .

Details

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

Examples

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Basic Examples (4)

Evaluate numerically:

Plot the function over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope (31)

Numerical Evaluation (5)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiDS efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix InverseJacobiDS function using MatrixFunction:

Specific Values (5)

Simple exact results are generated automatically:

Limiting values at the origin:

Value at infinity:

Find a real root of the equation :

Parity transformation is automatically applied:

Visualization (3)

Plot InverseJacobiDS for various values of the second parameter :

Plot InverseJacobiDS as a function of its parameter :

Plot the real part of :

Plot the imaginary part of

Function Properties (6)

InverseJacobiDS is not an analytic function:

It has both singularities and discontinuities:

is neither nondecreasing nor nonincreasing:

is injective:

is surjective:

is neither non-negative nor non-positive:

is neither convex nor concave:

Differentiation and Integration (5)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiDS with respect to the second argument :

Plot higher derivatives for :

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 :

InverseJacobiDS can be applied to a power series:

Function Identities and Simplifications (2)

InverseJacobiDS is the inverse function of JacobiDS:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features (2)

InverseJacobiDS threads elementwise over lists:

TraditionalForm formatting:

Applications (1)

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

Properties & Relations (1)

Obtain InverseJacobiDS from solving equations containing elliptic functions:

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