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InverseJacobiDC

InverseJacobiDC[v,m]

gives the inverse Jacobi elliptic function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
gives the value of for which .
InverseJacobiDC 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, InverseJacobiDC automatically evaluates to exact values.
InverseJacobiDC can be evaluated to arbitrary numerical precision.
InverseJacobiDC 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 (28)

Numerical Evaluation (5)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiDC efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix InverseJacobiDC function using MatrixFunction:

Specific Values (3)

Simple exact results are generated automatically:

Limiting value at infinity:

Find a real root of the equation :

Visualization (3)

Plot InverseJacobiDC for various values of the second parameter :

Plot InverseJacobiDC as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (6)

InverseJacobiDC is not an analytic function:

It has both singularities and discontinuities:

is nonincreasing on its real domain:

is injective:

is not surjective:

is non-negative on its real domain:

is neither convex nor concave on its real domain:

Differentiation (4)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiDC with respect to the second argument :

Higher derivatives:

Series Expansions (3)

Series expansion for around :

Plot the first three approximations for around :

Taylor expansion for :

Plot the first three approximations for around :

InverseJacobiDC can be applied to a power series:

Function Identities and Simplifications (2)

InverseJacobiDC is the inverse function of JacobiDC:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features (2)

InverseJacobiDC threads elementwise over lists:

TraditionalForm formatting:

Applications (1)

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

Properties & Relations (1)

Obtain InverseJacobiDC from solving equations containing elliptic functions:

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