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InverseJacobiCN

InverseJacobiCN[v,m]

gives the inverse Jacobi elliptic function .

Details

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

Examples

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Basic Examples (5)

Evaluate numerically:

Plot the function at different values of the modulus m over a subset of the reals:

Plot over a subset of the complexes:

Series expansions at the origin:

Series expansion at Infinity:

Scope (28)

Numerical Evaluation (6)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiCN efficiently at high precision:

InverseJacobiCN threads elementwise over lists:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix InverseJacobiCN 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 InverseJacobiCN for various values of the second parameter :

Plot InverseJacobiCN as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (6)

InverseJacobiCN 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:

Differentiate InverseJacobiCN with respect to the second argument :

Higher derivatives:

Series Expansions (2)

Taylor expansion for :

Plot the first three approximations for around :

Taylor expansion for :

Plot the first three approximations for around :

Function Identities and Simplifications (2)

InverseJacobiCN is the inverse function of JacobiCN:

Compose with the inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features (2)

InverseJacobiCN can be applied to a power series:

TraditionalForm formatting:

Generalizations & Extensions (1)

InverseJacobiCN can be applied to a power series:

Applications (1)

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

Properties & Relations (1)

Obtain InverseJacobiCN from solving equations containing elliptic functions:

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