# 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:

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:

## 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:

Wolfram Research (1988), InverseJacobiCN, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseJacobiCN.html.

#### Text

Wolfram Research (1988), InverseJacobiCN, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseJacobiCN.html.

#### CMS

Wolfram Language. 1988. "InverseJacobiCN." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseJacobiCN.html.

#### APA

Wolfram Language. (1988). InverseJacobiCN. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseJacobiCN.html

#### BibTeX

@misc{reference.wolfram_2024_inversejacobicn, author="Wolfram Research", title="{InverseJacobiCN}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/InverseJacobiCN.html}", note=[Accessed: 19-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_inversejacobicn, organization={Wolfram Research}, title={InverseJacobiCN}, year={1988}, url={https://reference.wolfram.com/language/ref/InverseJacobiCN.html}, note=[Accessed: 19-September-2024 ]}