# InverseJacobiDN

InverseJacobiDN[v,m]

gives the inverse Jacobi elliptic function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• gives the value of for which .
• InverseJacobiDN has branch cut discontinuities in the complex v plane with branch points at and infinity, and in the complex m plane with branch points and infinity.
• The inverse Jacobi elliptic functions are related to elliptic integrals.
• For certain special arguments, InverseJacobiDN automatically evaluates to exact values.
• InverseJacobiDN can be evaluated to arbitrary numerical precision.
• InverseJacobiDN automatically threads over lists.

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot the real part of the function:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(24)

### Numerical Evaluation(3)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiDN efficiently at high precision:

### Specific Values(3)

Simple exact results are generated automatically:

Value at infinity:

Find a real root of the equation :

### Visualization(3)

Plot InverseJacobiDN for various values of the second parameter :

Plot InverseJacobiDN as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(4)

InverseJacobiDN is not an analytic function:

It has both singularities and discontinuities:

is injective:

InverseJacobiDN is neither non-negative nor non-positive:

InverseJacobiDN is neither convex nor concave:

### Differentiation(4)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiDN with respect to the second argument :

Higher derivatives:

### Series Expansions(3)

Taylor expansion for around and :

Plot the first three approximations for around :

Series expansion for around :

InverseJacobiDN can be applied to a power series:

### Function Identities and Simplifications(2)

InverseJacobiDN is the inverse function of JacobiDN:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

## Applications(1)

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

## Properties & Relations(1)

Obtain InverseJacobiDN from solving equations containing elliptic functions:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2023_inversejacobidn, author="Wolfram Research", title="{InverseJacobiDN}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/InverseJacobiDN.html}", note=[Accessed: 22-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_inversejacobidn, organization={Wolfram Research}, title={InverseJacobiDN}, year={1988}, url={https://reference.wolfram.com/language/ref/InverseJacobiDN.html}, note=[Accessed: 22-April-2024 ]}