# InverseJacobiSN

InverseJacobiSN[v,m]

gives the inverse Jacobi elliptic function .

# Details

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

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot the function at different values of the modulus m:

Plot over a subset of the complexes:

Series expansions at the origin:

Series expansion at Infinity:

## Scope(29)

### Numerical Evaluation(5)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiSN efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix InverseJacobiSN function using MatrixFunction:

### Specific Values(4)

Simple exact values are generated automatically:

Value at infinity:

Find a real root of the equation :

Parity transformation is automatically applied:

### Visualization(3)

Plot InverseJacobiSN for various values of the second parameter :

Plot InverseJacobiSN as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(6)

InverseJacobiSN is not an analytic function:

It has both singularities and discontinuities:

is nondecreasing on its real domain:

is injective:

is not surjective:

is neither non-negative nor non-positive on its real domain:

is neither convex nor concave on its real domain:

### Differentiation and Integration(4)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiSN with respect to the second argument :

Definite integral of an odd function over an interval centered at the origin is 0:

### Series Expansions(2)

Taylor expansion for around :

Plot the first three approximations for around :

Taylor expansion for around :

Plot the first three approximations for around :

### Function Identities and Simplifications(2)

InverseJacobiSN is the inverse function of JacobiSN:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

### Other Features(3)

InverseJacobiSN can be applied to a power series:

## Generalizations & Extensions(1)

InverseJacobiSN 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 InverseJacobiSN from solving equations containing elliptic functions:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_inversejacobisn, author="Wolfram Research", title="{InverseJacobiSN}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/InverseJacobiSN.html}", note=[Accessed: 09-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_inversejacobisn, organization={Wolfram Research}, title={InverseJacobiSN}, year={1988}, url={https://reference.wolfram.com/language/ref/InverseJacobiSN.html}, note=[Accessed: 09-September-2024 ]}