# InverseJacobiSC

InverseJacobiSC[v,m]

gives the inverse Jacobi elliptic function .

# Details

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

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot the function over a subset of the reals:

Plot over a subset of the complexes:

Series expansions at the origin:

## Scope(27)

### Numerical Evaluation(3)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiSC efficiently at high precision:

### Specific Values(4)

Simple exact results are generated automatically:

Values at infinity:

Find a real root of the equation :

Parity transformation is automatically applied:

### Visualization(3)

Plot InverseJacobiSC for various values of the second parameter :

Plot InverseJacobiSC as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(6)

InverseJacobiSC 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:

is neither convex nor concave:

### Differentiation and Integration(4)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiSC with respect to the second argument :

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

### Series Expansions(3)

Taylor expansion for around :

Plot the first three approximations for around :

Taylor expansion for around :

Plot the first three approximations for around :

InverseJacobiSC can be applied to a power series:

### Function Identities and Simplifications(2)

InverseJacobiSC is the inverse function of JacobiSC:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

## Applications(2)

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

Construct lowpass elliptic filter for analog signal:

Compute filter ripple parameters and associate elliptic function parameter:

Use elliptic degree equation to find the ratio of the pass and the stop frequencies:

Compute corresponding stop frequency and elliptic parameters:

Compute ripple locations and poles and zeros of the transfer function:

Compute poles of the transfer function:

Assemble the transfer function:

Compare with the result of EllipticFilterModel:

## Properties & Relations(1)

Obtain InverseJacobiSC from solving equations containing elliptic functions:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2023_inversejacobisc, author="Wolfram Research", title="{InverseJacobiSC}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/InverseJacobiSC.html}", note=[Accessed: 17-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_inversejacobisc, organization={Wolfram Research}, title={InverseJacobiSC}, year={1988}, url={https://reference.wolfram.com/language/ref/InverseJacobiSC.html}, note=[Accessed: 17-April-2024 ]}