# InverseJacobiSD

InverseJacobiSD[v,m]

gives the inverse Jacobi elliptic function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• gives the value of for which .
• InverseJacobiSD 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, InverseJacobiSD automatically evaluates to exact values.
• InverseJacobiSD can be evaluated to arbitrary numerical precision.
• InverseJacobiSD 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 input tracks the precision of the output:

Evaluate for complex arguments:

Evaluate InverseJacobiSD efficiently at high precision:

### 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 InverseJacobiSD for various values of the second parameter :

Plot InverseJacobiSD as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(6)

InverseJacobiSD is not an analytic function:

It has both singularities and discontinuities: is nondecreasing: 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 InverseJacobiSD 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)

InverseJacobiSD is the inverse function of JacobiSD:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

### Other Features(3)

InverseJacobiSD can be applied to a power series:

## Generalizations & Extensions(1)

InverseJacobiSD 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 InverseJacobiSD from solving equations containing elliptic functions: 