InverseJacobiDS

InverseJacobiDS[v,m]

gives the inverse Jacobi elliptic function .

Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• gives the value of for which .
• InverseJacobiDS 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, InverseJacobiDS automatically evaluates to exact values.
• InverseJacobiDS can be evaluated to arbitrary numerical precision.
• InverseJacobiDS 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 expansion at the origin:

Scope(29)

Numerical Evaluation(3)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiDS efficiently at high precision:

Specific Values(5)

Simple exact results are generated automatically:

Limiting values at the origin:

Value at infinity:

Find a real root of the equation :

Parity transformation is automatically applied:

Visualization(3)

Plot InverseJacobiDS for various values of the second parameter :

Plot InverseJacobiDS as a function of its parameter :

Plot the real part of :

Plot the imaginary part of

Function Properties(6)

InverseJacobiDS is not an analytic function:

It has both singularities and discontinuities:

is neither nondecreasing nor nonincreasing:

is injective:

is surjective:

is neither non-negative nor non-positive:

is neither convex nor concave:

Differentiation and Integration(5)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiDS with respect to the second argument :

Plot higher derivatives for :

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 :

InverseJacobiDS can be applied to a power series:

Function Identities and Simplifications(2)

InverseJacobiDS is the inverse function of JacobiDS:

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 InverseJacobiDS from solving equations containing elliptic functions:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_inversejacobids, author="Wolfram Research", title="{InverseJacobiDS}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/InverseJacobiDS.html}", note=[Accessed: 22-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_inversejacobids, organization={Wolfram Research}, title={InverseJacobiDS}, year={1988}, url={https://reference.wolfram.com/language/ref/InverseJacobiDS.html}, note=[Accessed: 22-April-2024 ]}