InverseJacobiSN

InverseJacobiSN[v,m]

gives the inverse Jacobi elliptic function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • gives the value of u 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  (27)

Numerical Evaluation  (3)

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:

Specific Values  (4)

Simple exact values are generated automatically:

Value at infinity:

Find a real root of the equation TemplateBox[{x, {1, /, 3}}, InverseJacobiSN]=1:

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 TemplateBox[{z, 2}, InverseJacobiSN]:

Plot the imaginary part of TemplateBox[{z, 2}, InverseJacobiSN]:

Function Properties  (6)

InverseJacobiSN is not an analytic function:

It has both singularities and discontinuities:

TemplateBox[{x, {1, /, 3}}, InverseJacobiSN] is nondecreasing on its real domain:

TemplateBox[{x, {1, /, 3}}, InverseJacobiSN] is injective:

TemplateBox[{x, {1, /, 3}}, InverseJacobiSN] is not surjective:

TemplateBox[{x, {1, /, 3}}, InverseJacobiSN] is neither non-negative nor non-positive on its real domain:

TemplateBox[{x, {1, /, 3}}, InverseJacobiSN] 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 TemplateBox[{nu, m}, InverseJacobiSN] around :

Plot the first three approximations for TemplateBox[{nu, 2}, InverseJacobiSN] around :

Taylor expansion for TemplateBox[{nu, m}, InverseJacobiSN] around :

Plot the first three approximations for TemplateBox[{{1, /, 2}, m}, InverseJacobiSN] 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 threads elementwise over lists:

InverseJacobiSN can be applied to a power series:

TraditionalForm formatting:

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_2023_inversejacobisn, author="Wolfram Research", title="{InverseJacobiSN}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/InverseJacobiSN.html}", note=[Accessed: 18-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_inversejacobisn, organization={Wolfram Research}, title={InverseJacobiSN}, year={1988}, url={https://reference.wolfram.com/language/ref/InverseJacobiSN.html}, note=[Accessed: 18-March-2024 ]}