WOLFRAM

gives the inverse Jacobi elliptic function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • gives the value of u 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)Summary of the most common use cases

Evaluate numerically:

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Plot the function over a subset of the reals:

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Plot over a subset of the complexes:

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Series expansions at the origin:

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Scope  (29)Survey of the scope of standard use cases

Numerical Evaluation  (5)

Evaluate to high precision:

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

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Evaluate for complex arguments:

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Evaluate InverseJacobiSC efficiently at high precision:

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Compute average-case statistical intervals using Around:

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Compute the elementwise values of an array:

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Or compute the matrix InverseJacobiSC function using MatrixFunction:

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Specific Values  (4)

Simple exact results are generated automatically:

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Values at infinity:

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

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Parity transformation is automatically applied:

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Visualization  (3)

Plot InverseJacobiSC for various values of the second parameter :

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Plot InverseJacobiSC as a function of its parameter :

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Plot the real part of TemplateBox[{z, 2}, InverseJacobiSC]:

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

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Function Properties  (6)

InverseJacobiSC is not an analytic function:

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It has both singularities and discontinuities:

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TemplateBox[{x, 3}, InverseJacobiSC] is nondecreasing on its real domain:

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

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TemplateBox[{x, 3}, InverseJacobiSC] is not surjective:

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TemplateBox[{x, {1, /, 3}}, InverseJacobiSC] is neither non-negative nor non-positive:

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TemplateBox[{x, {1, /, 3}}, InverseJacobiSC] is neither convex nor concave:

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Differentiation and Integration  (4)

First derivative:

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Higher derivatives:

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Plot higher derivatives for :

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Differentiate InverseJacobiSC with respect to the second argument :

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

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Series Expansions  (3)

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

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

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

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Plot the first three approximations for TemplateBox[{nu, m}, InverseJacobiSC] around :

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InverseJacobiSC can be applied to a power series:

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Function Identities and Simplifications  (2)

InverseJacobiSC is the inverse function of JacobiSC:

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Compose with inverse function:

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Use PowerExpand to disregard multivaluedness of the inverse function:

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Other Features  (2)

InverseJacobiSC threads elementwise over lists:

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TraditionalForm formatting:

Applications  (2)Sample problems that can be solved with this function

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

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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:

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Compute corresponding stop frequency and elliptic parameters:

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

Compute poles of the transfer function:

Assemble the transfer function:

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Compare with the result of EllipticFilterModel:

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Properties & Relations  (1)Properties of the function, and connections to other functions

Obtain InverseJacobiSC from solving equations containing elliptic functions:

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

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.

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

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

BibTeX

@misc{reference.wolfram_2025_inversejacobisc, author="Wolfram Research", title="{InverseJacobiSC}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/InverseJacobiSC.html}", note=[Accessed: 06-June-2025 ]}

@misc{reference.wolfram_2025_inversejacobisc, author="Wolfram Research", title="{InverseJacobiSC}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/InverseJacobiSC.html}", note=[Accessed: 06-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_inversejacobisc, organization={Wolfram Research}, title={InverseJacobiSC}, year={1988}, url={https://reference.wolfram.com/language/ref/InverseJacobiSC.html}, note=[Accessed: 06-June-2025 ]}

@online{reference.wolfram_2025_inversejacobisc, organization={Wolfram Research}, title={InverseJacobiSC}, year={1988}, url={https://reference.wolfram.com/language/ref/InverseJacobiSC.html}, note=[Accessed: 06-June-2025 ]}