JacobiCS [u,m]
gives the Jacobi elliptic function .
JacobiCS
JacobiCS [u,m]
gives the Jacobi elliptic function .
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- , where .
- is a doubly periodic function in u with periods and , where is the elliptic integral EllipticK .
- JacobiCS is a meromorphic function in both arguments.
- For certain special arguments, JacobiCS automatically evaluates to exact values.
- JacobiCS can be evaluated to arbitrary numerical precision.
- JacobiCS automatically threads over lists.
Examples
open all close allBasic Examples (5)
Evaluate numerically:
Plot the function over a subset of the reals:
Plot over a subset of the complexes:
Series expansions at the origin:
Series expansion at a singular point:
Scope (34)
Numerical Evaluation (5)
Evaluate numerically to high precision:
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate JacobiCS efficiently at high precision:
Compute average-case statistical intervals using Around :
Compute the elementwise values of an array:
Or compute the matrix JacobiCS function using MatrixFunction :
Specific Values (3)
Simple exact values are generated automatically:
Some poles of JacobiCS :
Find a zero of TemplateBox[{x, {1, /, 3}}, JacobiCS]:
Visualization (3)
Function Properties (8)
JacobiCS is 2 TemplateBox[{m}, EllipticK]-periodic along the real axis:
JacobiCS is 4ⅈTemplateBox[{{1, -, m}}, EllipticK]-periodic along the imaginary axis:
JacobiCS is an odd function in its first argument:
JacobiCS is not an analytic function:
It has both singularities and discontinuities:
TemplateBox[{x, 3}, JacobiCS] is neither nondecreasing nor nonincreasing:
TemplateBox[{x, m}, JacobiCS] is not injective for any fixed :
It is injective for :
TemplateBox[{x, m}, JacobiCS] is not surjective for fixed :
It is surjective for :
JacobiCS is neither non-negative nor non-positive:
JacobiCS is neither convex nor concave:
Differentiation (3)
First derivative:
Higher derivatives:
Plot higher derivatives for :
Derivative with respect to :
Integration (3)
Indefinite integral of JacobiCS :
Definite integral of an odd function over the interval centered at the origin is 0:
More integrals:
Series Expansions (3)
Series expansion for TemplateBox[{x, {1, /, 3}}, JacobiCS]:
Plot the first three approximations for TemplateBox[{x, {1, /, 3}}, JacobiCS] around :
Taylor expansion for TemplateBox[{1, m}, JacobiCS]:
Plot the first three approximations for TemplateBox[{1, m}, JacobiCS] around :
JacobiCS can be applied to a power series:
Function Identities and Simplifications (3)
Primary definition:
Parity transformation and periodicity relations are automatically applied:
Automatic argument simplifications:
Function Representations (3)
Representation in terms of Cot of JacobiAmplitude :
Relation to other Jacobi elliptic functions:
TraditionalForm formatting:
Applications (5)
Hierarchy of solutions of the nonlinear diffusion equation :
Check:
Flow lines in a rectangular region with a current flowing from the lower‐right to the upper‐left corner:
Conformal map from a unit triangle to the unit disk:
Show points before and after the map:
Solution of the sinh‐Gordon equation :
Check the solution:
Plot the solution:
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 (2)
Compose with inverse functions:
Use PowerExpand to disregard multivaluedness of the inverse function:
Solve a transcendental equation:
Possible Issues (2)
Machine-precision input is insufficient to give the correct answer:
Currently only simple simplification rules are built in for Jacobi functions:
See Also
Tech Notes
Related Guides
Related Links
History
Introduced in 1988 (1.0)
Text
Wolfram Research (1988), JacobiCS, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiCS.html.
CMS
Wolfram Language. 1988. "JacobiCS." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiCS.html.
APA
Wolfram Language. (1988). JacobiCS. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiCS.html
BibTeX
@misc{reference.wolfram_2025_jacobics, author="Wolfram Research", title="{JacobiCS}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiCS.html}", note=[Accessed: 10-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_jacobics, organization={Wolfram Research}, title={JacobiCS}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiCS.html}, note=[Accessed: 10-January-2026]}