JacobiSC [u,m]
gives the Jacobi elliptic function .
JacobiSC
JacobiSC [u,m]
gives the Jacobi elliptic function .
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- , where .
- is a doubly periodic function in with periods and , where is the elliptic integral EllipticK .
- JacobiSC is a meromorphic function in both arguments.
- For certain special arguments, JacobiSC automatically evaluates to exact values.
- JacobiSC can be evaluated to arbitrary numerical precision.
- JacobiSC automatically threads over lists.
Examples
open all close allBasic 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 (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 JacobiSC efficiently at high precision:
Compute average-case statistical intervals using Around :
Compute the elementwise values of an array:
Or compute the matrix JacobiSC function using MatrixFunction :
Specific Values (3)
Visualization (3)
Function Properties (8)
JacobiSC is 2TemplateBox[{m}, EllipticK]-periodic along the real axis:
JacobiSC is 4ⅈTemplateBox[{{1, -, m}}, EllipticK]-periodic along the imaginary axis:
JacobiSC is an odd function in its first argument:
TemplateBox[{x, m}, JacobiSC] is an analytic function of for :
It is not, in general, analytic:
It has both singularities and discontinuities for :
TemplateBox[{x, 3}, JacobiSC] is neither nondecreasing nor nonincreasing:
JacobiSC is not injective for any fixed
It is injective for :
TemplateBox[{x, m}, JacobiSC] is not surjective for :
It is surjective for :
JacobiSC is neither non-negative nor non-positive:
JacobiSC is neither convex nor concave:
Differentiation (3)
First derivative:
Higher derivatives:
Plot higher derivatives for :
Derivative with respect to :
Series Expansions (3)
Taylor expansion for TemplateBox[{x, {1, /, 3}}, JacobiSC]:
Plot the first three approximations for TemplateBox[{x, {1, /, 3}}, JacobiSC] around :
Taylor expansion for TemplateBox[{1, m}, JacobiSC]:
Plot the first three approximations for TemplateBox[{1, m}, JacobiSC] around :
JacobiSC can be applied to a power series:
Function Identities and Simplifications (3)
Parity transformation and periodicity relations are automatically applied:
Identity involving JacobiNC :
Argument simplifications:
Function Representations (3)
Representation in terms of Tan of JacobiAmplitude :
Relation to other Jacobi elliptic functions:
TraditionalForm formatting:
Applications (3)
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:
Properties & Relations (3)
Compose with inverse functions:
Use PowerExpand to disregard multivaluedness of the inverse function:
Solve a transcendental equation:
JacobiSC can be represented with related elliptic functions:
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), JacobiSC, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiSC.html.
CMS
Wolfram Language. 1988. "JacobiSC." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiSC.html.
APA
Wolfram Language. (1988). JacobiSC. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiSC.html
BibTeX
@misc{reference.wolfram_2025_jacobisc, author="Wolfram Research", title="{JacobiSC}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiSC.html}", note=[Accessed: 16-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_jacobisc, organization={Wolfram Research}, title={JacobiSC}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiSC.html}, note=[Accessed: 16-November-2025]}