JacobiNS [u,m]
gives the Jacobi elliptic function .
JacobiNS
JacobiNS [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 .
- JacobiNS is a meromorphic function in both arguments.
- For certain special arguments, JacobiNS automatically evaluates to exact values.
- JacobiNS can be evaluated to arbitrary numerical precision.
- JacobiNS 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 about 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 JacobiNS efficiently at high precision:
Compute average-case statistical intervals using Around :
Compute the elementwise values of an array:
Or compute the matrix JacobiNS function using MatrixFunction :
Specific Values (3)
Visualization (3)
Function Properties (8)
JacobiNS is 4TemplateBox[{m}, EllipticK]-periodic along the real axis:
JacobiNS is 2ⅈTemplateBox[{{1, -, m}}, EllipticK]-periodic along the imaginary axis:
JacobiNS is an odd function in its first argument:
JacobiNS is not an analytic function:
It has both singularities and discontinuities:
TemplateBox[{x, 3}, JacobiNS] is neither nondecreasing nor nonincreasing:
TemplateBox[{x, m}, JacobiNS] is not injective for any fixed :
It is injective for :
JacobiNS is not surjective for any fixed :
JacobiNS is neither non-negative nor non-positive:
JacobiNS is neither convex nor concave:
Differentiation (3)
First derivative:
Higher derivatives:
Plot higher derivatives for :
Derivative with respect to :
Integration (3)
Indefinite integral of JacobiNS :
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}}, JacobiNS]:
Plot the first three approximations for TemplateBox[{x, {1, /, 3}}, JacobiNS] around :
Taylor expansion for TemplateBox[{1, m}, JacobiNS]:
Plot the first three approximations for TemplateBox[{1, m}, JacobiNS] around :
JacobiNS can be applied to power series:
Function Identities and Simplifications (3)
Parity transformation and periodicity relations are automatically applied:
Identity involving JacobiCS :
Argument simplifications:
Function Representations (3)
Representation in terms of Csc of JacobiAmplitude :
Relation to other Jacobi elliptic functions:
TraditionalForm formatting:
Applications (5)
Map a rectangle conformally onto the lower half‐plane:
Solution of the pendulum equation:
Check the solution:
Plot solutions:
Closed form of iterates of the Katsura–Fukuda map:
Compare the closed form with explicit iterations:
Plot a few hundred iterates:
Solution of the sinh‐Gordon equation :
Check the solution:
Plot the solution:
Hierarchy of solutions of the nonlinear diffusion equation :
Verify these functions:
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 may be 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), JacobiNS, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiNS.html.
CMS
Wolfram Language. 1988. "JacobiNS." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiNS.html.
APA
Wolfram Language. (1988). JacobiNS. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiNS.html
BibTeX
@misc{reference.wolfram_2025_jacobins, author="Wolfram Research", title="{JacobiNS}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiNS.html}", note=[Accessed: 09-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_jacobins, organization={Wolfram Research}, title={JacobiNS}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiNS.html}, note=[Accessed: 09-January-2026]}