gives the Jacobi elliptic function .


  • 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.


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Basic 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  (33)

Numerical Evaluation  (4)

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:

JacobiNS threads elementwise over lists:

Specific Values  (3)

Simple exact values are generated automatically:

Some poles of JacobiNS:

Find a local minimum of JacobiNS as a root of (d)/(dx)TemplateBox[{x, {1, /, 3}}, JacobiND]=0:

Visualization  (3)

Plot the JacobiNS functions for various parameter values:

Plot JacobiNS as a function of its parameter :

Plot the real part of TemplateBox[{z, {1, /, 2}}, JacobiNS]:

Plot the imaginary part of TemplateBox[{z, {1, /, 2}}, JacobiNS]:

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 halfplane:

Solution of the pendulum equation:

Check the solution:

Plot solutions:

Closed form of iterates of the KatsuraFukuda map:

Compare the closed form with explicit iterations:

Plot a few hundred iterates:

Solution of the sinhGordon 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:

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


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


Wolfram Language. 1988. "JacobiNS." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiNS.html.


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


@misc{reference.wolfram_2024_jacobins, author="Wolfram Research", title="{JacobiNS}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiNS.html}", note=[Accessed: 18-June-2024 ]}


@online{reference.wolfram_2024_jacobins, organization={Wolfram Research}, title={JacobiNS}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiNS.html}, note=[Accessed: 18-June-2024 ]}