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.
  • 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.


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Basic 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  (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 JacobiCS efficiently at high precision:

JacobiCS threads elementwise over lists:

Specific Values  (3)

Simple exact values are generated automatically:

Some poles of JacobiCS:

Find a zero of TemplateBox[{x, {1, /, 3}}, JacobiCS]:

Visualization  (3)

Plot the JacobiCS functions for various parameter values:

Plot JacobiCS as a function of its parameter :

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

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

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 :


Flow lines in a rectangular region with a current flowing from the lowerright to the upperleft corner:

Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

Solution of the sinhGordon 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:

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


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


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


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


@misc{reference.wolfram_2024_jacobics, author="Wolfram Research", title="{JacobiCS}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiCS.html}", note=[Accessed: 25-July-2024 ]}


@online{reference.wolfram_2024_jacobics, organization={Wolfram Research}, title={JacobiCS}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiCS.html}, note=[Accessed: 25-July-2024 ]}