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JacobiCN[u,m]

gives the Jacobi elliptic function .

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
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Function Properties  
Differentiation  
Integration  
Series Expansions  
Function Identities and Simplifications  
Function Representations  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
Related Guides
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History
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JacobiCN

JacobiCN[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.
  • JacobiCN is a meromorphic function in both arguments.
  • For certain special arguments, JacobiCN automatically evaluates to exact values.
  • JacobiCN can be evaluated to arbitrary numerical precision.
  • JacobiCN automatically threads over lists.

Examples

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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 about the origin:

Series expansion at a singular point:

Scope  (33)

Numerical Evaluation  (5)

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate JacobiCN efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix JacobiCN function using MatrixFunction:

Specific Values  (3)

Simple exact values are generated automatically:

Some poles of JacobiCN:

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

Visualization  (3)

Plot the JacobiCN functions for various parameter values:

Plot JacobiCN as a function of its parameter :

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

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

Function Properties  (8)

JacobiCN is 4 TemplateBox[{m}, EllipticK]-periodic along the real axis:

JacobiCN is 4ⅈTemplateBox[{{1, -, m}}, EllipticK]-periodic along the imaginary axis:

JacobiCN is an even function in its first argument:

JacobiCN is an analytic function of x:

It has no singularities or discontinuities:

TemplateBox[{x, {1, /, 3}}, JacobiCN] is neither nondecreasing nor nonincreasing:

TemplateBox[{x, m}, JacobiCN] is not injective for any fixed :

TemplateBox[{x, m}, JacobiCN] is not surjective for any fixed :

TemplateBox[{x, m}, JacobiCN] is non-negative for positive integer values of :

In general, it is neither non-negative nor non-positive:

JacobiCN is neither convex nor concave:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Integration  (3)

Indefinite integral of JacobiCN:

Definite integral of an even integrand over the interval centered at the origin:

This is twice the integral over half the interval:

More integrals:

Series Expansions  (3)

Taylor expansion for TemplateBox[{x, {1, /, 3}}, JacobiCN]:

Plot the first three approximations for TemplateBox[{x, {1, /, 3}}, JacobiCN] around :

Taylor expansion for TemplateBox[{1, m}, JacobiCN]:

Plot the first three approximations for TemplateBox[{1, m}, JacobiCN] around :

JacobiCN can be applied to power series:

Function Identities and Simplifications  (2)

Parity transformations and periodicity relations are automatically applied:

Identity involving JacobiSN:

Function Representations  (3)

Representation in terms of Cos of JacobiAmplitude:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting:

Applications  (9)

Cnoidal solution of the KdV equation:

Verify the solution:

Plot the solution:

Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

Solution of an anharmonic oscillator :

Plot various solutions:

Elliptic parametrization of an ellipse:

Plot using elliptic parametrization and circular parametrization:

Solution of Nahm equations:

Check that the solutions fulfill the Nahm equations:

Parametrization of a mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):

Plot the inflated balloon:

Use JacobiCN to map a point from a 2D simplex to a 3‐by‐3 correlation matrix:

Visualize the determinant over the simplex:

Verify that the determinant is maximal at the centroid of the simplex:

Plot the maximum value of the determinant as a function of the elliptic parameter :

Use the correlation matrix to define a 3D T copula distribution

Draw a sample:

Parameterization of an algebraic lemniscate:

Verify that the parametric function solves the algebraic equation for the lemniscate:

Periodic motion of 3 bodies, where the bodies chase each other on a common orbit with equal time‐spacing on a lemniscate:

The elliptic parameter is fixed by requiring that the center of mass of the 3-body system remains fixed at the origin:

Visualize the 3‐body configuration:

A 5-body configuration:

The 5-body configuration admits two elliptic parameters that fix the center of mass:

Parameterization of Costa's minimal surface [MathWorld]:

Properties & Relations  (4)

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the inverse function:

Evaluate as a result of applying Cos to JacobiAmplitude:

Solve a transcendental equation:

Integral:

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), JacobiCN, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiCN.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_jacobicn, author="Wolfram Research", title="{JacobiCN}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiCN.html}", note=[Accessed: 31-October-2025]}

BibLaTeX

@online{reference.wolfram_2025_jacobicn, organization={Wolfram Research}, title={JacobiCN}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiCN.html}, note=[Accessed: 31-October-2025]}