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.
  • 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.


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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  (26)

Numerical Evaluation  (4)

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:

JacobiCN threads elementwise over lists:

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[{{x, +, {ⅈ,  , y}}, {1, /, 2}}, JacobiCN]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}, {1, /, 2}}, JacobiCN]:

Function Properties  (2)

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:

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  (6)

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:

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:


Possible Issues  (2)

Machine-precision input is insufficient to give the correct answer:

Currently only simple simplification rules are built in for Jacobi functions:

Introduced in 1988