gives the Jacobi elliptic function .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • , where .
  • a is doubly periodic function in u with periods and , where is the elliptic integral EllipticK.
  • JacobiDN is a meromorphic function in both arguments.
  • For certain special arguments, JacobiDN automatically evaluates to exact values.
  • JacobiDN can be evaluated to arbitrary numerical precision.
  • JacobiDN automatically threads over lists.


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Basic Examples  (4)

Evaluate numerically:

Plot the function for several values of the modulus m over a subset of the reals:

Plot over a subset of the complexes:

Series expansions about the origin:

Scope  (33)

Numerical Evaluation  (4)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiDN efficiently at high precision:

JacobiDN threads elementwise over lists:

Specific Values  (3)

Simple exact values are generated automatically:

Some poles of JacobiDN:

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

Visualization  (3)

Plot the JacobiDN functions for various parameter values:

Plot JacobiDN as a function of its parameter :

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

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

Function Properties  (8)

JacobiDN is 2 TemplateBox[{m}, EllipticK]-periodic along the real axis:

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

JacobiDN is an even function in its first argument:

JacobiDN is an analytic function:

It has no singularities or discontinuities:

TemplateBox[{x, 3}, JacobiDN] is neither nondecreasing nor nonincreasing:

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

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

TemplateBox[{x, m}, JacobiDN] is non-negative for :

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

JacobiDN is neither convex nor concave:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Integration  (3)

Indefinite integral of JacobiDN:

Definite integral of JacobiDN 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}}, JacobiDN]:

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

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

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

JacobiDN can be applied to a power series:

Function Identities and Simplifications  (3)

Parity transformations and periodicity relations are automatically applied:

Identity involving JacobiSN:

Argument simplifications:

Function Representations  (3)

Differential representation:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting:

Applications  (9)

Cartesian coordinates of a pendulum:

Plot the time dependence of the coordinates:

Plot the trajectory:

Uniformization of a Fermat cubic :


Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

Solution of Nahm equations:

Check that the solutions fulfill the Nahm equations:

Periodic solution of the nonlinear Schrödinger equation :

Plot the solution:

Parametrize a lemniscate by arc length [more info]:

Show arc length parametrization and classical parametrization:

Zero modes of the periodic supersymmetric partner potentials:

Plot the zero modes:

Complex parametrization of a "sphere":

Plot real and imaginary parts:

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

Properties & Relations  (3)

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the inverse function:

Evaluate as a result of applying D 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:

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


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


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


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


@misc{reference.wolfram_2024_jacobidn, author="Wolfram Research", title="{JacobiDN}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiDN.html}", note=[Accessed: 27-May-2024 ]}


@online{reference.wolfram_2024_jacobidn, organization={Wolfram Research}, title={JacobiDN}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiDN.html}, note=[Accessed: 27-May-2024 ]}