gives the Jacobi elliptic function .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • , where .
  • is a doubly periodic function in with periods and , where is the elliptic integral EllipticK.
  • JacobiSD is a meromorphic function in both arguments.
  • For certain special arguments, JacobiSD automatically evaluates to exact values.
  • JacobiSD can be evaluated to arbitrary numerical precision.
  • JacobiSD 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 at 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 JacobiSD efficiently at high precision:

JacobiSD threads elementwise over lists:

Specific Values  (3)

Simple exact values are generated automatically:

Some poles of JacobiSD:

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

Visualization  (3)

Plot the JacobiSD functions for various values of parameter:

Plot JacobiSD as a function of its parameter :

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

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

Function Properties  (8)

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

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

JacobiSD is an odd function in its first argument:

TemplateBox[{x, m}, JacobiSD] is an analytic function for :

It is not in general analytic:

It has both singularities and discontinuities for :

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

JacobiSD is not injective for any fixed :

It is injective for :

TemplateBox[{x, m}, JacobiSD] is not surjective for :

It is surjective for :

JacobiSD is neither non-negative nor non-positive:

JacobiSD is neither convex nor concave:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Integration  (3)

Indefinite integral of JacobiSD:

Definite integral of an odd function over an interval centered at the origin:

More integrals:

Series Expansions  (3)

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

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

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

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

JacobiSD can be applied to a power series:

Function Identities and Simplifications  (3)

Parity transformation and periodicity relations are automatically applied:

Identity involving JacobiCD:

Automatic argument simplifications:

Function Representations  (3)

Representation in terms of Csc and JacobiAmplitude:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting:

Applications  (5)

Conformal map from a rectangle to the unit disk:

Visualize the map:

Generator for the hierarchy of solutions of the nonlinear diffusion equation :

Numerical check of the solutions:

Conformal map from an ellipse to the unit disk:

Visualize the map:

Cartesian coordinates of a pendulum:

Plot the time dependence of the coordinates:

Plot the trajectory:

Parameterization of Costa's minimal surface [MathWorld]:

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


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


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


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


@misc{reference.wolfram_2024_jacobisd, author="Wolfram Research", title="{JacobiSD}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiSD.html}", note=[Accessed: 27-May-2024 ]}


@online{reference.wolfram_2024_jacobisd, organization={Wolfram Research}, title={JacobiSD}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiSD.html}, note=[Accessed: 27-May-2024 ]}