JacobiCD
✖
JacobiCD
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.
- JacobiCD is a meromorphic function in both arguments.
- For certain special arguments, JacobiCD automatically evaluates to exact values.
- JacobiCD can be evaluated to arbitrary numerical precision.
- JacobiCD automatically threads over lists.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0cq1clmx-eermve

Plot the function over a subset of the reals:

https://wolfram.com/xid/0cq1clmx-f76t5

Plot over a subset of the complexes:

https://wolfram.com/xid/0cq1clmx-kiedlx

Series expansions at the origin:

https://wolfram.com/xid/0cq1clmx-17uo1


https://wolfram.com/xid/0cq1clmx-ku0xqm

Scope (34)Survey of the scope of standard use cases
Numerical Evaluation (5)

https://wolfram.com/xid/0cq1clmx-gk3z66

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

https://wolfram.com/xid/0cq1clmx-idhgdf

Evaluate for complex arguments:

https://wolfram.com/xid/0cq1clmx-f4wtrj

Evaluate JacobiCD efficiently at high precision:

https://wolfram.com/xid/0cq1clmx-di5gcr


https://wolfram.com/xid/0cq1clmx-bq2c6r

Compute average case statistical intervals using Around:

https://wolfram.com/xid/0cq1clmx-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0cq1clmx-thgd2

Or compute the matrix JacobiCD function using MatrixFunction:

https://wolfram.com/xid/0cq1clmx-o5jpo

Specific Values (3)
Simple exact answers are generated automatically:

https://wolfram.com/xid/0cq1clmx-qlzb8


https://wolfram.com/xid/0cq1clmx-wq506

Some poles of JacobiCD:

https://wolfram.com/xid/0cq1clmx-cw39qs


https://wolfram.com/xid/0cq1clmx-f2hrld


https://wolfram.com/xid/0cq1clmx-cj5txq

Visualization (3)
Plot the JacobiCD functions for various values of parameter:

https://wolfram.com/xid/0cq1clmx-ecj8m7

Plot JacobiCD as a function of its parameter :

https://wolfram.com/xid/0cq1clmx-du62z6


https://wolfram.com/xid/0cq1clmx-ouu484


https://wolfram.com/xid/0cq1clmx-ccvjds

Function Properties (8)
JacobiCD is -periodic along the real axis:

https://wolfram.com/xid/0cq1clmx-ewxrep

JacobiCD is -periodic along the imaginary axis:

https://wolfram.com/xid/0cq1clmx-w9fc2

JacobiCD is an even function in its first argument:

https://wolfram.com/xid/0cq1clmx-dnla5q

is an analytic function of
for
:

https://wolfram.com/xid/0cq1clmx-h5x4l2

It is not, in general, an analytic function:

https://wolfram.com/xid/0cq1clmx-o0fjjo

It has both singularities and discontinuities for :

https://wolfram.com/xid/0cq1clmx-mdtl3h


https://wolfram.com/xid/0cq1clmx-mn5jws

is neither nondecreasing nor nonincreasing:

https://wolfram.com/xid/0cq1clmx-nlz7s

is not injective for any fixed
:

https://wolfram.com/xid/0cq1clmx-poz8g


https://wolfram.com/xid/0cq1clmx-xtauq9


https://wolfram.com/xid/0cq1clmx-hglrpk


https://wolfram.com/xid/0cq1clmx-ctca0g

is not surjective for any fixed
:

https://wolfram.com/xid/0cq1clmx-65g2du


https://wolfram.com/xid/0cq1clmx-hdm869

JacobiCD is neither non-negative nor non-positive:

https://wolfram.com/xid/0cq1clmx-84dui

JacobiCD is neither convex nor concave:

https://wolfram.com/xid/0cq1clmx-8kku21

Differentiation (3)

https://wolfram.com/xid/0cq1clmx-mmas49


https://wolfram.com/xid/0cq1clmx-nfbe0l


https://wolfram.com/xid/0cq1clmx-fxwmfc


https://wolfram.com/xid/0cq1clmx-clw10k

Integration (3)
Indefinite integral of JacobiCD:

https://wolfram.com/xid/0cq1clmx-bz1zx0

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

https://wolfram.com/xid/0cq1clmx-ft0ejz

This is twice the integral over half the interval:

https://wolfram.com/xid/0cq1clmx-etmqww


https://wolfram.com/xid/0cq1clmx-c7etaq


https://wolfram.com/xid/0cq1clmx-bo4xic

Series Expansions (3)

https://wolfram.com/xid/0cq1clmx-ewr1h8

Plot the first three approximations for around
:

https://wolfram.com/xid/0cq1clmx-binhar


https://wolfram.com/xid/0cq1clmx-c7itxf

Plot the first three approximations for around
:

https://wolfram.com/xid/0cq1clmx-jkkunh

JacobiCD can be applied to a power series:

https://wolfram.com/xid/0cq1clmx-noub1v

Function Identities and Simplifications (3)

https://wolfram.com/xid/0cq1clmx-v8ak2

Parity transformations and periodicity relations are automatically applied:

https://wolfram.com/xid/0cq1clmx-fcv5op


https://wolfram.com/xid/0cq1clmx-b8xdyd

Automatic argument simplifications:

https://wolfram.com/xid/0cq1clmx-fgb1c1


https://wolfram.com/xid/0cq1clmx-ba5yxr

Function Representations (3)
Representation in terms of trigonometry functions and JacobiAmplitude:

https://wolfram.com/xid/0cq1clmx-dbb09w

Relation to other Jacobi elliptic functions:

https://wolfram.com/xid/0cq1clmx-ejwj5


https://wolfram.com/xid/0cq1clmx-gd2cwv

TraditionalForm formatting:

https://wolfram.com/xid/0cq1clmx-cbeg9n

Applications (4)Sample problems that can be solved with this function
Derivatives of Jacobi elliptic functions with respect to parameter :

https://wolfram.com/xid/0cq1clmx-d4ns6

Conformal map from a unit triangle to the unit disk:

https://wolfram.com/xid/0cq1clmx-v14wu
Show points before and after the map:

https://wolfram.com/xid/0cq1clmx-ifhzxc

https://wolfram.com/xid/0cq1clmx-io64zw

Solution of the Poisson–Boltzmann equation :

https://wolfram.com/xid/0cq1clmx-drlj8u
Check solution using series expansion:

https://wolfram.com/xid/0cq1clmx-frx0o2

Construct lowpass elliptic filter for analog signal:

https://wolfram.com/xid/0cq1clmx-bxw8jc
Compute filter ripple parameters and associate elliptic function parameter:

https://wolfram.com/xid/0cq1clmx-bm1kjg
Use elliptic degree equation to find the ratio of the pass and the stop frequencies:

https://wolfram.com/xid/0cq1clmx-b13u0f

Compute corresponding stop frequency and elliptic parameters:

https://wolfram.com/xid/0cq1clmx-wtjxg

Compute ripple locations and poles and zeros of the transfer function:

https://wolfram.com/xid/0cq1clmx-kvrem
Compute poles of the transfer function:

https://wolfram.com/xid/0cq1clmx-elkxl1
Assemble the transfer function:

https://wolfram.com/xid/0cq1clmx-fyal2v

https://wolfram.com/xid/0cq1clmx-bv4lrk

Compare with the result of EllipticFilterModel:

https://wolfram.com/xid/0cq1clmx-00j7s

https://wolfram.com/xid/0cq1clmx-ef2b35

Properties & Relations (3)Properties of the function, and connections to other functions
Compose with inverse functions:

https://wolfram.com/xid/0cq1clmx-crygdp

Use PowerExpand to disregard multivaluedness of the inverse function:

https://wolfram.com/xid/0cq1clmx-ei74r

Solve a transcendental equation:

https://wolfram.com/xid/0cq1clmx-fqvp5c



https://wolfram.com/xid/0cq1clmx-dy6wyi

Possible Issues (2)Common pitfalls and unexpected behavior
Machine-precision input is insufficient to give the correct answer:

https://wolfram.com/xid/0cq1clmx-epvdt2


https://wolfram.com/xid/0cq1clmx-cui73h

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

https://wolfram.com/xid/0cq1clmx-llrmb


https://wolfram.com/xid/0cq1clmx-bhctdh


https://wolfram.com/xid/0cq1clmx-bi24y3

Wolfram Research (1988), JacobiCD, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiCD.html.
Text
Wolfram Research (1988), JacobiCD, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiCD.html.
Wolfram Research (1988), JacobiCD, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiCD.html.
CMS
Wolfram Language. 1988. "JacobiCD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiCD.html.
Wolfram Language. 1988. "JacobiCD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiCD.html.
APA
Wolfram Language. (1988). JacobiCD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiCD.html
Wolfram Language. (1988). JacobiCD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiCD.html
BibTeX
@misc{reference.wolfram_2025_jacobicd, author="Wolfram Research", title="{JacobiCD}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiCD.html}", note=[Accessed: 30-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_jacobicd, organization={Wolfram Research}, title={JacobiCD}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiCD.html}, note=[Accessed: 30-May-2025
]}