JacobiZN
✖
JacobiZN
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
.
- Argument conventions for elliptic integrals are discussed in "Elliptic Integrals and Elliptic Functions".
is a singly periodic function in
with the period
, where
is the elliptic integral EllipticK. »
- JacobiZN is a meromorphic function in both arguments.
- For certain special arguments, JacobiZN automatically evaluates to exact values.
- JacobiZN can be evaluated to arbitrary numerical precision.
- JacobiZN automatically threads over lists.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
Scope (24)Survey of the scope of standard use cases
Numerical Evaluation (5)

https://wolfram.com/xid/0cg6ccs28-cgugaz

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

https://wolfram.com/xid/0cg6ccs28-idztux

Evaluate for complex arguments:

https://wolfram.com/xid/0cg6ccs28-fal6xf


https://wolfram.com/xid/0cg6ccs28-l0lxvj

Evaluate JacobiZN efficiently at higher precision:

https://wolfram.com/xid/0cg6ccs28-e4wx6


https://wolfram.com/xid/0cg6ccs28-d4fr5b

Compute average case statistical intervals using Around:

https://wolfram.com/xid/0cg6ccs28-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0cg6ccs28-thgd2

Or compute the matrix JacobiZN function using MatrixFunction:

https://wolfram.com/xid/0cg6ccs28-o5jpo

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

https://wolfram.com/xid/0cg6ccs28-e9wz2t


https://wolfram.com/xid/0cg6ccs28-e4b7vj

JacobiZN has poles coinciding with poles of JacobiDN:

https://wolfram.com/xid/0cg6ccs28-jdw3cj

Find a root of JacobiZN[u,2/3]=1/7:

https://wolfram.com/xid/0cg6ccs28-oqkkjd


https://wolfram.com/xid/0cg6ccs28-exerl8

Visualization (3)
Plot JacobiZN functions for various values of parameter m:

https://wolfram.com/xid/0cg6ccs28-be6dck

Plot JacobiZN as a function of its parameter m:

https://wolfram.com/xid/0cg6ccs28-bk5tn

Plot the real part of JacobiZN[x+y,1/2]:

https://wolfram.com/xid/0cg6ccs28-ktrk2e

Plot the imaginary part of JacobiZN[x+y,1/2]:

https://wolfram.com/xid/0cg6ccs28-ftny62

Function Properties (2)
JacobiZN is periodic with period :

https://wolfram.com/xid/0cg6ccs28-gu00j

JacobiZN is additive quasiperiodic with a quasiperiod of :

https://wolfram.com/xid/0cg6ccs28-dx7c9i

JacobiZN is an odd function:

https://wolfram.com/xid/0cg6ccs28-c7lgaz

Differentiation (3)

https://wolfram.com/xid/0cg6ccs28-fdh136


https://wolfram.com/xid/0cg6ccs28-b55psk

Plot derivatives for parameter :

https://wolfram.com/xid/0cg6ccs28-b96qxq

Derivative with respect to parameter m:

https://wolfram.com/xid/0cg6ccs28-27pne

Integration (1)
Series Expansions (3)
Series expansion for JacobiZN[u,1/3] around :

https://wolfram.com/xid/0cg6ccs28-hyk0ml

Plot three approximations for JacobiZN[u,1/3]:

https://wolfram.com/xid/0cg6ccs28-gd8xla

Taylor series for JacobiZN[2,m] around :

https://wolfram.com/xid/0cg6ccs28-fc25wz

Plot series approximations for JacobiZN[2,m]:

https://wolfram.com/xid/0cg6ccs28-lvss5u

JacobiZN can be applied to power series:

https://wolfram.com/xid/0cg6ccs28-i960j

Function Identities and Simplifications (2)
Parity transformation and quasiperiodicity relations are automatically applied:

https://wolfram.com/xid/0cg6ccs28-ba0hli


https://wolfram.com/xid/0cg6ccs28-hzl15t


https://wolfram.com/xid/0cg6ccs28-doyox2

Automatic argument simplification:

https://wolfram.com/xid/0cg6ccs28-dg3pn4


https://wolfram.com/xid/0cg6ccs28-fpmqh

Function Representations (2)
JacobiZN is related to JacobiZeta function:

https://wolfram.com/xid/0cg6ccs28-hygnpt

TraditionalForm formatting:

https://wolfram.com/xid/0cg6ccs28-lfj9au

Applications (4)Sample problems that can be solved with this function
Express derivatives of Neville theta functions:

https://wolfram.com/xid/0cg6ccs28-hvjxbq


https://wolfram.com/xid/0cg6ccs28-8agfu

Supersymmetric zero‐energy solution of the Schrödinger equation in a periodic potential:

https://wolfram.com/xid/0cg6ccs28-mt2fqv

https://wolfram.com/xid/0cg6ccs28-hjd350
Define a solution using JacobiZN:

https://wolfram.com/xid/0cg6ccs28-cxcdd
Check that the function defined previously solves the Schrödinger equation:

https://wolfram.com/xid/0cg6ccs28-d70bdn

Plot the superpotential, the potential and the wavefunction:

https://wolfram.com/xid/0cg6ccs28-dnisgj

Define a conformal map using JacobiZN:

https://wolfram.com/xid/0cg6ccs28-h1lc2g

https://wolfram.com/xid/0cg6ccs28-is8vuv

Parameterization of genus‐1 constant mean-curvature Wente torus:

https://wolfram.com/xid/0cg6ccs28-b3aqm3

https://wolfram.com/xid/0cg6ccs28-bavxcd

https://wolfram.com/xid/0cg6ccs28-dnbulz
Visualize 3‐lobe, 5‐lobe, 7‐lobe and 11‐lobe tori:

https://wolfram.com/xid/0cg6ccs28-cuf9xh

Properties & Relations (2)Properties of the function, and connections to other functions
JacobiZN is defined in terms of JacobiEpsilon:

https://wolfram.com/xid/0cg6ccs28-dz8sim

JacobiZN[u,m] is a meromorphic extension of :

https://wolfram.com/xid/0cg6ccs28-4sx36

Wolfram Research (2020), JacobiZN, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiZN.html.
Text
Wolfram Research (2020), JacobiZN, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiZN.html.
Wolfram Research (2020), JacobiZN, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiZN.html.
CMS
Wolfram Language. 2020. "JacobiZN." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiZN.html.
Wolfram Language. 2020. "JacobiZN." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiZN.html.
APA
Wolfram Language. (2020). JacobiZN. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiZN.html
Wolfram Language. (2020). JacobiZN. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiZN.html
BibTeX
@misc{reference.wolfram_2025_jacobizn, author="Wolfram Research", title="{JacobiZN}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiZN.html}", note=[Accessed: 11-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_jacobizn, organization={Wolfram Research}, title={JacobiZN}, year={2020}, url={https://reference.wolfram.com/language/ref/JacobiZN.html}, note=[Accessed: 11-July-2025
]}