# JacobiZN

JacobiZN[u,m]

gives the Jacobi zeta function .

# 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

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## Basic Examples(3)

Evaluate numerically:

## Scope(23)

### Numerical Evaluation(4)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiZN efficiently at higher precision:

### Specific Values(3)

Simple exact values are generated automatically:

JacobiZN has poles coinciding with poles of JacobiDN:

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

### Visualization(3)

Plot JacobiZN functions for various values of parameter m:

Plot JacobiZN as a function of its parameter m:

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

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

### Function Properties(2)

JacobiZN is periodic with period :

JacobiZN is additive quasiperiodic with a quasiperiod of :

JacobiZN is an odd function:

### Differentiation(3)

First derivative:

Higher-order derivatives:

Plot derivatives for parameter :

Derivative with respect to parameter m:

### Integration(1)

Indefinite integral of JacobiZN:

### Series Expansions(3)

Series expansion for JacobiZN[u,1/3] around :

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

Taylor series for JacobiZN[2,m] around :

Plot series approximations for JacobiZN[2,m]:

JacobiZN can be applied to power series:

### Function Identities and Simplifications(2)

Parity transformation and quasiperiodicity relations are automatically applied:

Automatic argument simplification:

### Function Representations(2)

JacobiZN is related to JacobiZeta function:

## Applications(4)

Express derivatives of Neville theta functions:

Supersymmetric zeroenergy solution of the Schrödinger equation in a periodic potential:

Define a solution using JacobiZN:

Check that the function defined previously solves the Schrödinger equation:

Plot the superpotential, the potential and the wavefunction:

Define a conformal map using JacobiZN:

Parameterization of genus1 constant mean-curvature Wente torus:

Visualize 3lobe, 5lobe, 7lobe and 11lobe tori:

## Properties & Relations(2)

JacobiZN is defined in terms of JacobiEpsilon:

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