# JacobiZeta

JacobiZeta[ϕ,m]

gives the Jacobi zeta function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The Jacobi zeta function is given in terms of elliptic integrals by .
• Argument conventions for elliptic integrals are discussed in "Elliptic Integrals and Elliptic Functions".
• JacobiZeta[ϕ,m] has branch cut discontinuities at and at .
• For certain special arguments, JacobiZeta automatically evaluates to exact values.
• JacobiZeta can be evaluated to arbitrary numerical precision.
• JacobiZeta automatically threads over lists.
• JacobiZeta can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

## Scope(30)

### Numerical Evaluation(5)

Evaluate to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate JacobiZeta efficiently at high precision:

JacobiZeta can be used with Interval and CenteredInterval objects:

### Specific Values(5)

Simple exact results are generated automatically:

Exact values after FunctionExpand is applied:

Value at infinity:

Find a local maximum as a root of :

JacobiZeta is an odd function with respect to the first argument:

### Visualization(3)

Plot JacobiZeta as a function of its first parameter :

Plot JacobiZeta as a function of its second parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(6)

JacobiZeta is not an analytic function:

However, for fixed , is an analytic function of :

Thus, for example, has no singularities or discontinuities: is neither nondecreasing nor nonincreasing: is not injective: is not surjective: is neither non-negative nor non-positive: is neither convex nor concave:

### Differentiation and Integration(4)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate with respect to its second parameter :

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

### Series Expansions(4)

Taylor expansion for JacobiZeta:

Plot the first three approximations for around :

Taylor expansion at the origin in the parameter :

Plot the first three approximations for around :

Find series expansions at a branch point:

JacobiZeta can be applied to a power series:

### Function Representations(3)

Primary definition:

Relation to other elliptictype functions:

## Applications(3)

Plot of the real part of JacobiZeta over the complex plane:

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

Check the Schrödinger equation:

Plot the superpotential, the potential and the wave function:

Define a conformal map:

## Properties & Relations(5)

Use FunctionExpand to express JacobiZeta in terms of incomplete elliptic integrals:

Expand special cases:

Some special cases require argument restrictions:

Numerically find a root of a transcendental equation:

For real arguments, if , then JacobiZN[u,m]JacobiZeta[ϕ,m] for :

JacobiZeta[ϕ,m] is real valued for real arguments subject to :

## Possible Issues(4)

Machine-precision input may be insufficient to give a correct answer:

A larger setting for \$MaxExtraPrecision may be needed: JacobiZeta, function of amplitude , is not to be confused with JacobiZN, sometimes denoted as and a function of elliptic argument :

The Wolfram Language JacobiZeta[ϕ,m] is a function of amplitude and uses the following definition:

JacobiZN[u,m] is a function of elliptic argument and uses the definition , where is JacobiEpsilon[u,m]:

To avoid confusion, JacobiZN uses a different TraditionalForm:

In traditional form, the vertical separator must be used: