JacobiZeta

JacobiZeta[ϕ,m]

gives the Jacobi zeta function TemplateBox[{phi, m}, JacobiZeta].

Details

Examples

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Basic Examples  (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion about the origin:

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 threads elementwise over lists:

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 (d)/(dphi)TemplateBox[{phi, {1, /, 2}}, JacobiZeta]=0:

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 TemplateBox[{{pi, /, 3}, z}, JacobiZeta]:

Plot the imaginary part of TemplateBox[{{pi, /, 3}, z}, JacobiZeta]:

Function Properties  (6)

JacobiZeta is not an analytic function:

However, for fixed , TemplateBox[{x, y}, JacobiZeta] is an analytic function of :

Thus, for example, TemplateBox[{x, {1, /, 3}}, JacobiZeta] has no singularities or discontinuities:

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

TemplateBox[{x, {1, /, 3}}, JacobiZeta] is not injective:

TemplateBox[{x, {1, /, 3}}, JacobiZeta] is not surjective:

TemplateBox[{x, {1, /, 2}}, JacobiZeta] is neither non-negative nor non-positive:

TemplateBox[{x, {1, /, 2}}, JacobiZeta] 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 TemplateBox[{phi, {1, /, 2}}, JacobiZeta] around :

Taylor expansion at the origin in the parameter :

Plot the first three approximations for TemplateBox[{{-, {pi, /, 3}}, m}, JacobiZeta] 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:

TraditionalForm formatting:

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 phi=TemplateBox[{u, m}, JacobiAmplitude], 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 TemplateBox[{phi, m}, JacobiZeta] 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 Z(u|m)=TemplateBox[{u, m}, JacobiEpsilon]-u TemplateBox[{m}, EllipticE]/TemplateBox[{m}, EllipticK], where TemplateBox[{u, m}, JacobiEpsilon] is JacobiEpsilon[u,m]:

To avoid confusion, JacobiZN uses a different TraditionalForm:

In traditional form, the vertical separator must be used:

Wolfram Research (1991), JacobiZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiZeta.html (updated 2020).

Text

Wolfram Research (1991), JacobiZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiZeta.html (updated 2020).

CMS

Wolfram Language. 1991. "JacobiZeta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/JacobiZeta.html.

APA

Wolfram Language. (1991). JacobiZeta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiZeta.html

BibTeX

@misc{reference.wolfram_2023_jacobizeta, author="Wolfram Research", title="{JacobiZeta}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiZeta.html}", note=[Accessed: 18-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_jacobizeta, organization={Wolfram Research}, title={JacobiZeta}, year={2020}, url={https://reference.wolfram.com/language/ref/JacobiZeta.html}, note=[Accessed: 18-March-2024 ]}