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".
  • For certain special arguments, JacobiZeta automatically evaluates to exact values.
  • JacobiZeta can be evaluated to arbitrary numerical precision.
  • JacobiZeta automatically threads over lists.

Examples

open allclose all

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  (23)

Numerical Evaluation  (4)

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:

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}, {x, +, {ⅈ,  , y}}}, JacobiZeta]:

Plot the imaginary part of TemplateBox[{{pi, /, 3}, {x, +, {ⅈ,  , y}}}, JacobiZeta]:

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  (3)

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:

Possible Issues  (4)

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

A larger setting for $MaxExtraPrecision may be needed:

The alternative definition requires JacobiAmplitude:

The Wolfram Language uses the following definition:

In traditional form the vertical separator must be used:

Introduced in 1991
 (2.0)