BUILT-IN MATHEMATICA SYMBOL

JacobiZeta

gives the Jacobi zeta function

.

MORE INFORMATION

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.

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Scope (9)

Evaluate for complex arguments and parameters:

Evaluate to high precision:

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

Simple exact results are generated automatically:

JacobiZeta threads element-wise over lists:

Series expansion at the origin in the parameter:

Find series expansions at branch points and branch cuts:

Find limits at branch cuts:

TraditionalForm formatting:

Generalizations & Extensions (2)

Infinite arguments give symbolic results:

JacobiZeta can be applied to a power series:

Applications (3)

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

Supersymmetric zero-energy 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 is insufficient to give a correct answer:

A larger setting for $MaxExtraPrecision can be needed:

The alternative definition

requires JacobiAmplitude:

Mathematica uses the following definition:

In traditional form the vertical separator must be used: