The best-known properties and formulas for Jacobi functions

Real values for real arguments

For real values of arguments and , the values of all Jacobi functions , , , , , , , , , , , , and are real (or infinity).

Simple values at zero

All thirteen Jacobi functions , , , , , , , , , , , , and have the following simple values at the origin:

Specific values for specialized parameter values

All Jacobi functions , , , , , , , , , , , , and can be represented through elementary functions when or . The twelve elliptic functions degenerate into trigonometric and hyperbolic functions:

All Jacobi functions , , , , , , , , , , , , and have very simple values at :

The twelve Jacobi functions , , , , , , , , , , , and have the following values at the half‐quarter‐period points:

The partial derivatives of all Jacobi functions , , , , , , , , , , , , and at the points , , or can be represented through trigonometric functions, for example:

Analyticity

All Jacobi functions , , , , , , , , , , , , and are analytical meromorphic functions of and that are defined over .

Poles and essential singularities

The amplitude function does not have poles and essential singularities with respect to and .

For fixed , all Jacobi functions , , , , , , , , , , , and have an infinite set of singular points, including simple poles in finite points and an essential singular point .

The following formulas describe the sets of the simple poles for the corresponding Jacobi functions:

The values of the residues of the Jacobi functions at the simple poles are given by the following formulas:

Branch points and branch cuts

For fixed , all Jacobi functions , , , , , , , , , , , and are meromorphic functions in that have no branch points and branch cuts.

For fixed , all Jacobi functions , , , , , , , , , , , and do not have branch points and branch cuts.

Periodicity

The Jacobi amplitude is a pseudo‐periodic function with respect to with period and pseudo‐period :

The Jacobi functions , , , and are doubly periodic functions with respect to with periods and . The Jacobi functions , , , and are doubly periodic functions with respect to with periods and . The Jacobi functions , , , and are doubly periodic functions with respect to with periods and . That periodicity can be described by the following formulas:

The periodicity of Jacobi functions follow from more general formulas that also describe quasi‐periodicity situations such as :

Parity and symmetry

All Jacobi functions , , , , , , , , , , , , and have mirror symmetry:

The Jacobi functions , , , , , and are even functions with respect to :

The Jacobi functions , , , , , , and are odd functions with respect to :

Series representations

The Jacobi functions , , , , , , , , , , , , and have the following series expansions at the point :

The Jacobi functions , , , , , , , , , , , , and have the following series expansions at the point :

The Jacobi functions , , , , , , , , , , , , and have the following series expansions at the point :

q-series representations

The Jacobi functions , , , , , , , , , , , , and have the following so-called q‐series representations:

where is the elliptic nome and is the complete elliptic integral.

Product representations

The twelve Jacobi functions , , , , , , , , , , , and have the following product representations:

where is the elliptic nome and is the complete elliptic integral.

Transformations

The amplitude function satisfies numerous relations that allow for transformations of its arguments, for example:

The twelve Jacobi functions , , , , , , , , , , , and with specific arguments can sometimes be represented through elliptic functions with other mostly simpler arguments, for example:

The twelve Jacobi functions , , , , , , , , , , , and with the argument complex can be represented through elliptic functions with arguments and , for example:

The twelve Jacobi functions , , , , , , , , , , , and satisfy the following half‐angle formulas:

The twelve Jacobi functions , , , , , , , , , , , and satisfy the following double-angle (or multiplication) formulas:

Identities

The twelve Jacobi functions , , , , , , , , , , , and satisfy the following nonlinear functional equations:

Simple representations of derivatives

The derivatives of all Jacobi functions , , , , , , , , , , , , and with respect to variable have rather simple and symmetrical representations that can be expressed through other Jacobi functions:

The derivatives of all Jacobi functions , , , , , , , , , , , , and with respect to variable have more complicated representations that include other Jacobi functions and the elliptic integral :

Integration

The indefinite integrals of the twelve Jacobi functions , , , , , , , , , , , and with respect to variable can be expressed through Jacobi and elementary functions by the following formulas:

Differential equations

The Jacobi amplitude satisfies the following differential equations:

All Jacobi functions , , , , , , , , , , , , and are special solutions of ordinary second-order nonlinear differential equations:

The twelve Jacobi functions , , , , , , , , , , , and satisfy very complicated ordinary differential equations with respect to variable , for example:

Zeros

All Jacobi functions , , , , , , , , , , , , and are equal to zero in the points , where is the complete elliptic integral of the first kind and , are even or odd integers: