The bestknown 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 :
qseries representations
The Jacobi functions , , , , , , , , , , , , and have the following socalled 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 doubleangle (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 secondorder 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:
