Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











JacobiCD






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiCD[z,m] > Introduction to the Jacobi elliptic functions





The best-known properties and formulas for Jacobi functions


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

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

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:

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

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:

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.

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 :

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 :

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 :

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

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

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

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

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:

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

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 :

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:

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:

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: