Introduction to the inverse Jacobi elliptic functions

General

Definitions of the inverse Jacobi functions

The inverses of the twelve Jacobi elliptic functions , , , , , , , , , , , and are defined by the following formulas:

It is obvious that the inverses of the twelve Jacobi elliptic functions are actually the definite elliptic integrals and can be expressed through the Legendre elliptic integrals.

A quick look at the inverse Jacobi functions

Here is a quick look at the graphics for the inverse Jacobi functions along the real axis.

Connections within the group of inverse Jacobi functions and with other related function groups

Representations through more general functions

Representations through related equivalent functions

Relations to inverse functions

Representations through other inverse Jacobi functions

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

Representations of through other inverse Jacobi functions are:

The best-known properties and formulas for inverse Jacobi functions

Simple values at zero

The inverse Jacobi functions , , , , , , , , , , , and have the following simple values at the origin:

Specific values for specialized parameter values

The inverse Jacobi functions , , , , , , , , , , , and can be represented through elementary functions when or . In these cases they degenerate into inverse trigonometric and inverse hyperbolic functions. If , they can be represented through the elliptic integrals and :

At the points , and , the inverse Jacobi functions , , , , , , , , , , , have the following representations through the elliptic integrals and :

At the points or , the inverse Jacobi functions , , , , , , , , , , , and have the following values:

Analyticity

The inverse Jacobi functions , , , , , , , , , , , and are analytical functions of and that are defined over .

Poles and essential singularities

The inverse Jacobi functions , , , , , , , , , , , and do not have poles and essential singularities with respect to and .

Branch points and branch cuts

For fixed , the point is the branch point for all twelve inverse Jacobi functions. Other branch points are the following: for , for , for , for , for , and for , for , for , for , for , and for , and for .

For fixed , the point is the branch point for all twelve inverse Jacobi functions. There are four or five other branch points that include the following: for , for , for , for , for , for , for , for , for , for , for , and for .

Parity and symmetry

The inverse Jacobi functions , , , , , , , , , , , and have mirror symmetry:

Nine inverse Jacobi functions , , , , , , , , have the following quasi‐reflection symmetry with respect to :

The other three inverse Jacobi functions , , and are odd functions with respect to :

Series representations

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

The previous expansions are the particular cases of the following series representations of the twelve inverse Jacobi functions near the point :

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

The previous expansions are the particular cases of the following series representations of the twelve inverse Jacobi functions near the point :

Integral representations

The inverse Jacobi functions , , , , , , , , , , , and have the following integral representations, which can be used for their definitions:

Transformations

Some inverse Jacobi functions satisfy additional formulas, for example:

Identities

The inverse Jacobi functions , , , , , , , , , , , and satisfy nonlinear functional equations:

Representations of derivatives

The derivatives of the inverse Jacobi functions , , , , , , , , , , , and with respect to variable can be expressed through direct and inverse Jacobi functions:

The previous formulas can be generalized to the following symbolic derivatives of the order with respect to variable :

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

The previous formulas can be generalized to the following symbolic derivatives of the order with respect to variable :

Integration

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

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

Differential equations

The twelve inverse Jacobi functions , , , , , , , , , , , and are the special solutions of the following second-order ordinary nonlinear differential equations:

Applications of the inverse Jacobi functions

Fields of application of the inverse Jacobi functions include most of the application areas of the direct functions. In many applications, the need for the inversion of the elliptic function fortunately does not arise. In cases where inversion is needed, the invese Jacobi elliptic functions are very useful tools for calculations.