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: