The bestknown 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 secondorder ordinary nonlinear differential equations:
