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The inverse Jacobi functions  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , and  have the following simple values at the origin:
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:
The inverse Jacobi functions  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , and  are analytical functions of  and  that are defined over  .
The inverse Jacobi functions  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , and  do not have poles and essential singularities with respect to  and  .
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  .
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  :
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  :
The inverse Jacobi functions  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , and  have the following integral representations, which can be used for their definitions:
Some inverse Jacobi functions satisfy additional formulas, for example:
The inverse Jacobi functions  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , and  satisfy nonlinear functional equations:
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  :
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:
The twelve inverse Jacobi functions  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , and  are the special solutions of the following second-order ordinary nonlinear differential equations:
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