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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:
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