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The incomplete elliptic integrals  ,  ,  and  are equal to  at the origin points:
The incomplete elliptic integrals  ,  ,  and  for particular argument values can be evaluated in closed forms, for example:
At infinities, the incomplete elliptic integrals  ,  ,  and  have the following values:
The incomplete elliptic integrals  ,  and  are analytical functions of  and  , which are defined over  .
The incomplete elliptic integral  is an analytical function of  ,  , and  , which is defined over  .
The incomplete elliptic integrals  ,  ,  and  do not have poles and essential singularities with respect to their variables.
For fixed  , the functions  ,  , and  have an infinite number of branch points at  and  .
They have complicated branch cut.
For fixed  ,  , the function  has two branch points at  and  . For fixed  ,  , the function  has an infinite number of branch points at  ,  and  . For fixed  ,  , the function  does not have branch points.
The incomplete elliptic integrals  ,  and  are quasi‐periodic functions with respect to  :
The incomplete elliptic integral  is a periodic function with respect to  with period  :
The incomplete elliptic integrals  ,  ,  and  have mirror symmetry:
The incomplete elliptic integrals  ,  ,  and  are odd functions with respect to  .
The incomplete elliptic integral  has the following series expansions at the point  :
The incomplete elliptic integrals  ,  ,  and  have the following series expansions at the point  :
The incomplete elliptic integrals  ,  ,  and  have the following series expansions at the point  :
The incomplete elliptic integrals  ,  and  have the following series expansions at the point  :
The incomplete elliptic integrals  and  have the following series expansions at the point  :
The incomplete elliptic integrals  and  have the following series expansions at the point  :
The incomplete elliptic integrals  ,  ,  and  can be represented through different kinds of series, for example:
where  is an elliptic nome and  is a complete elliptic integral.
The incomplete elliptic integrals  ,  ,  and  have the following integral representations:
The incomplete elliptic integrals  ,  ,  and  with linear arguments can sometimes be simplified, for example:
In some cases, simplification can be realized for more complicated arguments, for example:
Sums of the incomplete elliptic integrals  ,  ,  and  with different values  and  can be evaluated by the following summation formulas:
The incomplete elliptic integrals  ,  ,  , and  satisfy numerous identities, for example:
The first derivative of the incomplete elliptic integral  with respect to variable  has the following representation:
The previous formula can be generalized to the  derivative:
The first derivatives of the incomplete elliptic integrals  ,  ,  , and  with respect to variable  have the following simple representations:
The previous formulas can be generalized to the arbitrary-order symbolic derivatives:
The first derivatives of the incomplete elliptic integrals  ,  ,  and  with respect to variable  have the following representations:
The previous formulas can be generalized to the arbitrary-order symbolic derivatives:
The indefinite integral of the incomplete elliptic integral  with respect to variable  has the following representation through the infinite series that includes the Appell  hypergeometric function of two variables:
The indefinite integrals of all incomplete elliptic integrals  ,  ,  , and  with respect to variable  have the following representations:
The indefinite integrals of incomplete elliptic integrals  ,  , and  with respect to variable  can be expressed in closed forms by the following formulas:
The incomplete elliptic integrals  ,  ,  , and  satisfy the following differential equations:
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