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For real values of arguments  ,  , and  (with  ,  ,  ) the values of all complete elliptic integrals  ,  , and  are real.
All complete elliptic integrals  ,  , and  are equal to  at the origin:
All complete elliptic integrals  ,  , and  can be represented through elementary or other functions when  ,  , or  ,  or  , or  or  :
At any infinity, the complete elliptic integrals  ,  , and  have the following values:
The complete elliptic integrals  and  are analytical functions of  , which are defined over the whole complex  ‐plane.
The complete elliptic integral  is an analytical function of  and  , which is defined over  .
All complete elliptic integrals  ,  , and  do not have poles and essential singularities.
The complete elliptic integrals  and  have two branch points:  and  .
They are single‐valued functions on the  ‐plane cut along the interval  . They are continuous from below on the interval  :
For fixed  , the function  has two branch points at  and  . For fixed  , the function  has two branch points at  and  .
All complete elliptic integrals  ,  , and  are not periodical functions.
All complete elliptic integrals  ,  , and  have mirror symmetry:
All complete elliptic integrals  ,  , and  have the following series expansions at the point  :
The complete elliptic integrals  and  have the following series expansions at the point  :
The complete elliptic integrals  and  have the following series expansions at the point  :
The complete elliptic integral  has the following series expansions at the point  :
The complete elliptic integral  has the following series expansions at the point  :
The complete elliptic integral  has the following series expansions at the point  :
The complete elliptic integral  has the following series expansions at the point  :
The previous formulas can be rewritten in summed forms of the truncated series expansion near corresponding points  ,  , or  :
Some elliptic integrals have special series representations through the elliptic nome  and inverse Jacobi elliptic functions by the formulas:
The complete elliptic integrals  ,  , and  have the following integral representations:
The complete elliptic integrals  ,  , and  satisfy numerous identities, for example:
The first derivatives of all complete elliptic integrals  ,  , and  with respect to their variables can also be represented through complete elliptic integrals by the following formulas:
The symbolic  derivatives of all complete elliptic integrals  ,  , and  with respect to their variables can be represented through Gauss classical or regularized hypergeometric functions by the following formulas:
The indefinite integrals of all complete elliptic integrals  ,  , and  with respect to their variables can be expressed through complete elliptic integrals (or through hypergeometric functions of two variables) by the following formulas:
All complete elliptic integrals  ,  , and  satisfy ordinary linear differential equations:
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