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The elliptic exponent  , its derivative  , and the elliptic logarithm  have the following values at the origin point:
The elliptic exponent  has the following value at the specialized point  :
The elliptic exponent  and its derivative  are vector‐valued functions of  ,  , and  , which are analytic in each component, and they are defined over  .
The elliptic logarithm  is an analytical function of  ,  ,  ,  , which is defined in  .
The elliptic exponent  , its derivative  , and the elliptic logarithm  have complicated branch cuts.
The elliptic logarithm  does not have poles and essential singularities.
The elliptic exponent  , its derivative  , and the elliptic logarithm  do not have periodicity.
The elliptic exponent  , its derivative  , and the elliptic logarithm  have mirror symmetry:
The elliptic logarithm  has the following integral representation:
The elliptic exponent  satisfies the following identities including the complete elliptic integral  :
The first derivatives of elliptic exponent  and the elliptic logarithm  have the following representations:
The elliptic exponent  , its derivative  , and the elliptic logarithm  satisfy the following ordinary nonlinear differential equations:
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