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:
