The bestknown properties and formulas for elliptic exp and elliptic log
Values at zero
The elliptic exponent , its derivative , and the elliptic logarithm have the following values at the origin point:
Specific values for specialized parameter
The elliptic exponent has the following value at the specialized point :
Analyticity
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.
Poles and essential singularities
The elliptic logarithm does not have poles and essential singularities.
Periodicity
The elliptic exponent , its derivative , and the elliptic logarithm do not have periodicity.
Parity and symmetry
The elliptic exponent , its derivative , and the elliptic logarithm have mirror symmetry:
Integral representations
The elliptic logarithm has the following integral representation:
Identities
The elliptic exponent satisfies the following identities including the complete elliptic integral :
Simple representations of derivatives
The first derivatives of elliptic exponent and the elliptic logarithm have the following representations:
Differential equations
The elliptic exponent , its derivative , and the elliptic logarithm satisfy the following ordinary nonlinear differential equations:
