The best-known 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: