Browsing the wealth of functions available in *Mathematica*, one encounters two not-so-common functions: the "elliptic logarithm", which is an elliptic integral in another garb, and the "elliptic exponential", its inverse which is an elliptic function. The only substantial thing I know about this pair is that they can be expressed in terms of the Weierstrass ℘ function and its inverse (and in fact *Mathematica* internally uses the elliptic logarithm/exponential to compute the Weierstrass functions).

My questions on this pair are the following:

A bit of searching around seems to indicate that these are used in the theory of elliptic curves. Considering that the other elliptic integrals and the elliptic functions of Jacobi and Weierstrass have much wider applicability (e.g. in physics), why are these two seemingly marginalized? (Or, are there references where using these instead of the classical functions made for better exposition?)

Why exactly would one use an elliptic logarithm and/or an elliptic exponential? More precisely, why would not the other more common elliptic integrals/functions suffice that one has to resort to these? Do they satisfy more convenient identities than the classical one?