  In the theory of elliptic functions and elliptic curves, the Weierstrass
$zeta$ function (which is essentially an antiderivative of the Weierstrass
$\wp$ function) plays a prominent role. Although it is not an elliptic
function, Eisenstein constructed a simple (non-holomorphic) completion of this
form which is doubly periodic. This theorem has begun to play an important role
in the theory of harmonic Maass forms, and was crucial to work of Guerzhoy as
well as Alfes, Griffin, Ono, and the author. In particular, this simple
completion of $\zeta$ provides a powerful method to construct harmonic Maass
forms of weight zero which serve as canonical lifts under the differential
operator $\xi_{0}$ of weight 2 cusp forms, and this has been shown in to have
deep applications to determining vanishing criteria for central values and
derivatives of twisted Hasse-Weil $L$-functions for elliptic curves.
  Here we offer a new and motivated proof of Eisenstein's theorem, relying on
the basic theory of differential operators for Jacobi forms together with a
classical identity for the first quasi-period of a lattice. A quick inspection
of the proof shows that it also allows one to easily construct more general
non-holomorphic elliptic functions.
