A new construction of Eisenstein's completion of the Weierstrass zeta function (Q2790254)

Let \(\zeta_\Lambda(z)\) be the Weierstrass zeta function defined for a lattice \(\Lambda\) in \(\mathbb C\). The function \(\zeta_\Lambda(z)\) is not \(\Lambda\)-invariant with respect to the translation of \(z\). Eisenstein proved that ``Eisenstein's completion of \(\zeta_\Lambda(z)\)'' obtained from \(\zeta_\Lambda(z)\) by adding a linear form of \(z\) and \(\overline z\) is \(\Lambda\)-invariant. In this article, the author gives a new and interesting proof of Eisenstein's result by using a relation between the logarithmic derivative of a Jacobi form of weight \(1/2\) and \(\zeta_\Lambda(z)\) and a canonical raising operator on Jacobi forms.