  We derive new integral representations for objects arising in the classical
theory of elliptic functions: the Eisenstein series $E_s$, and Weierstrass'
$\wp$ and $\zeta$ functions. The derivations proceed from the Laplace-Mellin
transformation for multipoles, and an elementary lemma on the summation of 2D
geometric series. In addition, we present new results concerning the analytic
continuation of the Eisenstein series as an entire function in $s$, and the
value of the conditionally convergent series, denoted by $\widetilde{E}_2$
below, as a function of summation over increasingly large rectangles with
arbitrary fixed aspect ratio.
