Integral Representations for Elliptic Functions

Abstract

We derive new integral representations for objects arising in the classical theory of elliptic functions: the Eisenstein series $E_s$, and Weierstrass' $\wp$ and $ζ$ 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.