2004/09/13 by A. Dienstfrey, J. Huang, Dienstfrey, A. +1
Mathematics · #11M36 (secondary) #33E05 (primary) 33E20 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #math.CA #math.CV #msc:11M36 #msc:33E05 #msc:33E20
paper · pdf · doi:10.48550/arxiv.math/0409216
21 pages, 1 figure
arxiv created 2004/09/13 · arxiv updated 2009/12/01
We derive new integral representations for objects arising in the classical theory of elliptic functions: the Eisenstein series Es, 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 \widetildeE2 below, as a function of summation over increasingly large rectangles with arbitrary fixed aspect ratio.