Mathematics – Classical Analysis and ODEs
Scientific paper
2004-09-13
Mathematics
Classical Analysis and ODEs
21 pages, 1 figure
Scientific paper
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.
Dienstfrey A.
Huang Jianwei
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