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Integral representations for elliptic functions. (English) Zbl 1148.33017
Summary: We derive new integral representations for constituents of the classical theory of elliptic functions: the Eisenstein series, and Weierstrass’ $\wp$ and $\zeta$ functions. The derivations proceed from the Laplace-Mellin representation of multipoles, and an elementary lemma on the summation of 2D geometric series. In addition, we present results concerning the analytic continuation of the Eisenstein series to an entire function in the complex plane, and the value of the conditionally convergent series, denoted by ${\stackrel{˜}{E}}_{2}$ below, as a function of summation over increasingly large rectangles with arbitrary fixed aspect ratio.
##### MSC:
 3.3e+06 Elliptic functions and integrals