Integral representations for elliptic functions
Copyright policy )Integral representations for elliptic functions
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.
21 pages, 1 figure
- University of North Carolina at Chapel Hill United States
- National Institute of Standards and Technology United States
Mathematics - Complex Variables, Applied Mathematics, Elliptic functions, Planewave expansions, Lattice sums, Elliptic functions and integrals, Eisenstein series, Mathematics - Classical Analysis and ODEs, plane wave expansions, lattice sums, Classical Analysis and ODEs (math.CA), FOS: Mathematics, elliptic functions, Complex Variables (math.CV), 33E05 (primary) 33E20, 11M36 (secondary), Analysis
Mathematics - Complex Variables, Applied Mathematics, Elliptic functions, Planewave expansions, Lattice sums, Elliptic functions and integrals, Eisenstein series, Mathematics - Classical Analysis and ODEs, plane wave expansions, lattice sums, Classical Analysis and ODEs (math.CA), FOS: Mathematics, elliptic functions, Complex Variables (math.CV), 33E05 (primary) 33E20, 11M36 (secondary), Analysis
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