Integral representations for elliptic functions

We derive new integral representations for constituents of the classical theory of elliptic functions: the Eisenstein series, and Weierstrass' ℘ and ζ functions. The derivations proceed from the Laplace–Mellin representation of multipoles, and an elementary lemma on the summation of 2D geometri...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications Vol. 316; no. 1; pp. 142 - 160
Main Authors: Dienstfrey, Andrew, Huang, Jingfang
Format: Journal Article
Language:English
Published: San Diego, CA Elsevier Inc 01.04.2006
Elsevier
Subjects:
Eisenstein series
Elliptic functions
Exact sciences and technology
Functions of a complex variable
Lattice sums
Mathematical analysis
Mathematics
Planewave expansions
Sciences and techniques of general use
Several complex variables and analytic spaces
Special functions
Planewave expansions
Eisenstein series
Elliptic functions
Lattice sums
Complex function
Lattice sum
Analytic continuation
Elliptic function
Weierstrass functions
Mathematical analysis
Plane wave
Entire function
Integral representation
Summation
ISSN:0022-247X, 1096-0813
Online Access:Get full text
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Summary:We derive new integral representations for constituents of the classical theory of elliptic functions: the Eisenstein series, and Weierstrass' ℘ and ζ 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 E ˜ 2 below, as a function of summation over increasingly large rectangles with arbitrary fixed aspect ratio. 1 1 Contribution of US Government, not subject to copyright.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2005.04.058