- Title
- Lattice sums arising from the Poisson equation
- Creator
- Bailey, D. H.; Borwein, J. M.; Crandall, R. E.; Zucker, I. J.
- Relation
- Journal of Physics A: Mathematical and Theoretical Vol. 46, Issue 11
- Publisher Link
- http://dx.doi.org/10.1088/1751-8113/46/11/115201
- Publisher
- Institute of Physics Publishing
- Resource Type
- journal article
- Date
- 2013
- Description
- In recent times, attention has been directed to the problem of solving the Poisson equation, either in engineering scenarios (computational) or in regard to crystal structure (theoretical). Herein we study a class of lattice sums that amount to Poisson solutions, namely the n-dimensional forms ⏀n(r₁,...,rn) = 1/π²[formula could not be replicated]. By virtue of striking connections with Jacobi ϑ-function values, we are able to develop new closed forms for certain values of the coordinates rk, and extend such analysis to similar lattice sums. A primary result is that for rational x, y, the natural potential ⏀²(x, y) is 1/π log A where A is an algebraic number. Various extensions and explicit evaluations are given. Such work is made possible by number-theoretical analysis, symbolic computation and experimental mathematics, including extensive numerical computations using up to 20,000-digit arithmetic.
- Subject
- lattice sums; Poisson equation; Madelung constants
- Identifier
- http://hdl.handle.net/1959.13/940012
- Identifier
- uon:12924
- Identifier
- ISSN:1751-8113
- Rights
- This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at http://dx.doi.org/10.1088/1751-8113/46/11/115201
- Language
- eng
- Full Text
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