# Lattice sums arising from the Poisson equation

Bailey, David H. and Borwein, Jonathan M. and Crandall, Richard E. and Zucker, I. John (2013) Lattice sums arising from the Poisson equation. ournal of Physics A: Mathematical and Theoretical, 46 .

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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. By virtue of striking connections with Jacobi theta-function values, we are able to develop new closed forms for certain values, and extend such analysis to similar lattice sums. A primary result is that for rational x; y, the natural two dimensional potential is of the form $\log(A)/\pi$ 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.
Item Type: Article 11-xx Number theory > 11Mxx Zeta and $L$-functions: analytic theory34-xx Ordinary differential equations > 34Bxx Boundary value problems UNSPECIFIED Jonathan Borwein 10 Nov 2012 09:07 28 Nov 2014 15:18 https://docserver.carma.newcastle.edu.au/id/eprint/1458