- Title
- Compressed lattice sums arising from the Poisson equation
- Creator
- Bailey, David H.; Borwein, Jonathan M.
- Relation
- Boundary Value Problems Vol. 2013, Issue 75
- Publisher Link
- http://dx.doi.org/10.1186/1687-2770-2013-75
- Publisher
- SpringerOpen
- Resource Type
- journal article
- Date
- 2013
- Description
- Purpose: In recent years attention has been directed to the problem of solving the Poisson equation, either in engineering scenarios (computational) or in regard to crystal structure (theoretical). Methods: In (Bailey et al. in J. Phys. A, Math. Theor. 46:115201, 2013, doi:10.1088/1751-8113/46/11/115201) we studied a class of lattice sums that amount to solutions of Poisson’s equation, utilizing some striking connections between these sums and Jacobi ϑ-function values, together with high-precision numerical computations and the PSLQ algorithm to find certain polynomials associated with these sums. We take a similar approach in this study. Results: We were able to develop new closed forms for certain solutions and to extend such analysis to related lattice sums. We also alluded to results for the compressed sum ϕ2(x,y,d):=1π2∑m,n∈Ocos(πmx)cos(πnd√y)m2+dn2,(1) where d>0, x, y are real numbers and Odenotes the odd integers. In this paper we first survey the earlier work and then discuss the sum (1) more completely. Conclusions: As in the previous study, we find some surprisingly simple closed-form evaluations of these sums. In particular, we find that in some cases these sums are given by 1/π logA, where A is an algebraic number. These evaluations suggest that a deep theory interconnects all such summations.
- Subject
- lattice sums; Poisson equation; experimental mathematics; high-precision computation
- Identifier
- http://hdl.handle.net/1959.13/940198
- Identifier
- uon:12969
- Identifier
- ISSN:1687-2762
- Language
- eng
- Full Text
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