Distributive and analytic properties of lattice sums

McPhedran, RC; Smith, GH; Nicorovici, NA; Botten, LC

Distributive and analytic properties of lattice sums

McPhedran, RC Smith, GH Nicorovici, NA Botten, LC

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Publication Type:: Journal Article
Citation:: Journal of Mathematical Physics, 2004, 45 (7), pp. 2560 - 2578
Issue Date:: 2004-07-01

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Field	Value	Language
dc.contributor.author	McPhedran, RC	en_US
dc.contributor.author	Smith, GH	en_US
dc.contributor.author	Nicorovici, NA	en_US
dc.contributor.author	Botten, LC https://orcid.org/0000-0001-6471-6954	en_US
dc.date.issued	2004-07-01	en_US
dc.identifier.citation	Journal of Mathematical Physics, 2004, 45 (7), pp. 2560 - 2578	en_US
dc.identifier.issn	0022-2488	en_US
dc.identifier.uri	http://hdl.handle.net/10453/3420
dc.description.abstract	We use sums over Bessel functions of the first kind to derive a convenient form of the Poisson summation identity relating sums over direct lattices in two dimensions to sums over reciprocal lattices. After three simple examples of the use of the identity, we consider sums over complex powers of the radial distance to lattice points, and also sums incorporating factors exp(4imψp) depending on angles of lattice points. We study the distribution of zeros of lattice sums, and show two which seemingly obey the Riemann hypothesis, and a third which does not. We provide a reflection formula for angular lattice sums, and a Macdonald function sum for the lowest order angular lattice sum. © 2004 American Institute of Physics.	en_US
dc.relation	http://purl.org/au-research/grants/arc/LE0347499
dc.relation.ispartof	Journal of Mathematical Physics	en_US
dc.relation.isbasedon	10.1063/1.1755861	en_US
dc.subject.classification	Mathematical Physics	en_US
dc.title	Distributive and analytic properties of lattice sums	en_US
dc.type	Journal Article
utslib.citation.volume	7	en_US
utslib.citation.volume	45	en_US
utslib.for	0205 Optical Physics	en_US
utslib.for	0105 Mathematical Physics	en_US
utslib.for	01 Mathematical Sciences	en_US
utslib.for	02 Physical Sciences	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Science
pubs.organisational-group	/University of Technology Sydney/Faculty of Science/School of Mathematical and Physical Sciences
utslib.copyright.status	closed_access
pubs.issue	7	en_US
pubs.publication-status	Published	en_US
pubs.volume	45	en_US

Abstract:

We use sums over Bessel functions of the first kind to derive a convenient form of the Poisson summation identity relating sums over direct lattices in two dimensions to sums over reciprocal lattices. After three simple examples of the use of the identity, we consider sums over complex powers of the radial distance to lattice points, and also sums incorporating factors exp(4imψp) depending on angles of lattice points. We study the distribution of zeros of lattice sums, and show two which seemingly obey the Riemann hypothesis, and a third which does not. We provide a reflection formula for angular lattice sums, and a Macdonald function sum for the lowest order angular lattice sum. © 2004 American Institute of Physics.

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http://hdl.handle.net/10453/3420