Schlömilch series and grating sums

McPhedran, RC; Nicorovici, NA; Botten, LC

Schlömilch series and grating sums

McPhedran, RC Nicorovici, NA Botten, LC

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Publication Type:: Journal Article
Citation:: Journal of Physics A: Mathematical and General, 2005, 38 (39), pp. 8353 - 8366
Issue Date:: 2005-09-30

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Field	Value	Language
dc.contributor.author	McPhedran, RC	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	2005-09-30	en_US
dc.identifier.citation	Journal of Physics A: Mathematical and General, 2005, 38 (39), pp. 8353 - 8366	en_US
dc.identifier.issn	0305-4470	en_US
dc.identifier.uri	http://hdl.handle.net/10453/489
dc.description.abstract	We consider sums over the set of positive integers relevant to construction of periodic Green's functions for diffraction gratings and similar problems, and provide a general formula for a combination of Bessel functions of complex order and complex powers of distance from the origin. This general formula is investigated in a number of particular cases, and in particular we provide expressions which enable sums of functions with Neumann series to be re-expressed as combinations of hypergeometric series. We also investigate sums of Neumann functions of integer order, using analytic continuation techniques to provide formulae for their evaluation which we demonstrate are accurate and efficient in both the high and low frequency regions. We also exhibit sums which may be evaluated analytically, and recurrence formulae linking sums. © 2005 IOP Publishing Ltd.	en_US
dc.relation.ispartof	Journal of Physics A: Mathematical and General	en_US
dc.relation.isbasedon	10.1088/0305-4470/38/39/004	en_US
dc.subject.classification	Mathematical Physics	en_US
dc.title	Schlömilch series and grating sums	en_US
dc.type	Journal Article
utslib.citation.volume	39	en_US
utslib.citation.volume	38	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	39	en_US
pubs.publication-status	Published	en_US
pubs.volume	38	en_US

Abstract:

We consider sums over the set of positive integers relevant to construction of periodic Green's functions for diffraction gratings and similar problems, and provide a general formula for a combination of Bessel functions of complex order and complex powers of distance from the origin. This general formula is investigated in a number of particular cases, and in particular we provide expressions which enable sums of functions with Neumann series to be re-expressed as combinations of hypergeometric series. We also investigate sums of Neumann functions of integer order, using analytic continuation techniques to provide formulae for their evaluation which we demonstrate are accurate and efficient in both the high and low frequency regions. We also exhibit sums which may be evaluated analytically, and recurrence formulae linking sums. © 2005 IOP Publishing Ltd.

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http://hdl.handle.net/10453/489