Schlömilch series and grating sums
- Publication Type:
- Journal Article
- Journal of Physics A: Mathematical and General, 2005, 38 (39), pp. 8353 - 8366
- Issue Date:
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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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