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Publication Citation: Lattice Sums and the Two-dimensional, Periodic Green s Function for the Helmholtz Equation

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Author(s): Andrew M. Dienstfrey; Fengbo Hang; Jingfang Huang;
Title: Lattice Sums and the Two-dimensional, Periodic Green s Function for the Helmholtz Equation
Published: January 16, 2001
Abstract: Many algorithms that are currently used for the solution of the Helmholtz equation in periodic domains require the evaluation of the Green s function, G(x, x0). The fact that the natural representation of G via the method of images gives rise to a conditionally convergent series whose direct evaluation is prohibitive has inspired the search for more efficient procedures for evaluating this Green s function. Recently, the evaluation of G through the lattice-sum representation has proven to be both accurate and fast. As a consequence, the computation of the requisite, also conditionally convergent, lattice sums has become an active area of research. We describe a new integral representation for these sums, and compare our results with other techniques for evaluating similar quantities.
Citation: Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences
Volume: 457
Pages: pp. 67 - 85
Keywords: Helmholtz equation; lattice sums; plane-wave expansion
Research Areas: Math
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