LINTON, C.M., 2015. Two-dimensional, phase modulated lattice sums with application to the Helmholtz Green's function. Journal of Mathematical Physics, 56 (1), 013505.
A class of two-dimensional phase modulated lattice sums in which the denominator is an indefinite quadratic polynomial Q is expressed in terms of a single, exponentially convergent series of elementary functions. This expression provides an extremely efficient method for the computation of the quasi-periodic Green's function for the Helmholtz equation that arises in a number of physical contexts when studying wave propagation through a doubly periodic medium. For a class of sums in which Q is positive definite, our new result can be used to generate representations in terms of Θ-functions which are significant generalisations of known results.
The following article appeared in Journal of Mathematical Physics, 2015, 56 (1), 013505 and may be found at http://dx.doi.org/10.1063/1.4905732. Copyright 2015 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.