%0 DATA
%A Christopher, Linton
%D 2015
%T Two-dimensional, phase modulated lattice sums with application to the Helmholtz Green's function
%U https://repository.lboro.ac.uk/articles/Two-dimensional_phase_modulated_lattice_sums_with_application_to_the_Helmholtz_Green_s_function/9385340
%2 https://repository.lboro.ac.uk/ndownloader/files/16998533
%K untagged
%X 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.