## 1. Introduction

[2] Many numerical techniques in electromagnetics, such as moment methods, integral-equation techniques, boundary element methods, and so on, make use of Green's functions for particular geometries. Recent advances in these methods have been due to the improvements in terms of accuracy and efficiency related to the computation of the cited Green's functions. To obtain higher rates of convergence, a bunch of series acceleration techniques has been proposed in the technical literature. For instance, the Poisson summation formula for free-space periodic Green's functions was the preferred method in the 1980s [*Lampe et al.*, 1985], where it was applied to 2-D and 3-D Green's functions for the Laplace and the Helmholtz equations. The Ewald summation technique, originally proposed by Ewald in 1921, was used to accelerate the convergence rate of 3D periodic Green's functions [*Jordan et al.*, 1986]. Recently, this technique has been applied to solve the radiation of a line of sources in 2-D and 3-D (see *Capolino et al.* [2005] and *Capolino et al.* [2007], respectively).

[3] The rectangular waveguide geometry has been widely used in many practical applications. Therefore, the efficient and accurate evaluation of Green's functions for such a topology has been profoundly investigated. For instance, the Ewald method was successfully proposed to evaluate such functions for both rectangular cavities [*Park et al.*, 1998] and waveguides [*Park and Nam*, 1998]. More recently, in the work of *Quesada-Pereira et al.* [2007], Green's functions for the parallel-plate waveguide have been further accelerated using Kummer's transformation: this has been advantageous for the efficient study of inductive obstacles in rectangular waveguides. In this paper, we focus on improving the numerical efficiency related to the precise computation of 2-D Green's function for the Poisson equation in rectangular waveguides.

[4] The scalar Green's function considered in this paper is widely used within the frame of several numerical techniques, such as the well known Boundary Integral Resonant Mode Expansion (BI-RME) method [see *Conciauro et al.*, 1984]. This hybrid method allows one to obtain the modal chart of arbitrarily shaped waveguides, i.e., waveguides with metal perturbations in their rectangular cross section, as well as the full-wave analysis of arbitrary *H* and *E* plane waveguide components and junctions [*Conciauro et al.*, 1996a, 1996b; *Arcioni et al.*, 1997; *Arcioni et al.*, 1999; *Conciauro et al.*, 2000]. Therefore, the aim of this paper is to accelerate the computation of the involved scalar Green's function by preserving the prescribed accuracy. For such purposes, a closed-form expression written in terms of Jacobian elliptic functions with complex argument will be considered. A good tutorial on the properties of elliptic functions can be found in the work of *Orchard and Willson* [1997], in which the problem of rapid evaluation of such functions with real argument has been discussed. In this paper, the required elliptic functions have been accurately evaluated using the fast recursive algorithm, which will be thoroughly described.

[5] The first closed expression for the scalar Green's function in a rectangular waveguide domain was written in 1924, through the works carried out in the field of mathematical physics. Such analytical expression (without summations) is defined in terms of a special function called Weierstrass *σ* function, which is also related to the elliptic functions theory [*Courant and Hilbert*, 1989]. There is an alternative closed solution, also based on the elliptic functions, where the source and the field point contributions are explicitly separated [*Morse and Feshbach*, 1953]. Such solution obtained after applying a conformal mapping to the original rectangular domain is advantageous for multiple evaluations of Green's function in a grid of point values. However, the lack of knowledge of efficient algorithms for evaluating the involved elliptic functions has prevented their direct use up until now. This paper is intended to fill this gap by providing easy-to-program algorithms for the fast and accurate computation of this closed expression for the scalar Green's function, either individually or in a double grid of source and field points. The singular behavior of Green's function will also be extracted, with the aim of preserving computational stability when used in practice. A set of numerical examples that successfully confirms the wide reduction obtained in terms of computational effort when preserving the accuracy degree has been included.