#### 3.1. Expressions for the Regularized Principal Diagonal Dyad _{pd}^{∞}

[21] The function erfc(*ν*) denotes the complementary error function [*Abramowitz and Stegun*, 1965] of argument *β*^{±} = *R*_{n}*E* ± *ik*/(2*E*). The adopted Ewald parameter *E* is [*Kustepeli and Martin*, 2000]

which is chosen based on optimizing the total number of necessary terms in both the scalar spatial and spectral series, since with this choice they both exhibit the same Gaussian convergence rate. Note that the spatial 1/*R* singularity is fully represented by the *n* = 0 term of the spatial sum.

[23] In (19) a prime (*f*′) denotes a derivative of *f* in (17) with respect to its argument *R*_{n}, whose expression is given in Appendix B.

[24] Then, we accordingly split the dyadic GF in (3) as

where

where “type” is either spectral or spatial. The first term of the dyad in (21) is proportional to the scalar GF which is given in (15) for the spectral and in (16) for the spatial type. The terms ∇∇*G*_{spectral}^{∞} and ∇∇*G*_{spatial}^{∞} are derived based on (15) and (16) leading to

and

where

and _{n} = **R**_{n}/*R*_{n}. Furthermore, *f*″ and *f*‴ are the second and third derivatives of *f* in (17) with respect to its argument *R*_{n}, respectively, whose expressions are given in Appendix B. Similarly to what discussed for the scalar GF, here ∇∇ _{spatial}^{∞}(**r**_{0}, **r**_{0}, **k**_{B}) = [∇∇*G*_{spatial}^{∞}(**r**, **r**_{0}, **k**_{B}) − ∇∇*G*(**r**, **r**_{0})]. In its computation we need to evaluate the limit because both ∇∇*G*_{spatial}^{∞} and ∇∇*G* have singular terms at **r** = **r**_{0} of the kind 1/*R*_{0}^{p} with *p* = 1, 2, 3. After performing the limit for **r** **r**_{0} (see details in the Appendix A) one has

#### 3.2. Expressions for the Regularized Antidiagonal Dyad _{ad}^{∞}

[25] Similarly to what done for the principal diagonal dyad in the previous section, also the dyadic GF in (7) is split into the hybrid sum of spectral and spatial dyadic terms

through the Ewald method, where

and “type” is either spectral or spatial. The terms ∇*G*_{spectral}^{∞} and ∇*G*_{spatial}^{∞} are derived based on (15) and (16) leading to

#### 3.3. Convergence of the Scalar GFs and of the Principal and Antidiagonal Blocks of the Dyadic GF

[27] As stated in section 3.1, the Ewald method allows for obtaining series with Gaussian convergence rate, so only a handful of summation terms are needed to achieve convergence. In order to verify this, we define the relative error as

with *K*_{spectral} either *G*_{spectral}^{∞}(**r**_{0}, **r**_{0}, **k**_{B}), or ∇*G*_{spectral,i}^{∞}(**r**_{0}, **r**_{0}, **k**_{B}), or ∇∇*G*_{spectral,ij}^{∞}(**r**_{0}, **r**_{0}, **k**_{B}) and _{spatial} either _{spatial}^{∞}(**r**_{0}, **r**_{0}, **k**_{B}), or ∇ _{spatial,i}^{∞}(**r**_{0}, **r**_{0}, **k**_{B}), or ∇∇ _{spatial,ij}^{∞} (**r**_{0}, **r**_{0}, **k**_{B}), and *i*, *j* = *x*, *y*, *z* indicating the vector and dyad components. In (31), *K*_{spectral}^{exact} and _{spatial}^{exact} are evaluated with a sufficiently large number of terms to achieve high numerical accuracy, whereas *K*_{spectral}^{N} and _{spatial}^{N} are evaluated with *n*_{1} = *n*_{2} = *n*_{3} = 0, ±1, …, ±*N* terms only, as was done in *Capolino et al.* [2005, 2007]. As shown in section 2, the two principal blocks in (12) are proportional to one regularized principal diagonal dyad _{pd}^{∞} whereas the two antidiagonal blocks are proportional to one regularized antidiagonal dyad _{ad}^{∞}. This fact implies that it is sufficient to check the convergence rate of _{pd}^{∞} and _{ad}^{∞} for a given wave vector **k**_{B}.

[30] Note that, as expected, scalar spectral and spatial components in Figures 2 and 3 converge very rapidly, with about the same rate (i.e., same number of summation terms *N*). The same applies to the shown dyadic quantities. For example, in Figures 2 and 3, Err < 10^{−8} for all the series when using *N* = 2. Note that even *N* = 1 would already guarantee a relative error Err < 10^{−3}.

[32] Last, from a numerical point of view, one can notice that still in the provided numerical examples only a relatively small number of summation terms (*N* = 2) are needed to achieve convergence when using the Ewald method proposed here, with respect to the fully spatial (singular when computed at the dipole location, as in the present case) or the fully spectral (which needs an infinite number of terms to reconstruct the singularity when computed at the dipole location, as in the present case) counterparts.