A Modification of the Kummer's Method for Efficient Computation of the 2-D and 3-D Green's Functions for 1-D Periodic Structures

A new modification of the Kummer's method of
M
th order for 2 ≤
M
≤ 6 is proposed for efficient summation of the spectral and spatial series representing the 2-D and 3-D Green's functions, respectively, for 1-D periodic structures in homogeneous media. The modification is based on transformation of the auxiliary series consisting of asymptotic terms of the original series and subsequently subtracted from the latter into a new series which, unlike the previous one, allows its summation in closed form. As a result, there are obtained new representations of the Green's functions in question consisting of rapidly converging difference series whose terms decay with rate
n
^{-(M + 1)}
as
n
→ ∞, as well as new rigorous analytic expressions for the sums of the transformed auxiliary series. Some numerical examples and comparisons characterizing the effectiveness of the proposed method are also presented and discussed.